ML20247M304

Three-dimensional Buckling Analysis for AP600 Standard Plant Containment Vessel. Rept Undtd

ML20247M304
Person / Time
Site:	05200003
Issue date:	04/09/1998
From:	Fanous F, Greimann L, Safar S AMES LABORATORY, ENERGY & MINERAL RESOURCES RESEARCH
To:	NRC
Shared Package
ML20247M262	List: ML20247M259 ML20247M276 ML20247M291 ML20247M304 ML20247M325
References
NUDOCS 9805260237
Download: ML20247M304 (137)

Text

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l Three-dimensional Buckling Analysis for the AP600 Standard Plant Containment Vessel Prepared by L. Greimann F. Fanous, S. Safar, H. El-Arabaty, A. Khalil, D. Bluhm Ames Laboratory Iowa State University Prepared for U.S. Nuclear Regulatory Commission 3BP *188 s 3888830a Enclosure 3 A PDR

ABSTRACT A review of the effect of concentrated and asymmetric loads on the buckling performance of the Westinghouse AP600 steel containment was accomplished. The buckling and stress analysis were conducted using the finite difference softwares BOSOR4 and

' BOSOR5, respectively in a previous report submitted by the authors to the NRC. Load combinations of the ASME Code service limit categories were established as per the Safety Review Plan (SRP) Section 3.8.2. The stress analysis revealed that all stress intensities were within allowable limits. A minimum factor of safety against buckling of 2.02 was computed when the containment was subjected to a load combination of external pressure of 3.0 psig, uniform temperature of 120 F, dead load and seismic loads at Service Limit C.

In this report, the buckling analysis of AP600 containment was conducted using a three-dimensional finite element model using the general purpose program ABAQUS. A three-dimensional model that includes major penetrations, ring stiffeners, crane girder and crane bridge was constructed. The selection of the elements size was based on a mesh sensitivity study on an axially compressed imperfect cylinder. Two-hundred and seventy-three free vibration modes were extracted to ensure a sufficient total modal effective masses in the X, Y and Z directions. The maximum meridional seismic stress resultants were computed by the Response Spectmm method based on the structure response spectra for the Safe Shutdown Earthquake (SSE). The effects of high-frequency modes and closely spaced modes were included in the calculations. Two potential buck:ing regions were identified at which the compressive meridional stress resultants were peaked due to seismic loads. A set of equivalent static loads was developed for each region to bound the maximum compressive meridional stress resultants at that region. The factor of safety against buckling was determined for each region using a quasi-static approach in which the equivalent seismic static loads were combined with eternal pressure of 3 psig, uniform temperature of 120*F and dead loads at Service Limit C. The minimum factor of safety was 2.54 with the first buckle occurring at o_ oase below the upper equipment l hatch. The analysis indicated that all penetrations were adequately reinforced. A three-dimensional model of a longitudinal sector of the AP600 containment was constructed to investigate buckling due to asymmetric temperature during an emergency cooling event.

Thermal stress analysis was verified by comparing the computed thermal stresses to those obtained from a strength of material model and from BOSOR4 program. The worst temperature distribution was determined by conducting a sensitivity study on the effect of the number of strips, width of the dry strip and geometric imperfection. The minimum buckling factor of safety was 7.16. The buckling factor of safety for the design basis loads composed of a rise in temperature, internal pressure of 45 psig and dead loads was computed. Two cases of temperature loading were considered. In Case 1, a uniform l temperature rise of 280 F was assumed, whereas in Case 2, an asymmetric temperature profile due to the emergency cooling was applied. The buckling factors of safety were t

3.J2 and 3.03 for Case 1 and Case 2, respectively. In both cases the containment

- collapsed due to tensile stresses. All the buckling analyses conducted in this report revealed that the containment design satisfies the ASME code minimum requirements.

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i CONTENTS '

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ABSTRACT.............................................................................................................................iii LI ST OF TAB LES . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . .. . . . .. . . . . . .. . . .. .. . . . . . . . . . . . . . .. . .

LISTOFFIGURES..................................................................................................................

A C KN OWLE DOM ENTS . . . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .

EX EC UTI VE S UMM A R Y . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . .. . . . . . . . . .. .

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INTR ODU CTION . . . . . . . . .. . .. . . . . . . . . . . . . . ... . . .. . . . . . . . . . . . . . . . ... . . . .. .. . . . . . . . . .. .

1.1 Background......................................................................................................3  !

1.2 Summary of Work Presented in Ref.1.2.......................................................... 3 '

l.3 Objectives.........................................................................................................4 1.4 Description of AP600 Containment Vessel ..................................................... 4 1.5 S oft ware Selection.......... ...... .. ... ...... .. ... . . .. .. . ... ................. ... .. .... ...... . ... ... ..... .. .. 5 1.6 M e th odol ogy . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . .. . . . . .

1.7 A cceptance Criteri a ....... ... .. .. ... . . ... . .. ... . ... . ... ... . . . .......... . .... .... . .. ... . ... . ... .... ... . .. . . . .. 6 1.7.1 Section NE 3222.1................................................................................6 r

1.7.2 AS ME Code Case N-2 84 ........ .. . ... ... ............. ............ ... . ....... .. ...... .... .... 7 1.7.3 Reg ulat ory Guide 1.57 .. .... ..... ........ . .... ........... .. ... ... .......... . ......... . .... .. .. . 7 1.8 References........................................................................................................7 2.

VERIFICATION OF BUCKLING ANALYSIS IN ABAQUS.................................... 9 l 1

2.1 In t rod uc ti o n . . . . . .. . . . . . . .. .. . . . ... . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . .. . ... .. .. .. .. .

2.2 B ackground to B uckling of Shells.............. ..................................................... 9 2.3 Elastic B uekling Analysis in AB AQUS ......................................................... 10 2.3.1 Axially Loaded Circular Cylinder ...................................................... 10 2.3.1.1 Theoretical Solution ......................................................... 10  !

2.3.2.2 Finite Element Solution..................................................... I 1

. 2.4 Inelastic Buckling Analysis in ABAQUS ....................................................... I 1 l

2.5 Mesh Se n sitivity St udy. ....... . ................................................. ................. .. ...... 12  !

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2.5.1 M odel Geometry................... .................... .. ............ .......... .. ............. ... 12 2.5.2 Finite Element Results ......................................... ............................. 13 v

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b.gt 2.6 Summary........................................................................................................13 l 2.7 References.....................................................................................................14 l

l 3. THREE-DIMENSIONAL MODEL PARAMETERS................................................ 15 3.1 In trod uc tion . . . . . .. .. . . . . . . .. . . . .. . .. . . . . . .. .... .. . . . .. . .. .. . .. .. . . . .. .. . .. . .. . .. .. . .. .. . . . . ... . .

3.2 Modeling Proced ure s. ......... .. ........ ........................ ......... ... . .... .......... .. ... ... .... ... 15 1

3.2.1 . Upper Ellipsoidal Head ...................................................... ............... 15 3.2.2 Cylindrical Containment .................................................................... 16 3.2.3 Lower Ellipsoidal Base ...................................................................... 16 l

3.2.4 Crane Girde r . . . .. . . . . . . . . .. . .. .. . . . .. . .. . . .. . . . . . . . .. . .. . .. . . .. . . . . . . . . . . . . . . .. . . . . . ..

3.2.5 Crane B rid ge. . . . . . . . . . ... . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . .. .. . .. . . . . . . . . . . . . . . . . .. . . .. .. . . . 17 3.2.6 S ti ffe ners . . . . . . . . . . . . .. . . . . . .. . . . . . .. . . . .. . . ... . . . . . . . . . . .. .. . . . . . . . . . . . . . . . . . . . . ..

3.2.7 Penetration B arrels ............... ..... . ....... ... ......... .. ... .. ...... ... ... . .... . .......... .. 17 3.3 Me s h Ge ne rati on . . . .. . . . . . . . .. . . . . .. . . . .... . . . . . .. . . . . . . . . .... . . . . . . . . .. . . .. . . . . . . . . .. . .

L 3.3.1 S olid Modelin g ...... ........ ...... . ... ..... . ....... ... . .. ... . .. .. ...... ....... ........ . ...... .... . I 8 3.3.2 As sembling the Model . ....... .......... ....... ..... . .... ... ... . ... .... . .. . .. . ...... .. . . . . . . . I 8

! 3.4 Imperfection Parameters... .... .... .. ......................... ................. .............. .......... . . 18 i 3.5 M ateri al Properties .......... ....... .... .. ................ .... ......... ....... ..... .... .... ..... .. . ... ...... 19 3.6 CheckRun......................................................................................................19 1

i 3.6.1 Linear Static Analysis with Uniform Pressure...................................19 3.6.2 Nonlinear Buckling Analysis of Containment Sector i Around the Pene trations ..................................................................... 19

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3.7 References.....................................................................................................20

4. RESPONSE SPECTRUM ANALYSIS OF AP600 CONTAINMENT................... . 21 4.1 In trod uc ti on . .. . .. . . . .. . . . . .. .. . . .. . . . . . . .. . .. . . . . .. . .. . . . .. . . .. . . . .. . . . .. . . . .. . . . . . . .. . . . . .

4.2 M odes of Vibration .... ................ . ...... .. ...... ... .... .......... ........... ..... .............. . .. . . 21 4.2.1 Theoretical Background ..................................................................... 21  ;

4.2.2 Finite Element Solution...................................................................... 22 '

4.2.3 Comparison of Results with Axisymmetric Model............................ 23 4.3 Response Spectrum Analysis . ....................................................................... 24 VI

HEE 4.3.1 Modal Respon ses.. ..... ........ ...... . .. ........... ..... .... .... ... . .... ..... .. . .. .... ..... .. ... 2. 4 4.3.2 High-Frequency Modes . ................. .... .. . . .............. ........ .. ...... ... . ........ . 24 4.3.3 Modal and Directional Combination...................................... ........... 25 4.3.4 Discu ssion of Res ults .. .... ......... ............................. ..... . ... . .. ........... . .. .. . 25 4.3.5 Potential Buckling Region ................................................................. 26 4.4 Eq uiv ale nt Static Loads ........................... . ........... ....... .... . ........... .... .... .. . . ...... . 26 4.4.1 Purpose............................................................................................26 4.4.2 Potential B uckling Region 1 .............................................................. 27 4.4.3 Potential B uckling Region 2 ..... ........................................................ 27 4.5 References.....................................................................................................28

5. INELASTIC BUCKLING OF SYSTEM AP600 CONTAINMENT I AT LEV EL C B D A4 . . . . ... ..... ..... .... . ......... ....... . . . .. . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . ..i .

I 5.1 In trod uc tion .. . .. . . . . . .. . ... .... . . .. . . ...... . ... . . . . ... . . . . .. . . . . . . . . .. . . . .. .. . . . . . . . . .. .. l 5.2 M e th odol ogy . .. . . . . . . .. . . . . . . . ... . . . . . . . . .. . . . .. . . . . . . . . . . .. . . . . . . . .. .. . . . . . . . . . . . . . ... . . . . . . . . . .. . .. .I 5.3 Load Case ( 1 ) . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . .. . . . . . . . . . . . . . . . . j 5.4 Load Case: (2) . . . . . . . . . .. . . . . . . . . . .. . . .. . . .. . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . .

l 5.5 S u mmary and Discussion . .... .. ....... ....... .. .. . ..... . ..... .. . ..... . . . . . . .......... .... . ......... ... 31 5.6 References......................................................................................................31 1

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6. BUCKLING OF AP600 DUE TO ASYMMETRIC TEMPERATURE.................... 33 i

6.1 In t rod uc ti on . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .

6.2 Development of the Finite Element Model .................................................... 34 6.2.1 Model Parameters ....... ......................... .................. . ...... ............ .. .. .. . ... 34 ,

1 6.2.1.1 Upper Ellipsoidal Head .................................................... 34 6.2.1.2 Cylindrical Containment .................................................. 34 6.2.1.3 Lower Ellipsoidal Head............................................. ...... 35

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6.2.1.4 S tiffe n e rs .. . . . . . . . . .. .. . . .. . . . . ... . . .. . .. . . . ... . . . . . . . . .. . . .. . . . . ... . . . . . .i 6.2.1.5 Crane G irder .. .. ........................................................ . . ....... 3 5 6.2.2 Asymmetric Temperature Loading..................................................... 35 6.2.3 Pressure and Gravity Loading ............................................................ 36 6.3 Thermal Stress Analy sis ......................... .. .... .. ......... .... ....... ................ ....... .... . 36 Vil

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6.3.1 Comparison with BOSOR4 Results ................................................... 36 6.3.2. Comparison with an Approximate Strength of Material Model ........ 36 6.4 Elastic Thermal Buckling Analysis ................................................................ 37 6.4.1 Asymmetric Temperature Loading..................................................... 37 6.4.2 Assessment of the Worst Meridian Approach.................................... 38 6.5 Inelastic Thermal Buckling Analysis ............................................................. 38 6.5.1. Axisymmetric Imperfections ...... ....................................................... 3 8 6.5.2 Three-Dimensional Imperfections...................................................... 39 6.5.3 Dry S trip Width . .... .. ..... ..... ...... ... .. . ... ... .... ...... ... ................... ... .. ......... 40 '

6.5.4 N umber of S trips . .. ....... ... .................. .................... ....... ........... .... .. ...... 40 6.5.5 Concl u sions . . . . . . . . . ._ . . ... .. . . ... . . . . . . . ... . . . . .. . . . .. . . . . .. . . .. . . . . . . . . . . . . . . . . .. . .

6.6 Load Co mbin ation DB A 1 .......... .. ...... .. .............. ..................................... ....... 40 6.6.1 Case 1, Uniform Temperature ............................................................ 40 6.6.2 Case 2, Striping Condition ...................................................... .......... 41 6.7 Summary........................................................................................................41 6.8 References......................................................................................................41

7. S UMMARY AND CONCLU SIONS ........ .. . . .. ...... .... . ......... ... . ..... . ................ .. . .. ... .. . .. 4 3 7.1 Summary....................................................................................................43 7.2 Conc l u si o n s . . . . . . .. . . . . . . . . . . . . . . . . . . . .. .. . . .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . .

7.3 Recomme ndations ... . ....... .... .... .... ....... . ... ... ...... ............. ....... . ........ . ..... ... .... .. . .. 46 APPENDIX A. - COMPARISON BETWEEN ANSYS, ABAQUS AND STAGS............. 47 APPENDIX B.

• MODE FREQUENCY ANALYSIS RESULTS........................................ 51 l

V111 l

LIST OF TABLES Pagg Table 1.1 Buckling Factors of Safety Obtained from Axisymmetric Analysis............ 63 Table 1.2 Weight of the AP600 Attachments............................................................... 63 Table 1.3 Factors of Safety for ASME Service Limits........................ ........................ 64 Table 2.1 Buekling Modes of a Perfect Circular Cylinder........................................... 64 Table 2.2 Five Mesh Parameters ................................................................................. . 65 Table 2.3 Results of Mesh Sensitivity Study................................................................ 65 Table 4.1 Analytical Solution for Natural Frequencies and Mode Shapes of a Cl ampe d- Cl amped Cylin de r . . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 66 Table 4.2 Frequencies of First Horizontal and Venical Modes Obtained by AB AQ U S an d B O S OR4 .. ....... . ... ... ... . . .. . . ... .. .. .. .... ...... .... .. .... . . .... . . ... .. ... .. . . . .. 66 Table 4.3 Percentage of Total Effective Modal Masses from Total Mass................ .. 66 Table 6.1 Element Sizes in the Three-Dimensional Model........... .............................. 67 i

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LIST OF FIGURES I

l E. age Fig.1.1 Elevation of AP600 Steel Containment (Penetration Shown Distorted) ..................................................................... 68 i i

Fig.1.2 Orientation of Major Penetrations................................................... ............ 69 Fig.1.3 Details of the 22 ft. Diameter Equipment Hatch.......................................... 70 Fig.1.4 Details of the 16 ft. Diameter Equipment Hatch...................................... ... 71 Fig. 2.1 General Load Displacement Behavior of a Shell ......................................... 72 I

Fig. 2.2 Finite Element Model of Cylindrical Shell .................................................. 73 l Fig. 2.3 Buckling Modes of a Perfect Cylinder .............................................. .......... 74 Fig. 2.4 Stress-strain Curve Used in the Analysis ..................................................... 75 i

Fig. 2.5 Mesh Configuration of a Perfect Shell ......................... ............................... 76 Fig. 2.6 Prebuckling De formed Shape....................................................................... 77 Fig. 2.7 Deformed Shape of the Cylindrical Sectors ............ .................................... 78 l Fig. 2.8 Results of the Mesh Sensitivity Study.................................................. ....... 79 l

Fig. 3.1 Descritization of AP600 Containment ......................................................... 80 l

Fig. 3.2 Three-Dimensional Model of AP600 Containment Vessel.......................... 81 I Fig. 3.3 Finite Element Model of the Upper Ellipsoidal Head............ ..................... 82 i

Fig. 3.4 Portion of the Cylindrical Part Showing the Transition Zone...................... 82 j 1

i Fig. 3.5 Finite Element Model of the Crane Girder................................... ............. . 83 l Fig. 3.6 Crane Bridge and Wheel System.................................................................. 83 Fig. 3.7 Finite Element Idealization of Stiffeners.... .............. ........... ...................... 84 Fig. 3.8 ANSYS Solid Modeling in Penetration Area.... ......................... ................ 85 X

L _ _ _ _ _ _ _ __ - - . - - - - - - - - - - - - - - - - - - - - - ---- ------------J

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Fig. 3.9 Deformed Shape of the Containment due to Internal Pressure.................... 86 Fig. 3.10 Finite Element Model in the Penetration Area............................................. 87 Fig. 3.11 Deformed Shape due to Axial Compression .......... ..................................... 88

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Fig. 3.12 Load-Deflection Curve at the top of the Tested Portion .............................. 88 l

Fig. 4.1 Fundamental Frequency of a Freely Vibrating Cylinder 3 with Clamped Ends ...... .... ............ .. .... .... .. ....... ... ...... ...... .... ............. ... .. ... . .... 89 l

Fig. 4.2 Three-Dimensional Model Classification According to M as s De n s i ty . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. .. . . . ... . . . . . . . . . . . . . . . . . . . . . . .. . . . ... . . .. . . . . . . . . . . . . . . . . . . . .

-) i Fig. 4.3 First Five Free Vibration Modes .................................................................. 91  ;

Fig. 4.4 Selected Four Modes from the First Sixty Modes of Vibration................... 92 Fig. 4.5 Accumulative Effective Modal Mass Versus Frequency............................. 93 Fig. 4.6 Modes 109, 179, 180, 249 . .. ......... .... .. . ........... . ............................. .............. . 94 Fig. 4.7 Contour Plot of Nim Stress Resultants ...................................................... 95 i

Fig. 4.8 Distribution of N m iat t he B ase . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . . .

1 Fig. 4.9 Distribution of N i at Azimuth 67 and 73.125*...................................... % i i

Fig. 4.10 - Distribution of Ni m at Azimuth 107 and 126 ......................................... 97  !

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Fig. 4.11 Contour Plot of N 2 Stress Resultants ....................................................... 98 ,

1 Fig. 4.12 Potential Buckling Regions .......................................................................... 98 Fig. 4.13 Comparison of N im Stress Resultants at 0 = 2 86. 875 * ............................. 99 Fig. 4.14 Contour Plot of Ni due to Equivalent Static Loads...... ............................. 100  !

' Fig. 4.15 - Equivalent Static Loads of Potential Buckling Region 2.............................101 L '

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Fig. 4.16 Contour Plot of Ni due to Equivalent Static Loads at Region 2..................102 xi I

P.aan Fig. 4.17 Comparison of Ni and Ni due to Equivalent Static Loadt at the Upper Equipment Hatch .............. . ...................................................... 103 Fig. 4.18 . Comparison of Ni and Ni due to Equivalent Static Loads at the Lower Equipment Hatch.... ................................................................. 103 Fig. 4.19 Comparison of N i and Ni due to Equivalent Static Loads at the Upper Air Lock. . ............ .......... .......................... ............... .. .. ... .. ... .. ... 104 Fig. 4.20 Comparison of Ni and NI due to Equivalent Static Loads at the Lower Air Lock ..................... ... ........................... ..... .......... ... ............ 104 Fig. 4.21 Comparison of Ni . and N due i to Equivalent Static Loads

- in the Thin Shell Around the Upper Equipment Hatch Rein forcing Collar. . .......... .... . . ............ . ... ........ ......... .... .......... ...... . . ... . ... ... . . ... 105 Fig. 5.1 Stress Strain Relation at T = 120'F......... ................................................... 106 i Fig. 5.2 Deformed Shape at A = 2.54 (Load Case 1) ................................................. 107 Fig. 5.3 Load Deflection Curves at Points A and B (Load Case 1)...........................,108 Fig. 5.4 Contour Lines of Von Mises Stresses at A = 2.54 (Load Case 1) ................109 Fig. 5.5 Von Mises Stresses at Points A and B (Load Case 1)................................... I10 Fig.5.6 Detailed View of the Deformed Shape at A = 2.62 (Load Case 2)................ I11 Fig. 5.7 - Load Deflection Curves at Points C and D (Load Case 2)............................ I12 l Fig. 5.8 . Contour Lines of Von Mises Stresses at A = 2.62 (Load Case 2) ............ .... I13 Fig. 6.1 - Partitioning of the Wedge Model into Eight Regions for Meshing .............. I14 ,

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Fig. 6.2 Finite Element Model of the Top Ellipsoidal Head ...................................... I 14 '

Fig. 6.3 Modeling of Upper Stiffener (Elev.170'-0")................... ............................ I 15 i Fig. 6.4 Modeling of Lower Stiffener (Elev.132'-3") ................................................ I 15 l Fig. 6.5 Modeling of Crane Girder ............................................................................. I 16 1

Fig. 6.6 Assembled Finite Element Model ................................................................. I 16 xii i

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.P.aage Fig. 6.7 Case 2 Temperature Distribution. Above Elev. 132'-3 " . .. .. . .. ..... . . ... . .... ... ..... . 1 17 Fig. 6.8. Meridional (N1) and Circumferential (N2) Stress Resultants Due to Case 2 Temperature Loading (Dry Strip) ......................................... 118 Fig. 6.9 Meridional (N1) and Circumferential (N2) Stress Resultants Due to Case 2 Temperature Loading (West Strip)........................................ I 19 Fig. 6.10 Approximate Theoretical Model ........... ...................................................... 120 Fig. 6.11 Elastic P sxling Modes Due to Striping...................................................... 120 Fig. 6.12 Elastic Buckling Modes Using Worst Meridian Approach..........................121 Fig. 6.13 Variation of Stress Resultants with Load Factor (Theoretical Model Results) .......... ......................... .................................... 122 Fig. 6.14 Variation of Stress Resultants with Load Factor (Finite Ele me nt Resuts) ......... ........................ ............ . .. . . .. . ... ...... .... ... ... ..... . . 122 Fig. 6.15 Effect of Thr:;e-Dimensional Imperfections on B uc klin g & fety Factor ................................. ................ . ..... .. .............. ...... .. . 12 3 Fig. 6.16 Variation of Temperature Distribution with d.............................................. 123 Fig. 6.17 Effect of Striping Waves Number on Buckling Safety Factor.....................124 ,

Fig. 6.18 Deformed Shape (DB A 1, Temperature Case 1)........................................... 124 Fig. 6.19 Deformed Shape (DBA1, Temperature Case 2)

Perfe c t Case . . . . . . . . . . . . .. . . . . . . . . . .. . . .. . . . .. . . ... .. .. . . . . . . . . . . . .. . . . . . . . . . . .. . . . . . .

Fig. 6.20 Deformed Shape (DBA1, Temperature Case 2)

Imperfect Case, k = 4.0 ... .......... .... .......... .. ... ......... ........ ... . .... .. ... ... ..... . . ... .. . .. 125 i

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l ACKNOWLEDGMENTS l

l The authors would like to express their appreciation for the members of the U.S. Nuclear Regulatory Commission, Dr. Thomas Cheng (Project Officer), Dr. Seung lee (Structural l

Engineer), Mr. David Terao (Section Chief), Mr. Goutam Bagchi (Chief, Civil Engineering and Geosciences Branch), and Mr. Bernard L. Greiner, (Program Manager, Division of Engineering) for their help throughout the course of this work. The authors would also like to acknowledge the project secretaries, Ms. Jeanine Crosman and Ms.

Denise Wood, for word processor operations and secretarial services associated with this project. Additionally, they would like to thank Ms. Connie Bates, Administrative Specialist, for her assistance on financial planning and tracking.

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EXECUTIVE

SUMMARY

The objective of the present work is to review the effect of concentrated and asymmetrical loads on the buckling potential of the Westinghouse AP600 steel )

i containment vessel. In a companion report, the stress analysis and the buckling evaluation of the containment vessel were performed by using an axisymmetric finite difference model with BOSOR4 and BOSOR5 computer codes, respectively. Such analysis revealed that the load combination of an external pressure, gravity, temperature and seismic loads (DBA4 in Table 1.1) at level Service Limit C was the worst loading for buckling. A three-dimensional finite element model was developed to determine the buckling factor of safety for this load combination. The factor of safety against buckling due to asymmetric temperature at the loading condition pertaining to emergency cooling (DBAl in Table 1.1) at level Service Limit A was also computed. The analysis of the

' three-dimensional analysis was conducted- with the general purpose finite element program ABAQUS.

The clastic buckling procedure in ABAQUS was verified by solving the stability problem of an axially compressed perfect cylinder. Results were compared to theory. Inelastic buckling analysis which incorporates material and geometric nonlinearities was checked by introducing an axisymm:tric geometric imperfection to the cylinder. The constitutive relation was derived from the ASME Code Case N-284 plasticity reduction factor. The size of element were progressively halved in the hoop and meridional direction until a .

converged solution was obtained. Results were verified with the BOSORS and Code Case N-284 solutions. The findings of the element size sensitivity study were utilized in constructing the three-dimensional model of the containment. The area surrounding penetrations were modeled with a finer mesh than the remainder of the containment to secommodate to the expected high strew gradients in this area. After constructing the '

whole containment model, two static runs were conducted to check the validity of the i model. .l l

Free vibration modes were extracted by the subspace iteration technique. Two hundred f and seventy-three modes were extracted to ensure that the total effective modal mass was l

- sufficient in all directions. Response Spectmm analysis was conducted to obtain the  ;

maximum meridional seismic stress resultants based on the structure response spectra due l to the Safe Shutdown Earthquake (SSE). Modal responses and high-frequency modes responses were combined by the Ten Percent rule to account for modal coupling of closely spaced modes. Regions of high seismic stress resultants due to SSE were

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designated as potential buckling regions. Two sets of equivalent static loads were '

computed to bound maximum compressive meridional stress resultants at two recognized potential buckling regions.

Stability of the AP600 containment vessel due to loading combination of external pressure, gravity, temperature and seismic loads (DBA4 in Table 1.1) at level Service Limit C was conducted. The buckling assessment was accomplished by a quasi-static approach in which the interaction between the buckling inertia loads, ground excitation 1

inertm loads and other loads is neglected. Therefore, the containment was loaded with gravity loads, uniform temperature of 120*F, external pressure of 3.0 psig and the SSE ground excitation was replaced by the equivalent static loads. All loads were progressively increased until buckling. The minimum buckling factor of safety was 2.54 l which satisfied ASME Code minimum requirements. Buckling was initiated at the base below the upper equipment hatch. The reinforcement around all penetrations was shown to be adequate. local buckling due to concentrated loads at penetrations will take place at a load factor greater than 2.62.

Buckling of the containment vessel due to asymmetric temperature was investigated by another three-dimensional model. A longitudinal sector of the containment was modeled with a fine mesh. The temperature field generated by an emergency cooling event was assumed to vary from 200 F at a wet strip to 280 F at a dry strip. The model was loaded with such a temperatt.re field and the resulting thermal stresses were compared to BOSOR4 results and to a strength of material model. The effects of the number of strips, width of the dry strip per complete stripping wave and geometric imperfections were investigated on the inelastic buckling strength of the containment to obtain the worst case.

The minimum buckling load factor was 7.16 which revealed that buckling due to asymmetric temperature is immaterial and will not govern. The loading combination corresponding to emergency cooling (DBAl in Table 1.1) at Level Service Limit A was applied on the model. Two temperature cases were examined: Case 1 of uniform temperature of 280 F and Case 2 of stripping. All loads were progressively increased until buckling. In Case 1, the containment collapsed due to tensile yielding of the cylinder part at a load multiplier of 3.02. This value was not significantly affected when the circumferential variation of temperature of Case 2 was applied. Similar to Case 1, the containment collapsed at a load multiplier of 3.03. In both cases the tensile stresses generated by the internal pressure exceeded the compressive meridional stresses due to stripping. The analysis revealed that the containment design satisfies the ASME code minimum requirements.

i 2 4

1. INTRODUCTION

1.1 Background

The AP600 steel containment vessel is a thin cylindrical shell structure with an approximately 2:1 smooth elliptical head as shown in Fig.1.1. The cylindrical portion is provided with two T-ring stiffeners and an internal box stiffener which supports a polar crane. SA537-Class 2 steel plates are used to construct the containment vessel. The bottom of the containment vessel is enclosed by an elliptical head embedded into concrete foundation.

The AP600 nuclear containment vessel is designed to use passive means for emergency cooling. These include natural draft and water film evaporate cooling which are made possible by an airflow path in the annulus between the containment building and the shield building. In this system, water flows from the top ellipsoid dome down to the lower ring stiffener. At the same time air flows into inlets near the top of the shield building, downwards past a baffle wall in the shield building / containment annulus, then around the bottom of the baffle, upwards between the baffle and the containment vessel and out the chimney at the top of the shield building. A description of the relevant structures is provided in Sec. 3.8.4.1 of Ref.1.1.

As the containment vessel is subjected to various loading conditions, regions of compressive membrane forces develop in the steel containment vessel that may cause the shell to fail due to compression instability. In order for the containment vessel to perform its intended safety function and to sustain these loads, a sufficient margin of safety against buckling should exist.

1.2 Summary of Work Presented in Ref.1.2 In Ref.1.2 an axisymmetric stress and buckling analysis for the AP600 containment vessel was conducted to check the design adequacy against the ASME Code [1.13]

requirement and to investigate the margin of contaimnent stability beyond the Safe Shutdown Earthquak'e (SSE). The results of that axisymmetric study showed that all stress intensities were below the allowable limit as specified by the ASME Code, Section NE 3221. The axisymmetric analysis indicated that maximum compressive meridional stresses took place at Level C Service Limit when the containment vessel was subjected to the design basis accident load combination DBA4 (Table 1.1). Hence, a minimum buckling factor of safety of 2.02 was obtained at that load combination. For that load combination, the predicted buckling factors of safety do not satisfy the requirements of Section NE 3222.1 of the ASME Code [1.12]. However, they satisfy the Regulatory Guide 1.57 [1.11] and Code Case N-284 [1.13]. A three-dimensional analysis is required to consider the localized effects near penetrations and to gain further understanding of the buckling of the AP600 containment vessel.

1 3

r-A three-dimensional model is also required to determine the factor of safety against buckling at Level A Service Limit, where the containment is subjected to an asymmetric temperature loading caused by emergency cooling at the design basis combination, DBAl (Table 1.1). Asymmetric loads can not be included in the axisymmetric buckling analysis conducted with the BOSOR5 finite difference software.

1.3 Objectives The objectives of this work were to perform:

l (1) A three-dimensional buckling analysis to check the containment design i

against the ASME Design [1.13], I2 vel C and D safety requirements for DBA4 (Table 1.1).

(2) A three-dimensional analysis to determine the containment buckling factor of safety for asymmetric temperature loading caused by emergency cooling at level A Service Limit for DBA1 (Table 1.1).

1.4 Description of AP600 Containment Va==l An elevation view of the AP600 containment vessel is shown in Fig.1.1. The containment vessel consists of a cylindrical shell with an inner radius of 65 ft and a wall thickness of 1.625 in. The top of this cylinder is covered by a smooth elliptical dome while the bottom is enclosed by another elliptical dome that is embedded into a concrete foundation below an elevation of 100 ft. Between elevations 100 ft and 108 ft, concrete l

was on the inside portion only. The cylindrical portion of the containment vessel is provided with two T-ring stiffeners and one box-girder stiffener. The latter serves as a crane girder sopponing a crane bridge and a trolley that weighs 634.6 kips [1.3).

Two equipment hatches and two personnel air locks are located at different elevations within the cylindrical ponion of the AP600 containment vessel as shown in Fig.1.1 and 1.2. The centerline of the larger equipment hatch barrel is located at Elev.144'-6" and Azimuth 67 (measured from the North in the clockwise direction). The equipment hatch barrel is a circular cylinder with an inner diameter of 22 ft, a length of 5'-313/16" and a wall thickr:ss of 4 3/4". The total weight of this hatch is estimated to be 105,000 lb

[1.3]. Other details related to this hatch assembly is illustrated in Fig.1.3. The other equipment hatch has a cylindrical barrel with a 16 ft inside diameter and is location at Elev.112'-6" and Azimuth 126. Figure 1.4 illustrates details related to this hatch. The weight of the 16 ft equipment hatch was assumed to be 62,000 lb. [1.3]. Each personnel air lock has an inside diameter of 9'-10" and an assumed weight of 70,000 lb. Other attachments to the AP600 include the containment vessel airbaffle, walkway, heating-ventilation and cooling (HVAC) duct, cable tray, concrete on the external stiffener, and the containment vessel recirculation unit platform. The weight and the locations of these attachments are listed in Table 1.2.

4

1.5 - Software Selection The three-dimensional buckling analysis was conducted using the finite element techniques. The various capabilities of three finite element codes ANSYS [1.4],

ABAQUS [1.5], and STAGS [1.6], have been reviewed and compared. The comparison was based on types of shell elements, buckling and vibration analysis procedures, pre-and post-processing capabilities, hardware and software limitations, and computation time and storage requirements. A description of the three programs and a comparative study is presented in Appendix A. '

The general purpose finite element software ABAQUS, Version 5.3, was selected to perform the three-dimensional buckling analysis of the AP600 containment vessel

- because it consumes the least storage requirements and run time. However, the ANSYS software was also utilized to generate the finite element mesh around penetrations by '

means ofits solid modeling capabilities.

1.6 Methodology l

ABAQUS was previously validated by the authors in Ref.1.7. For further calibration, a cylindrical shell under uniform axial load was analyzed to verify clastic buckling analysis.

Both elephant foot and chess board buckling modes were calculated and results were '

compared to the classical theory. Inelastic buckling analysis was verified by comparing ABAQUS results to that of the widely recognized BOSOR5 finite difference software [1.2]. The inelastic buckling analysis results were also utilized to establish the adequate mesh size for modeling the AP600 containment vessel. -Based on the mesh )

i sensitivity study, the containment shell, equipment hatches, personnel air locks and stiffeners were modeled with quadratic and triangular shell elements. The mass of the penetrations and attachments was included in the model by specifying a suitable mass density.

The buckling analysis conducted by the ABAQUS program was accomplished for level C and D load combinations that include dead load (self weight, crane bridge and trolley),

uniform temperature, external pressure and seismic loads. A ufficient number of free vibration modes was extracted such that the total effective modal masses in the horizontal and vertical directions are sufficient. The modal responses of high-frequency modes was computed to incorporate the effect of the remainder effective mass not included in the L

extracted free vibration modes. Modal responses due to seismic input were determined by response spectrum analysis. Modal stress resultants were then combined by the i

Square Root of the Sum of the Squares (SRSS) method taking into account the effects of closely spaced modes. Sets of equivalent static loads were developed to produce a stress field that bound the SRSS maximum seismic stresses in potential buckling regions.

Pressure, thermal and dead loads were then added to these loads and the buckling behavior of the containment was predicted. Geometric imperfections as well as material residual stresses were incorporated in the analysis. Imperfection wave length and 5

L____-_____-__-____--___-__-_-_----- - - - _ - - - - - - - - - _ _ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -----------------O

amplitude were selected in accordance with tolerances specified in the ASME Code to yield a minimum buckling load.

The buckling strength of the AP600 containment vessel was also evaluated for the level A load combination that includes uniform and asymmetric temperature field, dead loads and internal pressure. The primary concern was to determine the buckling factor of safety of the cylindrical portion due to compressive stress resultants generated by the asymmetric temperature loading due to emergency cooling. Hence, the complete three-dimensional model was replaced by a wedge model. The minimum buckling load factor was determined by optimizing the geometric imperfection and temperature field parameters in accordance to test data [1.8] and the ASME code specifications.

1.7 Acceptance Criteria j

NRC SPR Section 3.8.2 [1.9) stipulates that the design and analysis procedure for the steel containment structures be in compliance with subsection NE of the ASME Code Section III [1.10] and with the Regulatory Guide 1.57 [1.11].

1.7.1 Section NE 3222.1 Section NE 3222.1 of the ASME Code [1.12] specifies the basic allowable compressive stress for the stability of structures as:

"The maximum buckling stress values to be used for the evaluation of instabilty shall be either of the following:

(a) One third the value of critical buckling stress determined by one of the methods given below.

(1) Rigorous analysis which considers the effects of gross and local  ;

buckling, geometric imperfections, nonlineanies, large deformations, and inenia forces (dynamic loads only). '

(2) Classical (linear) analysis reduced by margins which reflect the difference between theoretical and actual load capacities.

(3) Tests of physical models under conditions af restraint and loading the same as those to which the configuration is expected to be subjected.

(b) The value determined by the applicable rules of NE 3133."

The stability limits for various loading conditions, such as Design Conditions and Service Limits A, B, C, and D, have the factors of safety listed in Table 1.3. Method (a) (1) will be used in the present work.

6

i l

1.7.2 ASME Code Case N-284 l ASME Code Case N-284 [1.13] provides well-defined stability criteria for determining the structural adequacy against buckling of shells with mom complex geometries and loading conditions than those covered by Section NE 3133. The rules are based on linear elastic bifurcation buckling theory which has been reduced by knockdown factors to ,

account for the effect of imperfection, boundary conditions, material nonlinearities and residual stresses. The stability limits for the various loading conditions correspond to the

)

factors of safety shown in Table 1.3. The factors of safety are lower than those specified by NE 3222.2 of the Code, but are consistent with other ASME factors of safety for other failure criteria, e.g., yielding due to internal pressure [1.14].

1.7.3 Regulatory Guide 1.57 The Regulatory Guide 1.57 [1.11] delineates the acceptable design limits and appropriate loading combinations associated with normal operating condition, design condition and specified seismic events for the design of containment systems. The Regulatory Guide recognizes the design limits as specified in Section NE3222 of the ASME Code.

However, the Guide states that, "if a detailed analysis is performed, e.g., method (a) (1),

Note 7 to the regulatory position applies." Note 7 explicitly states that, "if a detailed l rigorous analysis of shells that contain the maximum allowable deviation from true theoretical form is performed for instability (buckling) due to loadings that induce  ;

compressive stresses, such analyses, considering inelastic behavior, should demonstrate l that a factor of at least two exits, between the critical buckling stress and the applied stress."

The factor of safety of two against buckling is not associated with a specific Service Limit. However, Regulatory Guide 1.57 states that "the loading combinations should encompass that loading which produces the greatest potential for shell-instability."

Hence, this factor can be associated with Level C and D Service Limits, which usually produce the greatest compressive stress in the shell since they are associated with the SSE event.  ;

1.8 References 1.1. "AP600 Standard Safety Analysis Report," document prepared for US NRC, San  ;

Francisco Operations Office, Document DE-ACO3-90 SF18495 dated June 26, ,

' 1992, Westinghouse Electric Corporation.  !

1.2 Greimann, et. al., "Axisymmetric Buckling Analysis for the AP600 Standard Plant Containment Vessel," NUREG - CR/6378, NRC, Washington, DC, Sept.1995.

1.3 " Seismic Model of Containment Vessel - AP600, Westinghouse," (calculation by Chicago Bridge and Iron CBI) of Lumped Mass Stick Model, Sht 3 A-14 through 3A-20, Sept.1992.

7 m

1.4 Swanson Analysis Systems, Inc., "ANSYS User's Manual," Houston, PA,1993.

1.5 Hibbit, Karlson and Sorensen, Inc., "ABAQUS User's Manual," Newark, CA, 1994.

1.6 " User's Manual for STAGS Computer Code," D266611, April 1972, Imckheed Missililes of Space Co., Sunnyvale, CA.

1.7 Greimann, L. et. al, " Buckling Evaluation of System 80+* Containment,"

NUREG- CR/6161, NRC, Washington DC, July 1994.

1.8 Fanto, S.V., and Piplica, E.J., "PCS Water Distribution Test Phase II Test Data Report," Doiiument ID:WCAP-13296, April 1993.

1.9 "U.S. NRC Standard Review Plan (SRP)," Section 3.8.2, Ref.1, July 1981, pp.

3.8.2.4 and 3.8.3.11.

1.10 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section NE 3220,1989.

1.11 Regulatory Guide 1.57, " Design Limits and Imading Combinations for Metal Primary Reactor Containment Systems Components," NRC, Washington, DC, June 1973.

1.12 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section NE 3222 and Table 3222-1,1989.

1.13 /, setican Srdety of Mechanical Engineers, Boiler and Pressure Vessel Code l Case N-284, Metal Containment Shell Buckling Design Methods," Supplement ,

1. 2 to Nuclear Code Case Book,1995, i 1.14 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code Appendix III, paragraphs 2100 and 3100.

l 8

2. VERIFICATION OF BUCKLING ANALYSIS IN ABAQUS j 1

2.1 Introduction In this chapter, the general theory of buckling of shells is briefly outlined. The elastic buckling procedure in ABAQUS [2.1] was verified by computing the eigen values and eigen modes of a perfect simply supported cylinder. Results were compared to the theomtical solution. ' The inelastic buckling analysis was previously verified by the authors in Ref. 2.2. Such analysis was used in a mesh sensitivity study to establish the suitable element size that yields accurate results and saves computational effort.

2.2 Backaround to Buckline of Shells A conceptual description of the instability of shells is given in [2.3]. Figure 2.1 schematically illustrates the possible load-displacement behavior of a shell. Imading may progress along a fundamental path OCP on which the deformed shape remains essentially 4 the same along, with only the magnitude of the displacement increasing. Eventually, an instability occurs at point P where a maximum load is reached. This point is variously referred as the limit point buckling, A,, or snap through or the plastic collapse load. For this behavior, the failed deflected shape differs from the initial shape only in magnitude.

For many shell configurations and loading, an alternative equilibrium path is also available. That is, a bifurcation point C exists at which two equilibrium paths are possible. At this point, the shell will follow the path of the least energy, path CD. This behavior is termed bifurcation buckling. The post buckling path is called a secondary path. Deformations along this path differ completely from the prebifurcation path, for example, non-symmetric buckling of an axially compressed cylinder in which wrinkles or lobes form around the circumference. Elastic buckling occurs if the shell is elastic at the .

bifurcation point, L , point C is below the proportional limit. If C is above the j proportional limit, inelastic (plastic) buckling occurs.

l If the shell has an initial imperfection, it will not reach the bifurcation point, A , and loading will progress along the fundamental path, OLF. The shell will deform in the l pattern of the initial imperfection without noticeably changing shape until it reaches a

)

L limit load identified as i t. No bifurcation occurs. However, if point C corresponds to

. bifurcation into a nonsymmetric buckling mode, the collapse at L will involve significant nonsymmetric displacement components [2.4]. The equilibrium path beyond point L will i be asymptotic to the post buckling path (See Fig. 2.1).

l The purpose of shell buckling analysis is to predict the load displacement behavior, e.g.,

path OCP or path OCD for perfect shells and path OLF for imperfect shells and their respective critical points (Fig. 2.1). The buckling analysis incorporates both material and geometric non-linearities. Nonlinear static analysis with gradually increasing k. ds to the point at which the structure loses its stability is employed. In this analysis, the structure is loaded step by step and at each load increment the equilibrium equations are solved by 9

iteration up to convergence. Singularities may occur somewhere along the equilibrium path, namely the critical points y and A. Hence the stability analysis requires a method capable of detecting critical points and tracing the equilibrium path beyond the critical points. The ABAQUS program uses the modified Riks method [2.5,2.6] to accomplish this purpose.

In the Riks method, a constraint equation is added to the equilibrium equations to select the length of the load increment in the load-deflection space. The applied load level becomes an additional variable. The modified Newton-Raphson method is applied to solve the system of equilibrium equations iteratively. The stability determinate and eigenvalues are checked during the solution to determine if a bifurcation load has been reached. The resulting technique detects the location of the bifurcation points and determines the solution in their regions, allows the limit points to be passed, improves convergence, and detects the location of h along the fundamental path [2.1].

2.3 Elastic Buckline Analysisin ABAOUS To estimate the critical (bifurcation) load of the elastic, linear stmetures that carry loads primarily by membrane action, classical eigenvalue buckling analysis is often used. TN eigenvalue problem can be described as follows [2.1,2.7]:

((k3 ]+A [kq)){9) = 0 (2.1) l l

where:

l

[ke] = elastic stiffness matrix of the structure

[kq] = initial stress and load stiffness matrix A = load multiplier (eigenvalue)

(9) = buckling mode shape (eigenvector).

The elastic buckling of an axially loaded perfect cylindrical shell was compared to the theoretical closed form solution in the following section.

! 2.3.1 Axially Loaded Circular Cylinder 2.3.1.1 TheoreticalSolution The critical buckling stress of an intermediate length cylinder without geometric or I material imperfection is given by [2.3];

E t

a, = -

(2.2) d3(1-v') r where:

o c= critical buckling stress 10

E = Young's Modulus t = shell thickness r = cylinder radius y = Poisson's ratio.

The associated buckling modes are composed of n full waves in the hoop direction and m half waves in the axial direction such that [2.3]:

n 2 ,32 2r 2 A2

= -43(1- v )

t (2.3)

A = ** (2.4)

I where 1 = length of the cylinder.

2.3.2.2 Finite Element Solution The finite element model, used in the analysis of the cylindrical shell shown in Fig. 2.2a, is illustrated in Fig. 2.2b. The cylinder has the same geometric dimensions of the AP600 containment cylindrical portion. - A segment that subtends an angle of 20 at the center was modeled using S8R5 shell elements. These elements are eight noded isoparametric curved shell elements with five degrees of freedom per node [2.1, 2.7]. Symmetry boundary conditions were applied along the two vertical edges. The upper and lower edges were simply supported. The axial load was applied at the upper edp as equivalent nodal forces. In the prebuckling stage, the radial displacements at the lower and upper edges were permitted to simulate the theoretical simplification of a uniform increase in diameter before buckling [2.3]. In the buckling stage, the radial displacement components were restrained at the upper and lower edges.

Figure 2.3 illustrates the first four buckling modes obtained from the finite element analysis. The first two modes are chess board buckling modes, whereas the third and forth are axisymmetric buckling modes. Table 2.1 compares the critical stesses obtained from the finite element and theoretical solutions. The percentage difference in the results was less than 0.2% of the theoretical solution. The buckled shape obtained from the finite element solution was also compared to theory in Table 2.1. Comparison confirms the good agreement of results.

2.4 Inelastic Bucklina Analysis in ABAOIJJ l

'. As' noted in Section 2.2, nonlinear buckling solutions can be achieved by using a nonlinear static analysis in which equilibrium equations are solved in a point wise l fashion up to the load at which the structure loses its stability. Two verification problems were solved using the modified Riks method in ABAQUS [2.1]: hinged arch with l external pressure and imperfect sphere with external pressure. The problems 11

configuration and results are described in detail in Ref. 2.6. Results were compared to numerical results in published literature [2.5] and to numerical results of the finite difference software BOSOR5. The comparison showed good agreement in the hinged arch problem. In the imperfect sphere problem, the bifurcation load computed by ABAQUS was higher than that computed by BOSORS by only 5% of the ABAQUS solution. The buckled shapes obtained from the two numerical solutions were in complete agreement. Both material and geometric non-linearities were incorporated in such analyses.

2.5 Mesh Sensitivity Study l The finite element solution offers an approximation to the exact solution. These -

approximation arise from the discritization of the continuous structures into subdivisions (elements), implementation of shape functions to describe the displacement and stress fields in an element and the computation of the element stiffness matrix by numerical integration. The accuracy of the numerical solution can be improved by decreasing the size of the elements, this is designated as h-refinement, and/or by increasing the order of the polynomial used to describe the behavior of each element which is known as p-refinement [2.6].

In this work, a mesh sensitivity study with h-refinement was conducted to obtain the convenient element size and aspect ratio that would produce accurate results and consume reasonable run time and storage requirements.

l 2.5.1 Model Geometry The mesh sensitivity study was conducted on a three-dimensional model of a cylinder of the same geometric dimensions as that of the AP600 cylindrical portion. A finite element model was constructed for the imperfect cylinder using the eight noded shell element S8R5 in ABAQUS. A segment of the cylinder that subtends an angle of 10 degrees at the center was modeled. Symmetry boundary conditions were applied at the two vertical l edges whereas simply supponed boundary conditions were applied at the lower and upper i edges. The axial pressure was applied as nodal forces at the upper edge. An imperfection

! sensitivity study was conducted on an axisymmetric model with the same geometric

! configuration using the BOSOR5 program. The results of that study showed that the ,

minimum buckling load was obtained for a sinusoidal imperfection shape with a wave .j l length of 3.0/fi. This value was used to generate the geometric imperfection in the  !

I thme-dimensional model used in this study. The stress strain curve of the material used is illustrated in Fig. 2.4. '

l Five mesh configurations A, B, C, D and E were studied (Fig. 2.5). The five meshes were constructed so that the area of the elements was progressively doubled from Mesh A to Mesh E. This was accomplished by doubling the size of the elements in the hoop direction in one mesh configuration followed by doubling the size of the elements in the axial direction in the other. Table 2.2 summarizes the parameters of the five meshes '

l 12

which were compared regarding the size of the elements in the hoop and axial directions, the element aspect ratio, the wave front, the root mean square (RMS) of the wave front, and the storage space required per analysis and per load increment.

2.5.2 Finite Element Results The nonlinear static analysis was conducted using the modified Riks method available in the ABAQUS program. Figure 2.6 illustrates the deformed shapes of the cylinder portion l at the limit load obtained using Mesh A to Mesh E, mspectively, As the cylinder was i gradually loaded, the deformed shape remained essentially identical to the imperfect shape but with a uniform increase in radial displacements magnitude along the entire l cylinder together with axial shortening. Once the limit load was reached, the deformations increased dramatically in a non-symmetric fashion. The respective buckled shapes illustrated in Fig. 2.7 are the non uniform incremental displacement components that appeared when the cylinder lost its stability. The buckled shape wave length is twice l the initial geometric imperfection wave length. This is consistent with Koiter's theory

[2.8]. The limit load, A ,tobtained from the five meshes was normalized by the classical buckling load of the perfect cylinder (Eq. 3.2).

i l Results were tabulated in Table 2.3 together with the percentage difference from l BOSOR5 results. Results were illustrated graphically in Fig. 2.8 and compared to limit i load factor obtained from the Code Case N-284 [2.9]. Figure 2.8 shows that mesh refinement leads to a better estimation of i andt the solution appears to be converging.

However, using a coarse mesh (Mesh E) overestimates it. At the same time, it can be inferred from Fig. 2.8 that doubling the size of the elements in the hoop direction had no significant effect on the solution accuracy. Based on the mesh sensitivity study, Meshes A to D are acceptable. However, the storage requirements and mn time were significantly reduced with the increase of the mesh size from A to D. Hence, in developing the three-dimensional model of AP600 containment vessel, the mesh size in _ l high stress concentration regions will be taken less than that of Mesh B (Mesh B has six elements per imperfection wave length), whemas in other areas the element size will not exceed that of Mesh D.

2.6 Summary The clastic and inelastic buckling analysis in ABAQUS was explored and verified. The l finite element results were verified by comparison to closed form solutions and other L

numerical solutions based on the finite diffemnce technique. Based on the mesh sensitivity study, element size less than that of Mesh B will be used in high stress ,

concentration areas of the AP600 containment vessel. The remainder of the containment, j the element size will be less than that of Mesh D. '

l 13 4

2.7 References 2.1. Hibbit, Karlson and Sorensen, Inc., "ABAQUS User's Manual," Newark, CA 1994.

2.2 . Greimann, et. al., " Buckling Evaluation of System 80+* Containment,"'

NUREG/CR-6161, NRC, Washington DC July 1994.

2.3 Timosher.ko, S. P. and Gere, J.M., " Theory of Elastic Stability," McGraw Hill,

-1982.

2.4 Bushnell, D., " Buckling of Shells - Pitfall for Designers," AIAA Journal, Vol. 9, No. 9, Sept.1981.

2.5 Riks, E., "An Incremental Approach to the Solution of Snapping and Buckling Problems," Journal of Solids and Structures, Vol.15, pp. 529-551,1978.

2.6 Crisfield, M. A., "A Fast Incremental / Iterative Solution Procedure That Handles Snap Through," Computers and Structures, Vol.13, pp. 55-62,1980.'

2.7 Cook, R. D., Malkus,- D. S. and Plesha, M. E., " Concepts and Applications of Finite Element Analysis," John Wiley and Sons,3rd Edition,1989.

2.8 Brush, D.O. and Almorth, B.O., " Buckling of Bars, Plates and Shells," ist Edition, McGraw Hill, New York,1975.

2.9 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code Case N-284, " Metal Containme F Shell Buckling Design Methods," Supplement #

2 to Nuclear Code Case Book, .%5.

i 14 i

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i

3. THREE-DIMENSIONAL MODEL PARAMETERS 3.1 Introduction In this chapter, the three-dimensional model of AP600 containment vessel was divided into several regions to facilitate the modeling process. The findings of the mesh sensitivity study accomplished in the previous chapter were used to model each region

. according to the expected stress field in that region. The area surrounding the penetrations was modeled with a fine mesh where high stress gradients are expected to occur. On the contrary, low stresses are expected at the upper dome and the remainder of the cylindrical portion away from the base and penetrations. At the same time, areas surrounding ring stiffeners remote from penetrations, and at crane support are not potential buckling regions. Therefore, such areas were modeled with a relatively coarse mesh. A preprocessor written in PASCAL was developed to compute model parameters such as nodal coordinates, element connectivity and properties. The output of the preprocessor is used as an input file for the ABAQUS analysis. The model was checked by conducting a linear static analysis of the containment when subjected to an internal pressure and an inelastic buckling analysis of the penetrations area when subjected to an axial load.

3.2 Modeline Procedure In general, quadratic shell elements (S8R5 in ABAQUS) were used throughout the model. However,in transition regions where it was necessary to reduce the size of the quadratic elements, the six-noded triangular elements (STRI65 in ABAQUS) were used together with S8R5 elements. Stiffener flanges were modeled using three-noded beam elements (B32), whereas the crane bridge was modeled using the two-noded beam element (B31).

To facilitate the modeling procedure, the containment was discretized as shown in Fig.

3.1. The cylindrical ponion was divided into two pans, Pan Al and Pan A2, with a transition zone in between. The upper and lower ellipsoidal heads, crane girder, ring stiffeners were modeled separately and assembled to the cylindrical portion. The model  !

contains 12%3 nodes and. 4282 elements. The total number of variables was 96270 with I a wave front equal to 1431. The assembled model is illustrated in Fig. 3.2. In the  ;

following sections, the various pans of the model are described.

L 3.2.1 Upper Ellipsoidal Head The upper ellipsoidal head was modeled using the eight-noded quadrilateral shell elements, S8R5. The axisymmetric analysis conducted in Ref. (3.5) revealed that i buckling took place at the base remote from the upper ellipsoidal head, hence it was modelled with a coarse mesh to minimize the model wave front and computation time.

The nominal element dimensions in the hoop and meridional directions were approximately 2.16and 1.56, respectively. All the nodes are located along the 15

principal spherical coordinates lines to avoid small elements in the top which may cause numerical instability. Figure 3.3 shows the finite element model for the upper ellipsoidal head.

\

3.2.2 Cylindrical Containment The cylindrical containment was divided into two parts as shown in Fig. 3.1. ' Area (Al) f includes all the four penetrations and their stiffening plates while Area (A2) encompasses the remaining area of the cylinder. Area (A1) subtends an angle 120" at the cylinder axis, and extends vertically from the embedment line (Elev.100') to the position of the upper ring stiffener at Elev.170'. This area is considered more critical than Area (A2) due to the presence of the penetrations and, therefore, was idealized with a finer mesh. The size of the elements used as the basic mesh of Area (Al) is approximately (0.3756 by 0.756). The element dimensions are close to those of Mesh A of the sensitivity study (Sec. 2.5) and have an aspect ratio of 2:1. Since high stress concentration are expected in the areas immediately adjacent to the penetrations (Fig. 3.1), a very fine mesh was utilized in which element size did not exceed (0.3756 by 0.756 ).

' l Area (A2) was idealized using an element size of (0.75 M by 1.56). This mesh was close to Mesh C of the sensitivity study (Sec. 2.5). Vertical and horizontal transition zones were created to achieve the transition between the different mesh sizes in Areas Al and A2. Figure 3.4 illustrates the finite element idealization of part of the cylindrical portion of the containment.

3.2.3 Lower Ellipsoidal Base The cylindrical containment ends at the bottom with an ellipsoidal shell, embedded in concrete, a small portion of the ellipsoidal shell (49.5 in. high) extends above the concrete l

base. Because high stress gradients were expected in this area, the ellipsoidal base was l l therefore, meshed with a mesh finer than that of the cylindrical portion. While the hoop i

dimensioas of the elements were kept the same, the meridional dimensions were halved.

Hence, the nominal element dimensions were (0.196 by 0.756) in the lower portion  ;

of Area (A1) and were (0.3756 by 1.56)in the bottom of Area (A2). The aspect l ratio of all the element of the ellipsoidal head elements is 4:1.

3.2.4 Crane Girder The crane girder was modeled using eight-noded shell elements, S8R5, as shown in Fig.

3.5. These were used to model the crane girder cross section together with its radial stiffeners, The holes in the inside of the vertical plate of the crane girder were ignored.

The number of the radial stiffeners was reduced from 80 (in the actual containment) to 48 in the finite element model to fit with the mesh size for the containment shell. An equivalent thickness was computed so as to provide the same total shear stiffness of the actual stiffeners.

16

t 3.2.5 Crane Bridge ,

t.

l The crane is composed of a rotating bridge carrying a trolley which extends diametrically l across the containment. The crane bridge was assumed to be parked at an azimuth of 10 .

The bridge was divided into two three-dimensional beam elements by an intermediate  ;

node at the trolley position, where the trolley mass was lumped. The trolley position was chosen to be the closest possible position to the containment. The crane bridge model is illustrated in Fig. 3.6.

The crane bridge is carried on a system of wheels which move on a circular rail connected l to the top of the crane girder. Each . set of wheels extends through an angle of l . approximately 43 degrees. The nodes of the wheel set at the crane rail location were l rigidly connected to each other using "the rigid beam multiple point constraint capabilities" of ABAQUS [3.1]. The connected nodes can therefore move together as a i rigid body, but no relative movement between the nodes is allowed. The wheel support

! system provides a means for transferring forces from the crane bridge to the crane girder in the radial and vertical directions, as well as in the tangential direction (due to the l wheel brakes and friction). The wheel set beam elements are pin connected to the crane l _ bridge beam and are pin connected to the crane shell model (which transfer forces only but no moments).

3.2.6 StifTeners L

The horizontal web plates of the ring stiffeners were modeled using the eight-noded quadratic shell elements, S8R5 (triangular elements were used in the areas of complex l details), whereas the flanges of the stiffener sections where modeled using the three-noded beam elements. The radial elements which brace the st i ffener sections were also modeled with B32 beam elements. Figure 3.7 shows the idealization of stiffeners used in

, the model.

t

3.2.7 Penetration Barrels The equipment hatches and personnel air lock barrels were modeled as short cylinders connected to the containment shell. The full length of the reinforced portion of the four i barrels was idealized in the model with S8R5 shell elements. The elements mass in each l barrel was computed so that the weight of the idealized barrel model is equal to that of the entire respective penetration assembly. The barrel cover was not modeled. However, when the containment is subjected to internal or external pressure, the pressure on the cover is replaced by nodal forces on the circumference of the barrel.

l 1

i 17

3.3 Mesh Generation l 3.3.1 Solid Modeling L

The containment area around the penetrations contains complex details such as the intersection of the equipment hatch and personnel lock barrels, the reinforcing collar around penetrations, the ring stiffeners, the junction of the cylindrical portion with the l lower dome, etc. Due to the complexity in this area, it was expedient to use some kind of l automatic meshing to save time and reduce errors in cmating the model. Since ABAQUS does not have such capabilities, ANSYS [3.2] solid modeling and automatic meshing techniques were used to mesh the areas around the penetrations (Sec. A.l.1). Figure 3.8 j illustrates some of the steps performed to create a model of the containment area around I the penetrations using solid modeling together with the created finite element model. All cylindrical surfaces were defined by two lines: the axis and the generator of the cylinder.

The intersection of the containment and the penetration barrels was computed by ANSYS and the enclosed elliptical part was deleted to model the opening. After defining the entire shape (See Fig. 3.8) of the penetrations area, every part was descritized by S8R5 or STRI65 shell elements by specifying the number of elements along the edges. The thickness of the elements surrounding each penetration (shown in dark color in Fig. 3.8) was set to its actual reinforcing collar thickness (Sec.1.3 of Ref. 3.5).

3.3.2 Assembling the Model Due the complexity of the model, it was found necessary to develop a special preprocessor for creating the ABAQUS input file. This preprocessor was developed in  !

PASCAL. The preprocessor creates the nodes and elements of the model in different areas (the uniform mesh portion of Area A1, Area A2, upper and lower domes, stiffeners, etc.) and reads in the mesh geometry in the penetration areas created using the ANSYS program. A special routine was developed to link the two models by unifying the node numbering at their intersection surfaces. This routine ensures the element connectivity in the finite element model. In addition several views from inside and outside were plotted and were carefully checked to verify the assembly of these regions.

3.4 Imperfection Paramstggg l An axisymmetric imperfection was incorporated into the three-dimensional finite element model of the AP600 containment. The imperfection was modeled as a sinusoidal wave with radial amplitude equals to half of the containment shell thickness, i.e.,0.8125 in. I and wave length of approximately 3.54i. The amplitude of the imperfection corresponds to the ASME Code [3.4] specified maximum deviation of one shell thickness. The imperfection wave length was based on the axisymmetric imperfection sensitivity analysis described in Ref. 3.5. It should be noted that each half-wave length of the imperfection contains approximately 4.5 elements which is larger than that of Mesh D of the mesh sensitivity study described in Sec. 2.5.

18

I I

3.5 Material ProDerties

! The stress strain relationship of the material (SA537-Class 2 steel) used in the three-dimensional finite element model was derived from the ASME Code Case N-284 (Ref.

l 3.6) plasticity reduction factor for the respective ambient temperatures. This has the effect ofincluding residual stresses in the analysis.

3.6 Check Run l 3.6.1 Linear Static Analysis with Uniform Pressure To verify the three-dimensional model of AP600 containment (Fig. 3.2) prior to the nonlinear buckling analysis, the model was loaded with uniform internal pressure of 1 psi.

No geometric or material imperfections were included in the model and the analysis was conducted using the small deflection theory. The deformed shape of the three-dimensional model is shown in Fig. 3.9. Elements connectivity was verified, since no displacement discontinuities were observed (See Fig. 3.9). For the smooth cylindrical pan away from penetrations and stiffeners, the meridional and hoop stress resultants can be computed using classical shell theory [3.3] as pr/2 and pr respectively (where p is the internal pressure and r is the radius of the cylinder). This yields 390 lb/in in the axial direction and 780 lb/m in the hoop direction. The axial and hoop stress resultants obtained from the finite element model varied from one pan of the model to the other and showed high stress concentration regions around the penetrations. However, in an area remote from the penetrations, the stress resultants were uniform and the magnitudes of the axial and hoop stress resultants were 404 lblin, and 793 lblin. receptively which is consistent to theory with a maximum percentage error of 3%.

3.6.2 Nonlinear Buckling Analysis of Containment Sector Around the Penetrations A sector of the model including the lower equipment hatch and the two personnel air locks was analyzed under axial compressive loading as shown in Fig. 3.10. This study was conducted to investigate the buckling behavior of the area around the penetrations ,

and to verify the finite element model in that area before performing the analysis with the  !

full model. The sector subtends an angle of 52.5 and extends venically from the l concrete base level at Elevation 100' up to Elevation 165'-9" (this sector is part of Area (A1) in Fig. 3.1). Full restrain was applied to all six degrees of freedom for nodes at the i base. Nodes at the upper edge were restrained from rotation about the circumferential axis, but were allowed to translate in the venical and radial directions. Symmetry l boundary conditions were applied along the two venical sides of the sector. An axisymmetric sinusoidal imperfection was used for the model (See Sec. 3.5). The axial compressive load was applied as concentrated nodal forces at the upper edge.

The nonlinear static analysis of the model was performed using the modified Riks method

[3.7]. Figure 3.11 shows the deformed shape of the cylinder sector just before the limit load was reached. The variation of the vertical deflection of the top of the sector with the 19 l

l L._________________

l load is illustrated in Fig. 3.12.' The limit load factor was 0.292 of the classical buckling load (Eq. 2.2). This factor is slightly lower than the factor of 0.315 obtained from

. BOSOR5 using an axisymmetric model for a cylinder under axial load but higher than O.252 which is the factor given by the Code Case N-284 design equations (3.6] for axially loaded compressed cylinders. Buckling took place in the form of sine waves in the hoop

- direction in the unreinforced shell above the upper personnel air lock reinforcing collar (Fig. 3.11) which indicates that the areas of the penetrations are adequately reinforced.

This conclusion will be further discussed in Chapter 5 of this report. This model, therefore, will be used in the next chapters to analyze the AP600 containment vessel.

3.7 References 3.1 Hibbitt, Karlsson and Sorensen, Inc., "ABAQUS User's Manual," Newark, CA i

1994.

3.2 Swanson Analysis Systems,Inc.,"ANSYS User's Manual," Houston, PA 1993 3.3 Timoshenko, S.P. and Gere, J.M., " Theory of Elastic Stability," McGraw Hill, 1982.

1

! 3.4 American Society of Mechanical Engineers, Boiler and Pressure Code, Section 4 NE 3222,1989. j l

3.5 Greimann, et. al., "Axisymmetric Buckling Analysis for the AP600 Standard Plant Containment Vessel," NUREG- CR/6378, NRC, Washington, DC, Sept.1995.

3.6 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code Case N-284, " Metal Containment Shell Buckling Design Methods," Supplement #

2 to Nuclear Code Case Book,1995.

3.7 Riks, E., "An Incremental Approach to the Solution of Snapping and Buckling Problems," Journal of Solids and Structures, Vol.15, pp. 529-551,1978.

l l

t i

20

4. RESPONSE SPECTRUM ANALYSIS OF AP600 CONTAINMENT l

4.1 Introduction When the containment vessel is subjected to seismic excitation, the generated stress field I will be time dependent. Hence, the containment vessel may exhibit a parametric instability excitation [4.1], or dynamic buckling in which the inertial forces due to seismic loading interact with the inenial forces associated with the deflections due to the i buckling motion [4.2]. To obtain the buckling factor of safety, a time history analysis I which incorporates both geometric and material nonlinearities is appropriate. However, due to its practicality, time history analysis was replaced by a static approach [4.3] in which the seismic excitation is replaced by a set of static forces that reproduce the maximum seismic stress resultants. Buckling inertial forces were neglected (Sec.1330 of 1 Ref. 4 4).

1 In this chapter, the response spectrum analysis of AP600 containment vessel was l accomplished. The objective was to compute the maximum SRSS of meridional and hoop stress resultants, Ni and N2 , respectively, due to the SSE ground excitation. The two horizontal components in the North-South and East-West directions, together with the vertical component of the SSE floor response spectra at El.100 ft. were used in the analysis. Two potential buckling regions were identified as those at which the maximum SRSS meridian stress resultants took place. Equivalent static forces were developed to regenerate the SRSS meridian stress resultants in each of the two potential buckling regions. The equivalent static forces were utilized in the buckling analysis presented in Chapter Five.

4.2 Modes of Vibration 4.2.1 Theoretical Background i Since the AP600 containment vessel is mainly composed of a circular cylindrical shell. l The free vibration of a clamped-clamped cylinder was investigated by the classical l approach as background information for the finite element analysis. The vibration of a cylinder may consist of any number of circumferential waves, n, and axial half waves, m.

Due to the complexity of such a problem, there is no exact closed form solution for natural frequencies and mode shapes. However, analytical solutions were based on either Raylieh's method [4.5,4.6] or on the assumption of some analytical simplifications. As a result, a cubic equation for natural frequencies can be obtained for a given pair of m and

n. The least root of this frequency equation, is the one which has practical importance.

Based on Donnell's equilibrium equations [4.5], the least frequency associated with any pair of m and n was computed as follows [4.5]:

E(n' + ((1- ty )(kr / L)') (4.1) pr[n'{(n + 1)(1- t) ) + n'(3- v)(1 + v)]

l 21

12 and p t

- k = 1.50x,2.5x,3.5x,. . . form = 1,2,3, . . . respectively where E, y and p are the Young's- modulus, Poisson's ratio and material density, respectively. A unit .value of m and n does not necessarily give the least fmquency.

However, for a given m, the value of w decreases with the incmase in n, until it reaches a minimum after which it increases again.

A numerical application Eq. (4.1) was conducted assuming a clamped-clamped cylinder of the same geometric dimensions of AP600 containment cylindrical part. Results are tabulated in Table 4.1 and illustrated in Fig. 4.1 for various values of m. The results indicate that the cylinder has numerous closely spaced modes composed of circumferential and axial waves. We can expect a similar phenomenon in the AP600 vessel. Therefore, it will be necessary to extract many modes for accurate representation of the containment dynamic behavior.

4.2.2 Finite Element Solution The three-dimensional model illustrated in Fig. 3.2 was utilized to compute the mode shapes and natural frequencies of AP600 containment vessel. A sinusoidal axisymmetric imperfection with a wave length of 4.0 6 and a peak-to-peak amplitude equals to one shell thickness was utilized (see Sec. 3.1 of Ref. 4.7). All shell elements in the model were classified according to their mass density in Fig. 4.2. Elements having the same mass density were given an identical color.

The subspace iteration method [4.3] was utilized to extract the first sixty free vibration modes (see Sec. 4.2.1). The first five modes are illustrated in Fig. 4.3. The first mode was local vibrations of the crane bridge in the East-West direction (i.e., Y direction) at a frequency of 3.69 Hz. The second mode is a cantilever mode of the containment vessel in the Nonh-South direction (i.e., X direction) at a frequency of 5.91 Hz. The forth and fifth modes comprise a cantilever action in the Y direction together with local vibrations of the upper equipment hatch as shown in Fig. 4.3. The remaining fifty-five modes were basically composed of circumferential waves with a sinusoidal vertical vibration of the lower stiffener at Elev.132'-3". These modes were closely spaced and occurred at a

! frequency range from 6.85 Hz. to 9.74 Hz. This is consistent with the theoretical solution I discussed in Sec. 4.2.1. The frequencies, participation factors and effective masses of the l first sixty modes are listed in Tables B.1, B.2 and B3 of Appendix B, respectively. The total effective mass [4.3], in the X and Y directions was 73% and 75% of the total mass, respectively. On the other hand, the total effective mass in the vertical direction (i.e., Z direction) was only 2.5 % of the total mass. This indicated that dominant vertical modes were not extracted. Hence, the computation of additional modes was required to meet the 4 requirements of Sec. 3.7.2 of the NRC Standard Review Plan, SRP [4.8].

22 i

i

i l

l Due to the limited computer disc space on the Alpha workstation used in conjunction I with the analysis, not more than sixty modes can be computed at a time. Therefore, the subspace iteration method with shifts [4.3] was utilized to extract higher frequency modes. The shifted stiffness matrix is computed by subtracting the product of the squared shift frequency and the mass matrix from the stiffness matrix. The subspace  !

iteration technique is then applied using the shifted stiffness matrix to determine the ten l closest mode shapes to the shift frequency. This procedure was repeated thirty-two times l with shift frequencies that ranged from 9.9 Hz to 20 Hz. The frequencies obtained in l

each step was checked to overlap with the previous step to ensure that no modes were l missed. The process was terminated when a sufficient total effective mass was reached in

all directions. The accumulative effective modal mass was plotted versus ' mode

!- frequency in Fig. 4.5. A sudden increase in the effective modal mass in the X, Y or Z direction (see Figs.1.1 and 1.2) at a certain frequency means that the corresponding mode l- shape is a significant vibration mode in that direction. Figure 4.5 shows that the first

cantilever mode in the X direction has a frequency of 5.91 Hz., whereas the first t

cantilever mode in the Y direction took place at a frequency of 6.4 Hz. (see Fig. 4.3) as  ;

previously noted. Similarly, the first vertical mode occurred at a frequency of 12.398 Hz (see Fig. 4.6). The second and third vertical modes had a frequency of 15.38 and 15.40 Hz, respectively. These two modes comprise sinusoidal circumferential waves together with vertical vibration of the upper dome (see Fig. 4.6). The forth vertical mode had a frequency of 18.82 Hz. The frequencies, participation factors and effective modal masses of the two hundred and thirty-seven extracted modes are listed in Tables B.1, B.2 and B.3 i of Appendix B, respectively. The cumulative modal effective masses were 86.74%,

j 88.91% and 73.8% of the total mass in the X, Y and Z directions, respectively.

f 4.2.3 Comparison of Results with Axisymmetric Model The three-dimensional finite element model results were compared to the results of an l

axisymmetric model in Ref. 4.7. The mass of all penetrations and attachments was  ;

smeared around the circumference in the axisymmetric model. The total mass of the model was 17720 lb.sec2/in [4.7], i.e., less than that of the three-dimensional model by only 1.7% (see Sec. 4.2.2). The vibration analysis was conducted using the BOSOR4 finite difference software. The frequencies of the first horizontal and vertical modes obtained by ABAQUS and BOSOR4 programs are compared in Table 4.2. Results are in  !

gwd agreement. However, the frequencies obtained by ABAQUS were generally less than those of the three-dimensional model (see Sec. 2.2 of Ref. 4.7). Mode shapes with local vibrations of penetrations cannot be computed by the axisymmetric model. The l total modal effective mass in the horizontal and vertical directions of the two numerical solutions were nearly identical as shown in Table 4.3.  ;

{ i 23

---u-- --.---- -------- ------- _ _ _ _ _- - -___ --__ ___ --_---__ - - - - - . - -

4.3 Response Spectrum Analysis The response spectmm analysis was conducted to compute the maximum meridional and hoop stress resultants due to the SSE ground excitation. The floor response spectra used in the analysis were illustrated in Fig. 2.16 of Ref. 4.7. The outline of the analysis is presented in the following sections.

4.3.1 Modal Responses Inenia forces, F *, associated with each mode, i, were applied as pseudostatic loads on the containment model. These inertia forces were computed as:

(F,') =[M] {@e}ml (4.2) where e, and m,is the mode shape and the circular frequency of mode i, respectively and

[M] is the mass matrix. The structure was then statically analyzed to determine the corresponding modal response, say the stress resultant,N*. The maximum seismic modal response in the i* mode, say a stress resultant Nj, i due to the j (j= X, Y or Z) eanhquake motion was computed as:

Nf# ji sij Nij = 2 (4.3)

I where pij is the participation factor of mode i in the direction j (Table B.2 of Appendix B), Sji is the spectral acceleration corresponding to mode i frequency for earthquake component in directionj.

4.3.2 High Frequency Modes In order to incorporate the response of high-frequency modes, the procedure outlined in Sec. 3.7.2 of the Standard Review Plan (SRP) [4.8] was implemented. The basic assumption is that the maximum response of high-frequency modes is not randomly phased in time. On the contrary, the maximum response in such modes will be essentially deterministic and the containment will respond to the modal inenial forces from the peak Zero Period Acceleration (ZPA) in a pseudostatic fashion. The fraction oflumped mass per degree of freedom which was not included in the extracted modes was computed as follows:

en = ( a Ga)-Sy (4.4) where et is the frection of; the remainder mass for degree of freedom (DOF) k, $a is the i* mode shape at DOF k, N is the total number of extracted Sy is the Kronecker delta, which is one if DOF k is in the direction of the earthquake, j, and zero if DOF k is a 24

rotation or not in the j direction. The pseudostatic inertial forces associated with the summation of all high-frequency modes for each DOF k was then computed as follows:

Py = ZPA). Mt. et (4.5) where Py is the inertia force to be applied at DOF k in the direction j, Mk is the lumped mass at DOF k. The ZPAj was conservatively assumed to be equal to the spectral acceleration of the j* spectrum corresponding to the frequency of the highest extracted mode (i.e.,20.3 Hz) (see Table A.1 of Appendix B), The pseudostatic inertia forces in each direction j were applied separately and the containment was statically analyzed in each case to determine stress resultants due seismic response in high-frequency modes.

4.3.3 Modal and Directional Combination The maximum modal responses determined by Eq. 4.3 and the high-frequency modal responses (Sec. 4.3.2) were combined by the ten percent rule [4.3] that accounts for closely spaced modes. The maximum combined response for the stress resultants was computed as follows:

Nomj= IN 2+2EIIN N u3 1 where i=k (4.6) e i k where No.) is the maximum stress resultant response due to ground excitation in the ja direction. Modes of vibration were considered closely spaced if their frequencies differ by less than ten percent. It is to be noted that the modal response of the high-frequency modes constituted 3.49%,2.69% and 4.46% from the total modal response in the X, Y and Z directions, respective. This means that the total response was not increased by more than 10% of its value when the high-frequency modal response was added. Hence, the requirements of Sec. 3.7.2 were satisfied.

The maximum stress resultant response of the containment, was then computed by combining the responses due to the three earthquake directions by the SRSS method [4.3]

as follows:

N,, = N2maxj (4.7) where No. is the maximum stress resultant.

4.3.4 Discussion of Results Figure 4.7 illustrates the contour plot of meridional stress resultants, Nim, in the vicinity of the penetrations where the subscript I designates the meridional direction of the shell.

25 1

L i __ _ ----- - ---------- --_ - - - - - -- ------ - ------- -- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - -

i

, The stress resultants in the remainder of the containment were significantly less than l those in this area and, hence, have less practical importance. The meridional stress resultant, Ni -, at the base were plotted versus the circumferential angle, 0, in Fig. 4.8. 1 The angle 6 was measured from the North (X direction, see Fig.1.2) and in the counter- '

clockwise direction. The maximum value was 5320 lb/in and took place at 0 equals to 286.875*. The distribution of N i along the meridians passing through the upper L equipment hatch axis (0 = 293') and the meridian at q equals to 286.875 were compared l in Fig. 4.9. The comparison shows a high stress concentration region located at the upper

equipment hatch reinforced collar. The maximum value of N i was 15670 lb/in (Fig.

l 4.7).

Figure 4.10 compares the distribution of Ni along the meridians passing through the two air locks and the lower equipment hatch. Meridional stresses were also concentrated l around these penetrations due to local masses effect, but the stresses were less than the meridional stresses near the upper equipment hatch.

The contour plot of the hoop stress resultants, N 2 - . in the penetrations area are j

illustrated in Fig. 4.11. The maximum value of N2 was 13300 lb/in at the reinforced i l collar of the upper equipment hatch. At the base, the maximum value was 4430 lb/in and I took place at 0 equals to 286.875 . The hoop stress resultants'were less than the l meridional stress resultants and were found to have a less significant effect on the buckling of the containment [4 2,4.7].

4.3.5 Potential Buckling Region The maximum N m i stress resultants were peaked at two regions (see Fig. 4.7): Region 1 near the base at 0 equals to 286.875*and Region 2 around all four penetrations. Figure l 4.12 illustrates these two regions. The highest potential for buckling will occur at one of l

thes: regions due to the compressive stress resultants. Hence, Regions 1 and 2 were designated as potential buckling regions. The factor of safety against buckling in each region was determined and compared to allowable values in Chapter 5.

1

( 4.4 Eaulvalent Static Loads '

l 4.4.1 Purpose

(

i The maximum SRSS stress resultants, Ni-, obtained from the response spectrum analysis (Fig. 4.7) can not be directly input into the ABAQUS program because they are not external forces but are internal forces (see Appendix A). As an alternative, these 1 internal forces will be reproduced in the two potential buckling regions by applying sets of equivalent external static loads on the containment [4.2]. Each loading set was l selected so that it produced a stress resultant field that bounded the maximum value of Ni in each buckling region. 'lhe stress resultant field in the remainder of the  ;

containment was less than the SRSS stress resultants to ensure that buckling will take I place in the potential buckling regions.

26 i

r 4.4.2 Potential Buckling Region 1 The maximum Nii of Region I was regenerated using the modal decomposition [4.2, 4.9] method. This method is based on expressing the SRSS stress resultants in the whole l region as a linear combination of modal stress resultants, N *, as follows:

{Ni- } = [N'] {Fi} (4.8) where P i is the linear combination coefficients of the individual modes that superimpose

- to produce the SRSS seismic response in Region 1. The vector Pi was obtained from Eqs. 4.8 by the method of least squa:es. The set of equivalent static loads was then computed by the linear combination of modal inertial forces as follows:

I

{F,} = (F*] {yi} (4.9)

~ '

where F,' are the inertia forces associated with mode i (Eq. 4.2) and {F,} is the set of i

equivalent static loads.  !

i The best and simplest bound of the meridional stress resoltants, N i , in Region I was obtained by using the most dominant cantilever modes two and four only (see Fig. 4.3). .

Remaining modes, when added, tended to cause locally high stress resultants in other '

regions. The equivalent static loads were applied on the containment and then the containment was statically analyzed to verify the analysis. Figure 4.13 illustrates the  ;

variation of the generated Ni along the meridian at 0 equals to 286.875' compared to the  !

respective N ,. stress resultants. The comparison indicates that maximum stresses at I the base (potential buckling Region 1) were enveloped. Note that the stresses generated )

at the equipment hatch were tensile and, hence, this loading case is only appropriate to  !

examine buckling at the base and not at penetrations. Figure 4.14 depicts the contour plot of N i due to equivalent static loads. The maximum compressive Ni took place in potential buckling Region 1, whereas tensile stresses were generated around penetrations.

Stresses in the remainder of the containment did not exceed the maximum SRSS stresses.

4.4.3 Potential Buckling Region 2 Attempts were also made to apply the modal decomposition method to reproduce the ~ {

maximum SRSS stress resultants, Ni -, in the reinforced collar around all penetrations i (Region 2). The resulting equivalent static loads produced stresses that enveloped the maximum SRSS stress resultants around all penetrations, however, they also produced  ;

high compressive stress at the base that exceeded the value of Ni,. in that region. The l buckling analysis of the containment with such loads would have caused buckling to occur in Region 1 at the base instead of Region 2. Consequently, a simple set of vertical line loads were applied at every penetration as shown in Fig. 4.15 to produce a compressive stress field in the meridional direction that envelops Ni , around every  !

27 i

penetration. The values of the line loads wem 7700 and 3900 lb/in for the upper and lower equipment hatches and were 5050 and 4950 lb/in for the upper and lower airlocks.

Figure 4.16 illustrates the contour plot of N idue to these equivalent static loads.

l Compressive stresses were only concentrated around penetrations, whereas a tensile stress field was generated in the remainder of the containment. Such a stress field is suitable to l examine local buckling near the penetrations. Figures 4.17 to 4.20 compare Ni due to the equivalent static loads around the four penetrations with the respective value of Ni m.

The figures indicate that peak values of Ni max were always enveloped by Ni due to this set of equivalent static loads. The distribution of Ni in the first row of elements in the i thin shell around the reinforced collar of the upper equipment hatch was also compared L

with the respective Ni m in Fig. 4.21. The figum indicates that all N im values were again enveloped as well by the generated Ni stress resultants. This confirms the validity of this set of the equivalent static loads (Fig. 4.15) to examine local buckling around penetrations.

4.5 References 4.1 Narayanan, R. and Roberts, T.M., " Structures Subjected to Dynamic loading:

Stability and Strength," Elsevier Applied Science, London,1991.

4.2 Greimann, L. et. al, " Buckling Evaluation of System 80+* Containment,"

NUREG- CR/6161, NRC, Washington DC, July 1994.

4.3 Regulatory Guide 1.92, " Combining Modal Responses and Spatial Components in Seismic Response Analysis," NRC Washington, DC, February 1976.

4.4 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code Case N-284, " Metal Containment Shell Buckling Design Methods," Supplement

1. 2 to Nuclear Code Case Book,1989.

4.5 Kraus, H., " Thin Elastic Shells," ist Edition, John Wiley and Sons Inc., New York,1%7.

4.6 Blevins, R.D., " Formulas for Natural Frequency and Mode Shape," ist Edition, Von Nostrand Reinhold Company, New York,1979.

4.7 Greimann. et. al., "Axisymmetric Buckling Analysis and Performance Beyond Design Basis for the AP600 Standard Plant Containment Vessel," NUREG-CR/6378, NRC, Washington, DC Sept.1995.

4.8 "U.S. NRC Standard Review Plan (SRP)," Section 3.8.2, Ref.1-July 1981, pp.

3.8.2.4 and 3.8.3.11.

4.9 Greimann, L., et al., " System 80+* Containment Structural Design Review,"

NUREG-CR/5957, NRC, Washington, DC, May 1993.

li l

28 l i

5. INELASTIC BUCKLING OF SYSTEM AP600 CONTAINMENT AT LEVEL C, DBA4 5.1 Introduction The axisymmetric analysis accomplished in Ref. 5.1 showed that the highest compressive meridional stress resultants occurred when the containment was subjected to the design basis accident load combination DBA4 (see Table 1.1) at Level C Service Limit.

Therefore, this load combination was analyzed here to determine the buckling factor of safety of the containment using the three-dimensional model. The loading configuration consists of an external pressure of 3 psig, uniform rise in temperature of 50 F, gravity loads and a set of static loads bounding the maximum effect of SSE ground excitation.

Gravity loads were applied to the model by specifying the value of the mass density of each element and the gravitational acceleration value and direction cosines. The program can automatically compute the weight of each element. Temperature was specified at each node in the model. The procedure and fimdings of the buckling analysis are presented and discussed in the following sections.

5.2 Methodology The buckling facior of safety of the containment was determined by a quasi-static approach. The containment was loaded with the equivalent static forces that regenerate the maximum SRSS meridional stress resultant, N imo (Chapter 4), rise in temperature, external pressure and gravity loads. The buckling analysis was conducted using the modified Riks method [5.2, 5.3), where all the loads are applied proportionally in increments and the solution is obtained by iterations up to convergence. Both geometric and material non-linearities were incorporated in the analysis. Figure 5.1 illustrates the constitutive relation utilized for the steel at a temperature 120 F (see Sec. 3.5). The yield stress was 59.2 ksi and the proportional limit stress was 32.5 ksi. The load deflection behavior and the growth of the Von Mises effective stresses [5.4] were monitored through out the containment during loading. This was done to identify the location of the first buckle and the respective buckling load factor. The analysis was terminated when the containment lost its stability and staned to unload.

The response spectrum analysis conducted in Chapter 4 indicated that there were two potential buckling regions due to seismic loading: Region I at the base below the upper equipment hatch and Region 2 around penetrations. Other loads in the load combination DBA4 (such as external pressure and uniform rise in temperature) produce a uniform stress field with high stress concentrations at the base (Region 1) and around penetrations (Region 2). The gradient stress field generated by gravity loads will also have peak values at the base (Region 1) and around penetrations (Region 2). In other words, the base

and the penetration region are potential buckling regions for each of the individual loads as well as the combined case.

29

Therefore, two load cases were considered, namely Load Case (1) and Load Case (2), to

.. compute the factor of safety against buckling at Regions I and 2, respectively, Each load case is composed of the equivalent static load for the respective potential buckling region (see Sec. 4.4) superimposed with an external pressure of 3 psig, uniform rise in temperature of 50'F and gravity loads (own weight plus dead load of crane). The buckling analysis results for the two load cases are described in the following sections.

5.3 Load Case (1)

The buckling strength of the containment was examined at the support (Region 1) where the maximum SRSS Nr , stress resultants occurred at Azimuth 73.125. The containment lost its stability at a load factor, At,2.54. The deformed shape at buckling is shown in Fig. 5.2.

The load factor was plotted versus radial displacements in two locations: Point A and Point B near the upper air lock reinforcing collar as shown in Fig. 5.3. This figure shows that the radial displacements at both Points A and B were increasing until the load factor reached 2.54, at which, the containment buckled and started unloading. Hence, the displacements at both points decreased sharply. The maximum radial displacement at Point B was larger than that of Point A because Point A was near the concrete support. In order to identify the location of first buckling, the Von Mises strosus were recorded at each load increment. Figure 5.4 illustrates the contour plot of the Von Mises stresses at Atequals to 2.54. The figure indicates that the peak value was 36 ksi at both Points A and B. Figure 5.5 shows the growth of the Von Mises stresses at these points. The figure shows that when At reached 2.54, stresses began to release at Point A and continue to build at Point B. This shows that buckling took place first at Point A where the maximum SRSS Ni stress resultants were enveloped by the proposed equivalent static forces.

5.4 Load Case (2)

In this case, the buckling was examined around the penetrations where high stress concentrations were recorded (Fig. 4.7). The containment was loaded by a set of axial loads shown in Fig. 4.15. The containment lost its stability at a load factor, Atequals to 2.62. Figure 5.6 shows the deformed shape at buckling. The maximum radial displacement took place at Point C in the thin shell below the upper equipment hatch reinforcing collar. Figure 5.7 illustrates the load versus the radial deflections at Point C and Point D, at the reinforcing collar of the equipment hatch. The figure indicates much larger radial displacements at Point C compared to Point D when the limit load, AL, was reached. The contour plot of the Von Mises stresses at A t is illustrated in Fig. 5.8. The peak stress was 40 ksi and took place at Point C at which the maximum radial displacement occurred. This confirms that buckling took place at Point C away from the upper equipment hatch penetration. It is to be noted that, however the maximum N imo stress resultants near penetrations were enveloped (Sec. 4.4.3), buckling took place in the thin shell away from penetrations. This means that the factor of safety against local 30 1

_ _ ___.___ _ _ _ __ ___ ~ - - - _ - - _ - - - - - - - - - - - - - - - - - - -

buckling at the penetrations is greater than 2.62. It is, therefore, concluded that all penetrations' areas are adequately reinforced.

This result agrees with the inelastic buckling analysis conducted on the penetrations area alone when subjected to axial compressive loads (Sec. 3.6.2). Buckling took place in the thin shell away from the reinforcing collar of the upper air lock (See Fig. 3.11).

5.5 Summary and Discussion In this chapter, the buckling factor of safety of AP600 containment was evaluated for level C Service Limit. Two load cases were considered. Load Case I to examine buckling at the base below the upper equipment hatch and Load Case 2 to examine the local buckling at penetrations. The minimum buckling factor of safety was 2.54 for Load Case 1. Buckling took place at the location of maximum Ni n, stress resultants. For Load Case 2, the buckling factor of safety was 2.62 and buckling occurred in the thin shell below the upper equipment hatch. Such a result showed that the factor of safety against local buckling at the penetrations would be greater than 2.62. Hence, the area of penetrations is adequately reinforced. Based on the above results, the containment design satisfies the requirements of Regulatory Guide 1.57 [5.5], ASME Code Case N-284 [5.6]

and ASME Section NE 3222.2 [5.7] which specify a minimum buckling safety factor of 2.0,1.67 and 2.5, respectively (Sec.1.7).

5.6 References 5.1 Greimann, L. et al., "Axisymmetric Buckling Analysis of Performance Beyond Design Basis for the AP600 Standard Plant Containment Vessel," NUREG-CR/6378, NRC, Washington, DC, Sept.1995.

5.2 Hibbitt, Karlsson and Sorensen, Inc., "ABAQUS User's Manual," Newark, CA, 1994.

5.3 Riks, E., "An Incremental Approach to the Solution of Snapping and Buckling Problems," Journal of Solids and Structures, Vol.15, pp. 529-551,1978.

5.4 Boresi, A.P. and Sidebottom, O.M., " Advanced Mechanics of Materials," Forth Edition, John Wiley, New York,1985.

5.5 Regulatory Guide 1.57, " Design Limits and Loading Combinations for Metal Primary Reactor Containment Systems Components," NRC, Washington, DC, June 1973.

31

I l

5.6 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code 1 Case N-284, "Metai Containment Shell Buckling Design Methods," Supplement

1. 2 to Nuclear Code Case Book,1980. t 5.7 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code >

Section NE3222.2,1989.

l l

I l

\

l l I i

4 I

, l l

I 32

6. BUCKLING OF AP600 DUE TO ASYMMETRIC TEMPERATURE l 6.1 Introduction At Level A Service Limit, the AP600 steel containment is subjected to the design basis accident load combination, DBAl (See Sec.1.2), which consists of the containment self weight, crane load, an internal pressure of 45 psig and a nominal temperature of 2807.

The ambient temperature is assumed to be 70T. For the purposes of analysis, three temperature cases were defined (Sec. 2.3.6 of Ref. 6.1). Case 1 assumes that the containment is subjected to a uniform temperature of 2807. In Case 2, the containment is cooled by the emergency cooling system. The resulting temperature field is assumed to vary in the hoop direction from 200T at the wet strips to 2807 at the dry strips. Case 3 is similar to Case 2 except for, a temperature gradient through the thickness of 30T in the dry region of the cylindrical portion. The circumferential variation of temperature produces compressive meridional stress resultants in the dry region which may cause the  ;

containment to buckle [6.2].  :

In Ref. [6.1], an axisymmetric model was used to determine the buckling factor of safety for Case 1 using the BOSOR5 finite difference program. The minimum factor of safety, A, was determined by conducting an in. perfection sensitivity analysis on the wave length parameter, K (Sec. 3.1 of Ref. 6.1). The value of A for DBA1 was 3.10. Failure was due to gross tensile yielding of the cylinder. The upper and lower domes were also isolated ,

from the rest of the containment and were analyzed by BOSOR5 to determine the factor of safety for each separately. The analyses yielded factors of safety of 3.52 and 4.90 for the upper and lower domes respectively. This indicates that tensile yielding of the cylinder is the controlling failure mode for DBAl for the Case 1 distribution. The buckling factor of safety for Case 2 could not be solved using BOSOR5 because of the stress gradients in the hoop direction, i.e., the loading is not axisymmetric. In addition, utilizing the worst meridian approach to solve this problem will lead to conservative results.

In this chapter, a fmite element model was developed to determine the buckling factor of safety for the Case 2 loading condition. The analysis was conducted using version 5.3 of ,

the finite element program, ABAQUS [6.3] The finite element model was verified by solving Case 1 and comparing the results with the BOSOR5 analysis results. A sensitivity study was conducted to establish the optimum configuration that will produce the minimum factor of safety. Three parameters were considered in the sensitivity study:

imperfection shape, number of strips and width of dry and wet strip. The controlling configuration was utilized to determine the minimum buckling factor of safety for Case 2.

Stress analysis for Imad Cases 2 and 3 (Sec. 2.3.6 of Ref. 6.1) demonstrated that the l stresses of Load Case 3 were bounded by those of Case 2.

l 33 i

f 6.2 hvelonment of the Finite Element Model This section describes the main features of the three-dimensional model used in the striping buckling analysis. The model parameters, such as nodal coordinates, elements connectivity and loading configurations, were computed by a preprocessor generated by the authors. The input to the preprocessor is the sinusoidal imperfection wave length parameters in the hoop and axial direction, number of strips and width of the dry strip.

The output of the preprocessoris the input file for ABAQUS analysis.

6.2.1 Model Parameters The finite element model was constructed using the quadratic eight noded S8R5 shell element in ABAQUS. In some areas, the triangular six noded STRI65 shell element was used. Three noded B32 beam elements were utilized to model the ring stiffeners. Only a wedge of the containment was modeled such tha: it subtends an angle 0 at the center (see Fig. 6.1). The two vertical edges of the model a e located at the middle of a dry and wet strip, respectively. The number of striping waves around the circumference, n, is determined by selecting the appropriate value of 0 according to the following relation:

n=E (6.1) 0 Mass densities listed in Table 2.1 of Ref. [6.1], were used in the finite element model.

The masses of all the penetrations and attachments were smeared around the circumference at the appropriate elevation. The total mass was checked with that of the axisymmetric model. Symmetry boundary conditions were applied on the two vertical edges since it was demonstrated that the symmetric buckling mode yields lower buckling factors of safety [6.4]. The nodes at the base of the containment model were fixed.

6.2.1.1 Upper Ellipsoidal Head The upper ellipsoidal head was divided into two regions as illustrated in Fig. 6.1. Region 1 extends vertically from the top down to Elev. 248'-7". A coarse mesh was used in that region because no stress concentrations were expected and the temperature was assumed to be uniform. Region 2 encompasses the remaining part of the ellipsoidal head. It was modeled with a finer mesh to capture high stress gradients anticipated in that region. A horizontal transition zone was created to achieve the transition between the two regions.

The finite element model of the upper ellipsoidal head is depicted in Fig. 6.2. Table 6.1 summarizes the elements sizes in the hoop and meridional directions and their respective aspect ratios.

6.2.1.2 Cylindrical Containment j

Five regions were defm' ed in the cylindrical containment as illustrated in Fig. 6.1. The regions wem numbered in an ascending order starting from Region 3 at Elev. 218'-8.5" 34 E_____________ . _ . _ _ _ _ _

l l

down to Region 7 at Elev.104'-1.5" The cylindrical containment is of primary interest in the three-dimensional model because it is a potential area for buckling due to nommiform temperature loading. Consequently, a fine mesh was used in all five regions. The elements size in the meridional direction did not exceed 0.706 [6.5]. Table 6.1 lists l the clements sizes and their respective aspect ratios in the five regions of the cylinder.

6.2.1.3 Lower Ellipsoidal Head The cylinder containment ends at the bottom with an ellipsoidal shell which is embedded in a concrete base. Although most of the ellipsoid is embedded in the concrete, a small portion of the ellipsoidal shell extends above the concrete base and high stress gradients may be expected in this area. Therefore, this region (designated as Region 8 in Table 6.1) was modeled with.the elements of an approximate size of 0.356 in the meridional and hoop directions.

6.2.1.4 StifTeners The horizontal web plates of the ring stiffeners were modeled with eight S8R5 shell elements. The flanges of the stiffeners section and radial stiffeners were modeled with eight B32 beam elements. The modeling methodology of the upper and lower stiffeners is depicted in Figures 6.3 and 6.4, respectively.

6.2.1.5 Crane Girder The flanges and web plate of the crane girder were modeled with S8R5 shell elements as shown in Fig. 6.5. Holes in the inside vertical plate of the crane girder were neglected.

Radial stiffeners were modeled using two beam elements B21 across the diagonal of the crane girder. These were used to substitute for the shear stiffness of the radial stiffeners (Sec. 2.2 of Ref. 6.1). Figure 6.6 illustrates the final assembled three-dimensional model. The model is composed of 893 elements,3710 nodes and a maximum wave front of258.

6.2.2 Asymmetric Temperature Loading Temperature is specified in ABAQUS as a predefined field at every node in the model

[6.4]. This option facilitated the description of the temperature distribution in a logical fashion. The top of the containment was assumed to be at a uniform temperature of 200"F down to an assumed Elevation of 248'-7" to resemble the cooling effect of water when it is sprayed from top to bottom. Staning from an Elev. 248'-7" and down to the lower stiffener, the temperature distribution around the circumference was varied from 280 F to 200 F as indicated by the dashed line in Fig. 6.7 for a dry strip width of 15 inches and a wet strip width of 34 inches. This temperature distribution was kept constant along each meridian. The temperature distribution utilized in the axisymmetric model (Sec. 2.3.6 of Ref 6.1) is also depicted in Fig. 6.7 for comparison. The difference between the two distributions was due to the limited number of Fourier series terms used 35

in specifying the temperature distribution in BOSOR4. A uniform temperature of'280 F was employed below Elev.132'-3". The temperature of the crane girder and the internal upper stiffener was assumed to be 280 F, whereas the external lower stiffener was assumed to b at 200 F due to the effect of accumulated water.

6.2.3 Pressure And Giavity Loading The internal pressme was applied in the form of a distributed surface load on the elements surface. Self weight of the containment was specified by defining the elements mass density (Table 2.1 of Ref. [6.1]) and the gravitational constant Figure 6.6 illustrates the classification of the elements in the model according to their mass density. Elements with the same mass density were plotted with an identical color. The crane load was applied in the form of concentrated forces at the nodes of the crane girder upper flange.

6.3 - Thermal Stress Analysis 6.3.1 Comparison With BOSOR4 Results The f' mite element model was verified by comparing thermal stress analysis results to I that obtained from BOSOR4 (Sec. 2.3.6 of Ref. 6.1). The same asymmetric temperature distribution used in BOSOR4 (solid line in Fig. 6.7) was employed for the purpose of comparison. Figures 6.8 and 6.9 illustrate the distribution of the meridional stress resultants, N i, and the hoop stress resultants, N2, along the meridians at the . middle of the  ;

dry strip and the wet strip, respectively. The distributions of Ni and N2 were in an 1 excellent agreement with BOSOR4 results (Sec. 2.3.6 of Ref. 6.1). The maximum I compressive Ni at the middle of the dry strip was'l1,700 lb/in. The maximum tensile Ni at the middle of the wet strip was 12,450 lb/in. No hoop stresses were generated in the  ;

cylinder and the top ellipsoidal head, except some local values at the stiffeners and the {

crane girder locations. A maximum compressive N2 of 56,500 lb/in occurred at the base compared to a value of 57,800 lb/in computed by BOSOR4.

6.3.2 Comparison With an Approximate Strength of Material Model l An approximate strength of materials type model was used for further verification of the numerical results. The theoretical model is composed of two steel strips of length, L, and thickness, t. The width of the two strips is b and b ,2 respectively, as shown in Fig. 6.10.

The two strips are rigidly connected at the top whereas the base is fixed. The modulus of 6

clasticity, E, is constant and equals 29x10 psi. The coefficient of thermal expansion, at, was assumed to be dependent on temperature as per the ASME code [6.6]. The axial i stiffness of the two strips, ki and k2 were computed as follows [6.7]: I k, = Eb,t L l (6.2) l k, = Eb,t L 1 36

The model was utilized to evaluate the stress resultants in each strip when one strip is heated to 280 F and the other to 200 F. The solution was based on the assumption of uniform axial stress'per strip (see Fig. 6.10 ), which means tl.at the rhear connection between the strips was neglected. From equilibrium of forces on any horizontal section, the total axial forces, Fi and F 2, acting on each strip (see Fig. 6.10) are equal and will be denoted as F. The compatibility relation is as follows [6.7]:

Ta:L-i Fk = Ta 2 2L+ F k

(6.3) i 2 Equation 6.3 was rearranged to obtain the value of the total force in each strip, F, as follows:

p, k,k (T2 i ai -T a2)L 2

(6.4)

(ki + k2 )

Using Eqs. 6.2 and 6.4, the value of F was computed to be 620 kips assuming that bi, b2 and t equals to 54.29,46.53 (i.e., n equals to 25) and 1.625 inches respectively, (see solid line in Fig. 6.7). The stress resultant in each strip was computed by dividing F by the respective strip width.' The compressive axial stress resultant, Ni , in the hot (dry) strip was 11,420 lb/in compared to 11,700 lb/in obtained from the three-dimensional element

{

solution. The tensile N iin the cool (wet) strip was 13,325 lb/in compared to 12,450 lb/in computed from the finite element model. In conclusion, the three-dimensional finite element results were in good agreement with both BOSOR and theoretical results.

6.4 Elastic Thermal Buckline Analysis I

6.4.1 Asymmetric Temperature Loading The elastic buckling factor of safety for striping was determined by solving an eigen value problem [6.8] using the perfect containment configuration and assuming an clastic material behavior. The temperature distribution (Sec. 6.2.2) across the circumference was selected such that the width of the dry and wet strips was 30 and 68 inches respectively, (dashed line in Fig. 6.7). This corresponds to n equals to 50 waves. Since buckling doe to striping is the primary objective of this analysis, the base was_ released in the radial direction to avoid buckling due to the high compressive hoop stresses near the base (see Figs. 6.8 and 6.9), as documented in Ref. 6.1. The maximum compressive stress in the dry region was 9.53 ksi. The clastic buckling factor of safety was 11.21. Buckling took place at the cylindrical portion as illustrated in Fig. 6.11. The wave length of the buckle was approximately equal to 1.685 in the axial direction and 2.756 in the circumferential direction.

37

(

6.4.2 Assessment of the Worst Meridian Approach The asymmetric temperature loading causes the meridional stress resultants to vary from compression in the dry region to tension in the wet region (Figs. 6.8 and 6.9). Such ,

circumferentially varying stress fields would make the use of the worst meridian approach l too conservative (Sec. 3.1 of Ref. 6.1). The worst meridian approach was evaluated by 1 computing the elastic buckling factor of safety due to a uniform compressive stress equals to the maximum compressive Ni induced by striping. The same model described in Sec.

6.4.1 was used except that temperature loading was replaced by a compressive axial stress of 9.53 ksi applied at Elev. 203'in the form of nodal forces in the meridional direction.

The buckling factor of safety was 6.79 compared to 11.21 for the striping case i summarized in Sec. 6.4.1. The buckled shape is illustrated in Fig. 6.12. The wave length of the buckle was 9.48Ei in the axial direction and 7.68 El in the circumferential I direction. The buckled shape was in good agreement with the classical solution [6.8].

This confirms that, in this case, steep circumferential stress gradients increases the buckling load by a factor of almost two, so that the worst meridian approach would yield  !

conservative results.

6.5 Inelastic Thermal Buckil== Analysis This section reviews the results of the inelastic bucking analysis of AP600 containment with geometric imperfections and when subjected to asymmetric temperature loading.

The buckling factor of ' safety,1, is dependent on three parameters: geometric imperfections, percentage of dry strip width per striping wave, d, and the number of striping waves around the circumference, n. The effect of each parameter on 1 was investigated to establish the worst configuration that leads to the least value of A.

6.5.1 Axisymmetric imperfections An axisymmetric sinusoidal imperfection [6.5] was used to perform an imperfection sensitivity analysis. The value of 1 was first determined for the inelastic perfect case to 4 be 8.98. For the imperfect case, the imperfection wave length parameter, K, (Sec. 3.1 of  ;

Ref. 6.1) was varied from 3.0 to 20.0. The loads were increased to a load multiple of fifteen and no buckling occurred. Hence, an axisymmetric imperfection prevented the occurrence of an instability due to striping. This phenomena will be explained using a modification to the theoretical model of Sec. 6.3.2.

When an imperfection is applied to the model of Fig. 6.10, the axial stiffness of the two strips will vary with F due to the beam-column effect [6.11]. The axial stiffness, ki, of the dry compressive strip decreases by the factor 4 i, whereas the axial stiffness, k2, of the wet tensile strip increases by the factor, A 2[6.11]:

l 38

i Ai = (1 F)

Fra A2 = (1 F) i Fr2 where g 2 EN 3 (6.5)

Fri =

L2 1

i

-j s'Eb,t*

Ft2 =

L*

Applying the factors A i and A 2 in Eq. 6.4, the following relation is obtained for the resultant axial force, F, in each strip: '

y , A, A,ki k:(7iai -T8 2 2 )L (6.6)

(Aiki + A2 k2 )

i Neglecting the variation of c1 with temperature, Eq. 6.6 is a quadratic equation in F and temperature difference, (T i - T2), designated as AT. Using the same geometric and physical configurations of the model as in Sec. 6.3.2, the stress resultants, Ni, in the dry strip were plotted versus AT in Fig. 6.13. Equation 6.4 was utilized to compute N i for the perfect case for comparison. In the perfect case, Ni is linearly proportional to A T.

Hence, membrane strain energy increases with AT to the point where it converts to bending strain energy at buckling when F reaches the elastic buckling. load, Fei, of 92,504 lb/in [6.12]. In the imperfect case, the increase in A T reduces the axial stiffness of the dry strip and increases that of the wet strip. This behavior limits the compressive stress resultants, as shown in Fig. 6.13, such that they are not sufficient to cause buckling.

The same behavior was observed in the finite element solution. Figure 6.14 illustrates the variation of the compressive Ni in the diy strip with increasing load factor. In the perfect case, buckling took place when Ni reached -82,628 lb/in at a load factor of 8.98. In the imperfect case was limited to -69,894 lb/in at load factor 12.6, and no buckling took place.

, 6.5.2 Three-DimensionalImperfections Geometric imperfections in the form of sinusoidal waves were also applied in the circumferential direction. The imperfection amplitude was restricted to the ASME l maximum limit of 0.8125 inches. No imperfections were applied in the axial direction as ,

l it was revealed (Sec. 6.5.1) that their presence limits buckling due to striping. The '

number of hoop imperfection waves, m, per striping wave was varied from 0 to 3, and the respective buckling factor of safety was computed. The analysis indicated that the buckling factor of safety increases with m, as shown in Fig. 6.15. Hoop imperfections, i.e., vertical corrugations, increase the moment of inertia about the axis of bending in the meridional direction and, thus, increase the buckling resistance.

39

l 6.5.3 Dry Strip Width The effect of the dry strip width percentage from the striping wave width, d, on A was 3 investigated. Three values of d were considered in the analysis: 20%,30% and 50%. The  !

assumed temperature distributions for each value of d are illustrated in Fig. 6.16. The  !

30% width corresponds to Fig. 6.7 and the results have been presented previously. When d was reduced to 20%, the maximum temperature at the middle of the hot strip was decreased to an assumed temperature of 260 F, to account for temperature diffusion through the hot strip (see Fig. 6.16). The value of A computed for d equals 20% was 12.2 compared to 8.98 corresponding to d equals 30%. When d was increased to 50%, the  ;

meridional stress resultants were reduced such that buckling did not take place at load  ;

factors below 15. The above discussion indicates that the worst condition corresponds to I d equals 30%. This temperature distribution was assumed in the remainder of this work.

A better approximation of the temperature distribution can be established by conducting i heat transfer analysis, which is beyond the scope of this work.  !

l 6.5.4 Number of Strips The effect of the number of strips around the circumference, n (Eq. 6.1), on the buckling safety factor was investigated. The value of n was changed [6.9] from 25 to 100 waves and the corresponding A was computed. The value of n was assumed to be in that range of experimental observations [6.9] Results are illustrated in Fig. 6.17. The compressive stress resultant produced due to striping is independent of n. However, reducing n decreases the Ni gradient in the hoop direction which leads to a lower value of A. The minimum value of A was 7.16 corresponding to n equals to 25.

6.5.5 Conclusions Tne previous discussion showed that the perfect configuration is the worst imperfection condition for this asymmetric temperature loading case only. The minimum buckling factor of safety was 7.16 corresponding to d equals 30% and n equals 25 waves, i.e., a l dry strip width of 60 in, and a wet strip width of 136 in. l 6.6 Load Combination DBA1 6.6.1 Case 1, Uniform Temperature The finite element model was utilized to determine the buckling factor of safety for DBAl load combination with uniform temperature of 280 F [6.1]. The model described l in Sec. 6.5 was used, except that the base was fixed. A sinusoidal axisymmetric imperfection was applied with imperfection wave length parameter, K, equals to 4.0 (Sec.

3.3 of Ref. 6.1). The containment collapsed axisymmetrically, due to tensile yielding of the cylinder, at a load multiplier of 3.02 compared to 3.10 obtained by BOSOR5. This confirms the validity of the numerical results. Figure 6.18 illustrates the deformed shape of the containment.

40

I l

l 6.6.2 Case 2, Striping Condition The 'relastic buckling analysis of Sec. 6.6.1 was repeated except with the uniform temg.ature field replaced by the asymmetric temperature distribution (Sec. 6.2.2).-

Twenty-five striping waves, n, were specified around the circumference with a dry strip width percentage, d, of 30%. The perfect containment was analyzed since Sec. 6.5.5 concluded this to be the controlling case. The factor of safety was 3.03 conesponding to tensile yielding of the cylinder. Figure 6.19 illustrates the deformed shape of the containment at collapse. Second, an axisymmetric imperfection with k equals to 4.0 was applied. The containment collapsed at load multiplier of 3.05 due to tensile yielding of the cylinder. The deformed shape is depicted in Fig 6.20. As for the Case 1 uniform temperature tensile stress resultants caused by the intemal pres ~ dominated the stress for the perfect and: imperfect case. Essentially identical factors w safety were obtained for the perfect and imperfect cases, indicating that the gross yielding mode is insensitive to geometric imperfections.

l 6.7 Summary The bucking analysis conducted in this chapter revealed that the asymmetric temperature loading is immaterial and will not govern. For the temperature striping loading, the lowest factor of safety as calculated to be 7.16. Tensile stresses induced by the pressure component of the DBAl loading would decrease the compressive stresses associated with striping and increase this factor of safety.

The controlling factor of safety for the DBAl load combination was 3.03 for the uniform temperature case for the three dimen;ional model and 3.10 from the BOSOR model.

These factors were controlled by tensile yielding of the cylinder due to internal pressure.

The tensile yielding mode was independent of geometric imperfections.

6.8 References 6.1. Greimann, et. al., "Axisymmetric Buckling Analysis for the AP600 Standard Plant Containment Vessel," NUREG-CR/6378, NRC, Washington, DC, Sept.1995.

6.2 Bushnell, D., and Smith, S., " Stress and Buckling of Nonuniformly Heated Cylindrical and Conical Shells," AIAA Journal,9(12), pp. 2314-2321, Dec.1971.

. 6.3 Hibbitt, Karslon and Sorensen, Inc., "ABAQUS User's Manual, Version 5.3,"

! Newark, CA 1994.

6.4 Hoff, N.J., Chao, C. C., and Madsen, W. A., " Buckling of a Thin-Walled Circular Cylindrical Shell Heated Along An Axial Strip," Journal of. Applied Mechanics, j pp. 253-258, June 1968. )

41

6.5 Greimann, L. et. al., " Buckling Evaluation of System 80+* Containment,"

NUREG/CR-6161, Washington, DC, June 1994.

6.6 "American Society of Mechanical Engineering, Boiler and Pressure Vessel Code Case N- 284," Supplement #2 to Nuclear Code Book,1980.

6.7 Boresi, A. P., and Sidebottom, O. M., " Advanced Mechanics of Materials," John Wiley and Sons, Fourth Edition,1984.

6.8 ' Cook, R. D., Mkus, D. S. and Plesha, M. E., " Concepts and Applications of Finite Element Analysis," Johnley and Sons, Third Edition,1989.

6.9 Timoshenko, S. P., and Gere, J. M., " Theory of Elastic Stability," McGraw Hill, 1982.

6.10 Fanto, S. V., and Piplica, E.J., "PCS Water Distribution Test Phase II Test Data 1 Report," Document ID: WCAP-13296, April 1993.

6.11 Salmon, C.G., and Johnson, J. E., " Steel Structures, Design and Behavior," Harper Collins, Third Edition,1990.

6.12 Bushnell, D., " Buckling of Shells- Pitfall For Designers," AIAA Journal,19(9),

pp. I183-1226, Sept.1981.

l l

42 i

7.

SUMMARY

AND CONCLUSIONS 7.1 Summary In a companion repon an axisymmetric model was utilized to review the design of the Westinghouse AP600 steel containment. The static and buciding analysis were conducted using the finite difference software BOSOR. Stresses resulting from individual load cases such as: dead load, temperature, wind and tornado, internal and  !

external pressure and seismic were computed using the stress analysis option in f

BOSOR4. The stresses from individual loads and seismic event were combined and j compared to allowable values as prescribed by the SRP Sec. 3.8.2 penaining to Design '

Conditiens and Service Limits A, C, and D classified by the ASME Code. This comparison indicated that the containment design was satisfactory for all Service Limits.

Maximum compressive meridional stresses were generated in the containment at Level Service Limit C at the load combination of the Design Basis Accident 4, DBA4. The buciding analysis was conducted by the BOSOR5 program. The minimum factor of safety was 2.02 associated with the DBA4 at 12 vel Service Limit C. Such analysis indicated that the containment design is satisfactory except for the requirements of the ASME code Sec. NE3222.1. The axisymmetric analysis was conservative.

The objective of this work is to investigate the effect of concentrated inertia forces associated with the equipment hatches and personnel air locks on the buckling strength of AP600 containment when subjected to the DBA4 load combination at Level Service Limit C. Such an analysis required the use of a three-dimensional numerical model. For l

that purpose, the finite element program ABAQUS was utilized. The three-dimensional i model was also used to determine the factor of safety against buckling at level Service Limit A when the containment is subjected to asymmetric temperature loading due to emergency cooling at the DBA1 loading condition. Such an asymmetric loading could not be analyzed by the BOSOR5 program which solves axisymmetric models subjected to ,

axisymmetric loads only. '

The elastic and inelastic buckling procedures in ABAQUS were verified. The elastic buckling analysis which involves the solution of an eigenvalue problem was verified by computing the buckling modes of an axially compressed perfect cylindrical shell. Results were in good agreement with those obtained using the closed form theoretical solutions.

l The inelastic buckling analysis in ABAQUS that incorporates material and geometric nonlinearities was previously verified by the authors of this report by solving two problems: uniformly loaded circular arch and axially compressed imperfect cylinder.

Results were in good agreement with those obtained by the BOSOR5 program. Such an analysis was utilized to conduct a mesh sensitivity study on an axially compressed

' imperfect cylinder to determine the size of elements that yields accurate results and save run-time and computational space. The h-refinement procedure was implemented in which the size of elements was progressively halved in the hoop and axial directions.

The finite element solution was compared to BOSOR5 results and the Code Case N-284 until a converged solution was obtained. The analysk indicated the use of a 0.58by 1.0 43

(

Esize elements in high stress concentration regions and a 1.0 Eby 2.0 6 size elements in the mmainder of the containment three-dimensional model. The smaller dimension of the elements was in the meridional direction. Results of the mesh sensitivity study also revealed that the solution accuracy was not affected when the aspect ratio of the elements reached two.

A preprocessor written in PASCAL was developed to compute nodal coordinates, element connectivity and material properties. The modeling procedure was accomplished by descritizing each part of the containment separately and then assembling all meshes to form the final complete model. The region of the cylindrical portion surrounding penetrations was modeled with a fine mesh, whereas the remainder of the cylinder was modeled with a coarser mesh as per the mesh sensitivity study. The complex ama around each penetration was modeled with the automatic solid modeling in the ANSYS finite element program. The model of each of the upper and lower domes, ring stiffeners and  ;

crane girder and bridge was assembled with the cylinder model. The entire model of the I containment was checked by conducting a linear static analysis of the containment with I internal pressum. . An inelastic buckling analysis. of the penetrations area was also conducted to verify the model in that area.

The developed three-dimensional model was utilized to determine the maximum responses due to SSE spectra using the response spectrum analysis. Two hundred and i thirty-seven modes were extracted by the subspace iteration technique with shifts to ensure that a sufficient modal effective mass was obtained in the X, Y and Z directions.

Model responses together with high-frequency modal responses, computed as per Sec.

3.7.2 of the SRP, were combined by the Ten Percent rule to account for modal coupling of closely spaced modes. Contour plots of the maximum meridional stmss resultants, Ni-, were utilized to identify potential buckling regions. Two regions were identified:

Region I was at the base below the upper equipment hatch and Region 2 was composed {

of the reinforced shell area around the four penetrations. Two sets of equivalent static l loads were developed to generate a stress field that bounds the maximum Ni stress resultants in each region. The generated stress field was compressive in the meridional direction in order to cause the worst buckling condition. Such stress fields were compared to the respective N im distribution in the potential buckling regions to validate the equivalent static loads. I The inelastic buckling analysis was conducted using the Riks method in ABAQUS to l determine the containment factor of safety against buckling for DBA4 of Level Service l l

Limit C. The loading configuration consists of an extemal pressure of 3 psig, uniform l rise in temperature of 50"F, gravity loads and the equivalent static loads of SSE spectra maximum effect. All loads wem progressively increased until buckling. Two load cases

- were applied to compute the buckling factor of safety in the' two potential buckling  :

regions. The minimum buckling factor of safety associated with Region I was 2.52. The first buckle was determined to occur in Region 1 by monitoring the growth of the Von Mises stresses and radial displacement- in the containment during loading. A buckling factor of safety of 2.62 was determined for Region 2. Buckling took place in the thin 44 L--_------_- - - - - - - - - - - - - - - - - - - - - ---- - - - - - - - ------ ------- - ------- ----- --------- ------ ------ - - -

I shell remote from penetrations. Hence, it was concluded that the containment was adequately reinforced at the penetrations area because the factor of safety against local buckling in the reinforced shell must be greater than 2.62. Such analysis indicated that the design of the containment satisfies the requirements of the Reg. Guide 1.57, ASME Code Case N-284 and ASME Section NE3222.1.

A three-dimensional model of a strip of the containment was utilized to determine the buckling factor of safety for loading condition DBAl at Level Service Limit A. The loading configuration consists of gravity loads, internal pressure of 45 psi and two possible cases of temperature distribution: Case 1, uniform temperature of 280*F and Case 2, at which the containment is cooled by the emergency cooling system. These Cases were analyzed by ABAQUS because nonaxisymmetric loads can not be modeled in BOSOR5. The temperature field was assumed to vary from 200 F at the wet strip to 280 F at the dry strip. Such temperature distribution produces a compressive meridional stress resultant in the dry strip which may cause the containment to buckle. The finite element model was verified by comparing thermal stresses due to stripping to those obtained from BOSOR4 numerical solution and a simple strength of material model.

Results were consistent. The worst meridian approach was shown to yield conservative results when the elastic buckling of the containment due to stripping was investigated.

{

This confirmed the use of the three-dimensional analysis to obtain realistic results. The i effect of geometric imperfections, dry strip width and number of strips on the inelastic j buckling strength of the containment was investigated. The sensitivity of the buckling- l load factor to each factor was investigated. The worst configuration was the perfect case with 25 strips around the circumference and 30% of the strip width being dry. For such a configuration, the buckling factor of safety was 7.16. This means that buckling due to stripping does not govem the design.

The buckling factor of safety for load combination DBAl of Level Service Limit A was

- computed. For Case 1, the containment collapsed at a load factor of 3.02 compared to 3.10 obtained by BOSOR5. The collapse was due to general yielding of the cylindrical portion. For Case 2, the worst configuration obtained for stripping was utilized, the' containment collapsed at a load multiplier of 3.03 due to general yielding of the cylindrical portion. The safety factor was insignificantly increased when axisymmetric i- imperfections were utilized.

7.2 Conclusions On the basis of the analyses performed herein, the following can be concluded:

(1)- The minimura buckling factor of safety was 2.54 for load combination DBA4 at Level Service Limit C which produces the maximum compressive meridional stresses in the containment. The calculated factor of safety satisfies Reg. Guide 1.57, ASME Code Case N-284 and ASME Sec. NE3222.1 minimum requirements. Hence, the design of the containment is satisfactory for buckling strength requirements.

I 45 i

(2) The containment is adequately reinforced around all penetrations. local buckling due to concentrated loads at penetration will take place at a load multiplier greater than 2.62 for load combination DBA4 at 12 vel Service Limit C.

(3) Buckling due to asymmetric temperature distribution due to emergency containment cooling does not govern the design of the AP600 containment. The minimum buckling factor of safety was 7.16.' Tensile stresses caused by the internal pressure of load combination DBA1 exceed compressive meridional stresses due to stripping and cause axisymmetric collapse of the containment due tensile yielding of the cylindrical portion. The minimum factor of safety for that load combination was 3.02 for the case of uniform temperature. Asymmetric temperature distribution and geometric imperfection had an insignificant effect on the factor of safety.

7.3 Recommendations In o:Jer to experience a better understanding of the containment performance, time integration analysis that incorporates material and geometric nonlinearities is recommended. In such analysis, the interaction of the inertia loads due to earthquake and buckling with other loads is included to obtain a better estimation of the containment response in the time domain.

46

APPENDIX A COMPARISON BETWEEN ANSYS, ABAQUS AND STAGS A.1 Description of the Finite Element Programs A.1.1 ANSYS The ANSYS program is a general-purpose program that can be used for almost any type of finite element analysis in virtually any industry. The latest release of the program is Revision 5.0 which includes major changes and several new features and capabilities.

The analysis procedures available on ANSYS include static analysis, buckling analysis, modal analysis (c'o mputing the response of the structure to dynamic loads by modal superposition), nonlinear analysis (including the effect of both geometrical and material nonlinearity), and dynamic analysis (determining the response of the structure to dynamic loads by direct integration). Nonlinear buckling analysis in ANSYS is a load-controlled solution based on Newton-Raphson technique in which the fundamental path only can be determined up to the critical point.

Flat and curved shell elements are available in ANSYS. Curved shell elements can take on a general second order polynomial shape and are therefore well suited to model doubly curved surfaces. Triangular and quadratic shell elements are available. Quadratic shell elements can be four-noded, eight-noded, or nine-noded with six degrees of freedom at j each node.  !

ANSYS contains a general substmeturing/superelement capability which may be used to reduce the execution time of the analysis. This feature is particularly useful in case of structures where a certain group of elements is repeated several times in the same pattern.

The superelements can be saved and reused in subsequent analyses.

The preprocessor defines geometric and material properties, loading configuration and kinematic conditions of the model. ANSYS has a powerful preprocessor which can be easily used in a batch mode or an interactive mode. The postprocessor illustrates output results after the solution such as displacements, reaction forces, stresses and strains.

ANSYS has high postprocessing capabilities for printing and plotting analysis results.

In addition, ANSYS Revision 5.0 has solid modeling and automatic mesh generation capabilities. Solid Modeling is done in ANSYS through a number of basic operations, where the model geometry is input and the resulting areas are meshed automatically by the program. The model geometry is input using a number of basic units (keypoints,

lines, areas, volumes), and boolean operations are used to obtain more complex shapes.

i Examples of available boolean operations are addition, subtraction, and intersection, of lines, areas, and volumes.

l 47

(.

i

. A.1.2 'ABAQUS i ABAQUS is a general-purpose finite element analysis pmgram. The program is highly oriented towards the nonlinear analysis of structures. The analysis procedures available _ ,

in ABAQUS include static analysis, buckling analysis, modal analysis, and dynamic j analysis. Nonlinear buckling analysis employs a nonlinear static analysis with gradually  ;

incmasing loads to the point of instability of the structure. ABAQUS utilizes ~ the modified Riks method, in whichia constraint equation is added to the equilibrium

)

equations to select the length of the incremental load step in the load-deflection ' space.  ;

The applied load level becomes an additional variable. This technique allows the prediction of post-buckling behavior, improves convergence, and detects locations of critical points along the fundamental path.

l Similar to ANSYS, flat 'and curved shell elements are available in ABAQUS. Thme types of quadratic shell elements exist: four-'n oded, eight-noded, or nine-noded with six degrees of freedom at each node. In addition, ABAQUS has shell elements (S8RS) in which the reduced integration procedure is utilized in the element formulation in order to reduce the number of degrees of freedom at each node from 6 to 5.; In those elements, in-

! plane rotational stiffness is eliminated and solution time is reduced. They provide

accuracy identical to shell elements which use six degrees of freedom per node.

The creation of substructures /superelements is possible in ABAQUS.- Superelements can

)

be created and saved to be used in subsequent analysis runs. This option can be very l effective in reducing the run time of certain problems. The preprocessor of ABAQUS is

{

limited only to batch mode usage, while the postprocessor has very good capabilities for  !

printing and plotting analysis msults. i The external loading configuration to which the response is sought can be defined as  ;

either, boundary conditions (such as support displacements and/or rotations), distributed  ;

l.

j loads (such as line loads on one dimensional elements or pressure on two dimensional j elements) or point loads (such as temperature and concentrated loads). An internal stress  ;

I field can not be directly applied as an initial condition for the structure except through a set of external loads that can be accepted by the program (i.e., either boundary conditions, distributed loads or point loads).

A.I.3 STAGS 2

STAGS is a finite element code for general-purpose nonlinear analysis of stiffened, thin- 1 shell structures of arbitrary shape and complexity. Its capabilities include stress, stability, i vibration, and transient analyses, with both material and geometric nonlinearities

.. permissible for all analysis types. STAGS is primarily intended for the analysis of shell structums. '

STAGS is routinely used for pre- and post-test verification of complex systems, especially those sensitive to initial geometric imperfections, which can be defined in i

48 L__ . - - - - - - - - - - - - - _ - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - -

STAGS with great ease and flexibility. The STAGS update of mid-1992 contains advanced nonlinear solution strategies, including equivalence transformation methods to obtain solutions during mode-jumping behavior that may occur in bifurcation and post-buckling response.

STAGS has been used extensively for the analysis of shells since the late 1960's. STAGS has played a major role in the design of all U.S. submarines over the past_20 years. The program was used for the buckling analysis of a space shuttle and for the analysis of nuclear reactor containment vessels.

The solution algorithms of STAGS range from simple load control to the advanced Riks-Crisfield path parameter control that enables traversal of limit points into the post-buckling regime, to equivalence transformation methods to obtain solutions during mode-jumping behavior that may occur in bifurcation and post-buckling response. The program allows the use of two load systems with different histories at the same time. Flexible restart procedures permit switching from one strategy to another during an analysis.

STAGS allows the use of substructures. The program output is also substructure oriented to allow the user to focus on that part of the model that is of greatest interest. Geometric imperfections car. be defined in a variety of ways, thereby permitting imperfection-sensitivity studies to be performed with minimal effon. Imperfections can be generated from linear combinations of previously-computed buckling modes, from a trigonometric expansion or from a user-written subroutine. A special random imperfection capability allows for randomly-generated amplitudes with specified standard deviation and mean values.

A.2. Comparison Between STAGS. ANSYS and ABAOUS STAGS has a number of advantages for use in the current study, which include a high level of credibility in the field of shell analysis, the ability to generate geometric imperfection in the model with minimal effort, and the ability to perform nonlinear analysis encompassing both the fundamental path and the post-buckling behavior of shells. However, use of STAGS requires the purchase (or licensing) of the program and about three months of training time for the research team, which is outside the range of both the funding and the time schedule of this project. Moreover, the pre- and post-processor capabilities of ANSYS and ABAQUS are far more developed than those of STAGS. ANSYS and ABAQUS are therefore recommended as two options for selection in this project.

A comparative study performed on both ANSYS and ABAQUS showed that for a very limited number oflarge problems analyzed using both programs, the time consumption of ABAQUS was about 50% that of ANSYS and the required storage was roughly about 70% that of ANSYS.

49

The shell elements available in ANSYS and ABAQUS are very similar. However, ABAQUS has additional shell elements where the in-plane rotational stiffness is eliminated leaving only five degrees of freedom at each node. This considerably reduces the analysis time and the required storage. ABAQUS also has the advantage of using the modified Riks method, which allows the prediction of the structural behavior both along the fundamental path and in the post-buckling stage.

The preprocessing and postprocessing capabilities of ANSYS are superior to those of ABAQUS. However, both programs have good pre- and post-processors.

A.3. Conclusion Based on the above-mentioned comparison, and because of the research team's extensive experience with the use of ABAQUS in previous projects, ABAQUS were used in this study for performing the finite element analysis of the AP600 containment. However, ANSYS was used as a preprocessor for ABAQUS for the development of some parts of the model. Areas of complicated geometry were modeled on ANSYS utilizing its solid modeling and automatic meshing capabilities, and were converted to ABAQUS format and linked to the rest of the model.

l l

l l

50

_ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ . _ _ _ . _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _____J

APPENDIX B MODE FREQUENCY ANALYSIS RESULTS Table B.1 Natural Frequencies of AP600 (

{

Mode Fnq(Hz) Mode Fnq(Hz) Mode Fml(Hz) Mode Fnq(Hz) Mode Fnq(Hz)

I 3.6859 33 8.6322 65 93921 97 J

11 3440 129 13 3030 2

l 5.9103 34 8 ]485 66 9.8587 98 11.5970 130 133310 I 3 6.0739 35 8.8262 67 9.8829 99 11.6130 131 133 360 4 63589 36 8.8694 68 9.9542 100 11.6430 132 133440 5 6.6476 37 8.9837 69 10.0010 101 11.6590 133 133890 6 6.8504 38 8.9872 70 10.0860 102 11.6740 134 13.6340 7 6.9049 39 9.0408 71 10.1250 103 113580 135 13.6880 8 6.9649 40 9.0613 72 102120 104 12.0990 136 13.7070 I 9 7.0101 41 9.0962 73 10 1530 105 12.1650 137 133140 i 10 1.0573 42 9.1759 74 103000 106 122400 138 13.8090 11 7.1680 43 9.2241 75 103460 107 121540 09 13.8400 12 7.1819 44 9.2695 76 103580 108 12 3440 140 13.8830 13 7.2806 45 93155 77 10.4110 109 123980 141 13.9400 14 13356 46 93866 78 103060 110 12.5410 142 13.9550 15 73482 47 9.4376 79 103930 lil 12.5740 143 13.9750 16 7.4406 48 9.4459 80 10.6240 l12 12 3830 144 13.9930 17 7.6303 49 9.4581 81 10.6250 113 123860 145 14.0210 18 73072 50 9.5169 82 10.6490 114 12.8930 146 14.0960 19 73664 51 9.5350 83 10.6570 115 12.9220 147 14.1020 20 7.8433 52 93458 84 10.6680 116 12.9500 148 14.1520 21 7. % 36 53 93585 85 101440 117 13.0590 149 14.1690 22 8.1170 54 93707 86 101720 118 13.0650 150 14.2340 23 8.1%3 55 93781 87 10.8330 119 13.1510 151 142420 24 8.2489 56 9.6158 88 10.9360 120 13.1660 152 14.2620 25 82741 57 9.6474 89 10.9450 121 13.1980 153 14 1880 26 83297 58 93083 90 10.9810 122 13.2250 154 14 3070 27 83395 59 9.7322 91 11.0490 123 13.2470 155 14 3290 28 83866 60 93408 92 11.0700 124 13.2810 156 14 3 350 l 29 841% 61 93544 93 11.1140 125 13 3100 157 14 3800 30 8 4937 62 93585 94 11.1970 126 13 3650 158 14.4030 31 8.5722 63 93687 95 l13000 127 13.4120 159 14.5520 32 83735 j

64 93822  % 113300 128 13.4390 160 14 3850 i l

l l 51 1 l

l

Table B.1 Natural Frequencies of AP600 ( Continued )

MWe Freq(th) MWe Freq(th) MWe freq(th) MWe Freq(th) MWe Freq(th) 161 14.5940 184 15.6110 207 16.7320 230 174490 253 19.0990 162 14.6230 185 15.6500 208 16.7590 231 17.6490 254 19.1410 163 14.6740 186 15.7270 209 16.7890 232 17.6650 255 19.1470 164 14.6780 186 15.7270 210 16.8M0 233 17.7210 256 19.1590 165 14.7090 186 15.7270 211 16.8720 234 17.7760 257 19.2970 166 14.7270 187 15.7390 212 16.9130 235 17.9080 258 193450 167 14.8300 190 15.9980 213 16.9250 236 17.9710 259 193620 168 14.8330 191 16.0400 214 16.% 30 237 18.0270 260 19.4810 169 14.8490 192 16.1370 215 17.0100 238 18.1270 261 19.4910 170 14.8920 193 16.1650 216 17.0800 239 18.1380 262 193130 171 15.0190 194 16.2200 217 17.0820 240 18.1630 263 193260 172 15.0720 195 16.2710 218 17.1810 241 18.1920 264 19.5550 173 15.1270 1% 163570 219 17.2000 242 18 3880 265 19.6820 174 15.1780 197 163760 220 17.2M0 243 18.4200 266 19.8490 175 15.2010 198 16 3960 221 17.2660 244 183080 267 19.9190 176 15.2290 199 16.4150 222 173050 245 183470 268 19.9740 177 153090 200 16.4570 223 173600 246 18.6530 269 20.0660 178 153770 201 16.4870 224 17.4540 247 18.6850 270 20.1490 179 153840 202 163170 225 17.4620 248 18.7730 271 20.1830 180 15 3960 203 16.5470 226 17.5170 249 18.8160 272 20.2170 181 15.5540 204 163860 227 17.5650 250 18.9500 2 73 20.2680 182 153740 205 16.6120 228 17.6030 251 19.0050 183 15.5960 206 16.6860 229 17.6150 252 19.0250 52

Table B.2 Modal Participation Factors in X, Y, and Z Directions Mode Participation Participation Y ParticipadonZ Mode PartidpationX Participation ParticiseZ i

l 2360k@ 1.4107ed0 3.1755e@ 65 3.2384e43 1.8967e42 4.9082e02 2 1.4137e@ 12300e41 -3.0969e42 66 1.2797c42 3.7765c42 3.1669e41 3 1.0137edo 1.562kd0 1.510k-02 67 7.0959ee -4.652k42 2.218k01 4 1.0466e-02 2.8297ed0 4.857k42 68 1.9 % 42 8.6190e@ 1.8571e-01 5 1.07kd0 2.2430edo 9.9469e4 69 8.4083e42 4.3513e42 7.4338e4 6 42337e42 1.859k41 3.7804e42 70 7.8741c44 4.0964e43 1.1495e 01 7 4.6439e@ 3.1878e42 -8.953k43 71 -1.279k42 3.9R42 -3.6874e41 8 1.0777c41 13874e01 3.2460e 02 72 1.6984e41 2.5987e 01 - 8.1594e41 9 2.1795e02 10995e42 1.0519e@ 73 2.6 h 42 2.0549e42 1.0905e41 10 1086te41 7.2817e 02 3.1242 74 3.9125e42 4.5439e42 4.0172e41 11 5.4445e41 1.2765e-01 6.2098e42 75 1.2574e42 -1484k42 39616e42 12 2.06Sle41 -4.5102e 03 2.4624e42 76 1.857k42 43288e02 3.6610e41 13 7.1580e42 2.9396e42 4.1208e-03 77 7.7860e 02 1.2086e41 3.6143e41 14 8.283k-02 43151e42 3.9069e42 78 2.0957e@ 2.9802e-02 9340le-03 15 1.8757e41 1.ll?2c-01 1.2672c42 79 9.80lle 02 23 M 42 il441c41 16 7.8975e42 4.2667e42 4.4378e44 80 3.4707e42 -9.5600ce 7.4449e 02 17 2.7395e 03 6.4447e42 1.0091e 02 81 2.5721e 02 7.8890e-02 1.2146e 01 18 18714e02 13052e-02 1.3341e42 82 il43k@ 4.0882e45 1.600k41 19 1.6489e41 2.4843e42 10066e 02 83 2.4772c 02 43248e43 62527e02 20 2.225k02 2.2292e 02 12 % 42 84 2.1991e-03 -11097e42 2.7031e-02 21 46065e42 1615e42 4.6556e@ 85 43633e42 4.4174e42 7.6256e-02 22 6.1410e 02 9.2162e42 2.931k 02 86 1.1754e-01 97 M 42 40144e 01 23 1.0 W 01 -4.1829e-02 2.6376e42  !? 1.6532c-02 13781e41 1.8376e 01 24 1.0449e 02 3.7720e4 4.0087e 04 88 2.9157e 02 1.4679e42 6.094k42 25 2825k42 4.4730e@ 1.684k-02 89 4.8539e 02 1.2024e 02 2.680le 01 26 2.1720e4 2.18M @ 3.8092e 04 90 1.0467e41 7.9709e-02 42M-01 27 1.6082e42 -93681e@ 46307e-03 91 -2.2460e 01 2.2389c41 49485e41 28 8.4749e42 -3.2612e4 7.4976e42 92 -2.241k41 13421c41 1.N65ed0 29 4.2749e42 93B4e4 41929e@ 93 1871k43 -4.9229e42 2.2212e41

) 30 13842e42 1.0527e 02 -9.3107e-03 94 9.4037c-02 9.9492e43 4.4995e41 31 1.1242 1.7673e4 73017c4 95 4.5521e-02 1.1780e-01 23579e-01 32 -9.0431e@ 1.8603e 03 8.5639e4  % 13006c-01 19711e.02 18870e-01 i

l 53 l

Table B.2 Modal Participation Factors in X, Y, and Z Directions ( Continued )

bde Participation Particip 6en Y Particip 6onZ Mode FarbeipationX r.% Y Participation 33 2.3064e42 -9.0543e@ 13565e-02 97 49469e42 1344k41 47695c41 34 6.9402e4 l#4k 03 22084e42 98 6.1911c4 47126c43 1.1919e42 35 -2.4 & 42 3.7962e42 -21575e42 99 6.10W4 4114e4 2.Ulk42 36 23908e4 44323e42 2.9675e42 100 10647e42 9.92 4 3.147e42 37 2.2484e43 31356ea 3.9544e03 101 1.8159e41 -ill81e42 62311e41 38 3.409k@ 93060e@ 1.2160e4 102 7.8585e4 3.6740e4 2.6393e42 39 6273ke 1.4Wle 02 -2.&7c45 103 1.5137c41 2.214k42 3.4889e41 40 633k44 -1343k42 1.1620e-02 104 2.3552e41 44959e42 7.7U6e41 41 5.6592ee 1.947k-02 43664e42 105 9.4578c 02 8.1320e@ 3.4101c41 42 43195e41 41327e-01 22h+00 106 -17489e41 1.0479e41 4050le+00 43 -53620e42 4445k-02 1.08W41 107 1.0611e-01 7.8289e 02 2.8079e 01 44 7.0943 -1.407k-02 1.8635e42 108 3.5121e41 4.1570e4 -1.1138 4 0

! 45

_ j 6.2070e43 7.0173e44 3.8149e42 109 44h 41 2.9659e42 all25e+w l 4 8.855k41 -9.8132e41 1.9433eA0 110 3.7806e41 -23544e41 13238eA0 j

47 12582e42 41831e03 46199e42 111 1.9080e41- 1.000e41 496k-01 I 48 8.2985e42 458De 01 1.96&+00 112 3.242k-02 l]I32e@ 1.017e41 49 -9.M21e6 -7.7457e42 2.20$le-01 113 7.5659c 02 63832e42 2.8 h 41 50 -5]IW41 53068e42 5.851k-02 114 13350c41 1.5345e41 5.0786c-01 51 5.80k41 41405e41 8.6795e41 115 2.0162e-01 2121k-01 12849e-01 52 1.457c42 4.27W-02 1.1 % 41 116 4.8921c4 2.6995e42 41392e42 1

i 53 6.16&B 4.2659e@ 6.8039e-02 117 -1.2253c-01 2]D6e42 -2.3881e41

. 54 -19277c-01 63079e42 46192c 02 118 -23211e42 -13247e-02 l.1740e-01 1 55 2]424e4! -9336k42 13414e41 119 2.6h-02 103h42 48287e42 56 47070e.02 5.ll60e-02 1.2770e42 120 2.475e-02 9.6684eG 1.9h02 i

57 1.4W 01 4.4 M 41 1.6300e+00 121 18421e-02 -7.6169e@ 11163e42 58 196k42 33655e42 1.779kB 122 1.1872e-01 -9.4067e-02 il515e-03 59 1364e-02 2.8674e41 23902e+00 123 4.5194e-(2 -12479c-02 1947k43 60 46972e44 18141c-03 -2.2120e41 124 4.0bO2 1.18W 02 2.658k41 j

6! 1.4311c42 1.5495e-02 1.20S8e 01 125 -9.0 h 02 2.5424e 02 -33664e41 62 1.4840e42 18M02 1.9386e41 126 -13144e 02 13220e-01 Alls 3e41 0 1.340e42 -1.8749c 02 1.2824e41 127 1.6867e 01 -7.6335e42 33447e41 64 -53U5ed 2.6605e4 15302e42 128 1.015k 01 4.1488e@ 6.2020e41 54

Table B.2 Medal Participation Factors in X, Y, and Z Directions ( Continued )

Mode Participation ParticipatiaY hrliciptiaZ Mode Par %+nX Participation ParticipatiaZ 129 1]464e-01 3.8114e42 11360e-02 161 5.1072e42 63517e42 43630e-01 130 14292c4 73522c8 2391k42 162 53 h 02 49136c42 -9.9642e 02 131 46761e43 -9.8296e44 2.67k42 163 3.2609e-02 3486te 02 8.8556e43 132 1.08W41 43970e-02 4.1752e42 164 4.8837e43 -it$12e43 1425k42 133 40005e42 1.1583e41 74124e41 165 13911c-02 1.176k41 1.0807e41 134 -9.8127e4 -24759e42 11498e41 166 43053e42 13600e41 6.2662e42 135 il E 02 2.9095e42 -33924e41 167 108W42 18959e42 48730e42 136 1.2418e41 3.54 & 02 1.1643e+00 168 1.8518e42 7.7541e.03 16467e42 137 1.5161e41 -44878e42 -1.544Sc+00 169 6.2437c42 -94697e02 7.4314e-02 138 13809e4 -44894e42 3.4291e41 170 1.ll24e41 40925e41 -9.8307e42 139 -4jl937e-03 2]640e0' 4.8418e42 171 14696e42 40970e42 6.5551e-02 140 1.0322e42 1.4446eG 4.45W42 172 1.8987e41 -3.0241e42 4.9374e-01 141 498Me-02 -7.4651e.02 13611e41 173 2.21W41 6.7202e 02 4.8669e-01 142 33040e42 -1.3409e42 6.172k41 174 23949e-01 41197e 01 33852e 01 143 1.0153e-02 3.5654e42 1.lh01 175 -2.6527e41 1.1358e-01 4.9547e 01 144 4073te-02 -63768e4 3.684k-01 176 10732e41 44247e 02 24195e 01 145 530k42 33046e42 4.570k41 177 5]312c41 13632e-03 -13210e+00 146 1.151k-02 1.36W 02 -34630e-01 178 162h41 47171e42 -5.4376e-01 147 8.4084e4 3.6398e42 -2.9430e41 179 43799e-01 4090le42 44954e+00 l

148 1.2321e42 1.4086e42 1.420le 01 180 45987e-01 7.6176e 02 1/053e+00 I 149 9.9578e43 3.4730e42 .l.2379e41 181 -3]63k-01 45 2 01 1.1352e+00 1 150 1.1602e 02 23046e42 53399e42 182 -1.3754e 01 8.5638e 02 2.6892e41 151 -1.6400e-02 11606e42 il10k-02 183 -1.3092e 0) 6.0389e 02 43020e41 152 12776e42 1.1828e42 3.0628c42 184 10745e41 1.2385e-01 -24195e 01 153 2.3100e-02 1.8085e41 13628c41 185 138 & 01 2.4921e-02 49436e 01 154 2.8th42 2.2508e-02 -1.0ll7e41 186 2.2303e-01 41642e-01 7.9751e41 155 2.1026ee 9].W 44 14121e@ 186 42.10k.01 dlN2e 01 7.9751e41 156 2.0038e-03 2 VS7e 0.1 -3.1048e-03 186 2.230k-01 41642e-01 19751c-01 157 6387k 02 -14.118e-01 49725e-01 187 1.8800e+02 1.5787e+01 1.8900e+02 158 All63e-01 23908e41 13987e41 190 23826e-01 2.4918e 02 -i9435e-01 159 1.0800e41 1.4806e 01 437lle-02 191 -2.2304c 01 3.lN5e41 19746c41 160 l.2711e-02 33180e42 2.1359e42 192 42559e-02 23932e-01 8.0724e-01 t

l 55

Table B.2 Modal Participation Factors in X, Y, and Z Directions ( Continued )

Mode PendetionX P.Gdp. tion Y ParticipationMode Par @tionX Participation Participation 193 4.50lk41 1.1623e42 13496cd0 225 -14577e41 17192e42 1.2216c41 194 1.8820e41 2.91N41 8.6648e41 226 3.6179c41 -4.386'e41 8.6812e01 195 -135W42 -15158e42 3.8765e-02 227 2.0740e41 7.356k 02 2.ll82e41 1% 9.997k42 9.6886e42 1.2320e41 228 2.1610c41 47144e+00 7.9206e 01 197 -9.1752e 03 4.4753e41 -6.5797c42 229 -14696e42 3.8886e41 -2.6976e42 198 1.9030e 01 1.6322c41 1.0880e40 230 1.5473e+00 14606e41 3.7450e41 199 1.2659c41 -7.4522e42 2.5838e41 231 -2.5975e41 3.090k41 -1231k41 200 13150e-01 18200e02 3.4137e41 232 33057e+00 18689e+00 63926e-01 201 18915e-02 19960e42 1.9661e41 233 -14018e42 4.2641e 02 13453e 01 202 1.369k41 3.80W 02 33071e01 234 3.037le+00 3.2659e 01 4.2626e42 203 7.5705e42 10752e42 13180e41 235 -7.4464e+00 3.0557e+00 2.1828e40 204 8.4153e 02 1.2597e-01 11042e-02 236 -4.0865e43 -1.2697e 01 4.0928e 02 205 1.411k-01 1]Dik-02 2.lD93e 01 237 19608e-02 19050e42 2.020ie41 206 21130e41 43516e42 -12581e41 238 1.0548e41 13987e02 1.8545e41 207 130 5 01 9.851k42 17104e01 239 2.5289e-03 1.2307e-02 -1))60e 01 208 2.9h03 1.9551e-02 1.157le41 240 23015e41 8.94E01 1.521k41 209 1.2046e+00 1.04W41 13408e41 241 27484e41 1.8 h 0! 3.0120e41 210 1.2951e41 6.4603e 02 63391e-02 242 1.9414e41 45705e41 ad507e42 211 239S4e+00 19759e+00 -2.1630e+00 243 4.8182e-01 6.835k41 3.665k 01 212 2.2616e42 2.2455e41 3.0417e41 244 4.210le41 7.9247e 01 -3.4895c41 213 73915e+00 1.4613e+00 4.2709e 01 245 -93080e41 8.4 % 01 23013c41 214 4.0051e41 1.1206e+00 l

2.1226e+00 246 1.0729e+00 -9365k41 -1.8119e42 215 1.4638e42 1]!60e 02 1.4463e-01 247 7.2612e41 14757e41 91319e42 216 1.0264e41 1.7305e42 1.2612e-01 248 -1.0h00 1.0Mk-01 -11163e41 217 1.8772c41 4.1876e-01 1.8772e 01 249 19078e-01 9.6486e41 1.ll72e-02 218 6.0264e 02 3.6809e42 1.8826e-01 250 1.9159e-03 7.0633e 01 -4.4607e.02 219 2.5831e41 2.5831e41 3.4140e41 251 4.7254e-01 9.0078e-01 -l.8857e42 220 -43703e+00 1.6660e+00 -1.0591e+00 252 6.4309e-02 4.5857e-01 1.1060c-02 221 23412e+00 2]340e41 4.2661e-01 253 9.9583e-04 -2.0553e41 1.6694e-02 222 -9.4712e+00 29016e+00 11624c+00 254 7.4781e-02 -2.4927e 02 2.8515e42 223 -1.IID5e41 .53556e41 1.0572cA0 255 15012e42 4.0192c42 7.0070c43 224 1.2629e-01 -12559c-02 1.2255e41 256 8.6991e-02 23677e02 2.0397e 02 56

Table B.2 Modal Panicipation Factors in X, Y, and Z Directions ( Continued )

Mode Partidpation X Partuip 6enY Partmpation Z 157 -1A97k 01 23511e41 5.8448c42 258 2.9787e 01 12847e41 -7.2210e42 259 1.1269e41 -13291c-01 2A565e02 260 2.0019e41 1.9976e41 4.2621c41 261 f

-5.2954e44 -5.7821e-02 -12409e41 262 3.8464e42 5.1555e-02 23959e41 263 2.7719e41 -4.2412e-01 7.9515e41 264 -5.5600e41 8.7105e42 -5.0656e41 t

265 9.8748c42 23442c41 1.2900e41 266 43380e42 2.1280e41 3A160e41 267 5.6499e42 1.1914e41 4.1496e-01 268 L.6537e41 8.7292e41 4399k41 269 -3.9825e41 3.9628e42 5.225 Seal 270 1361k.02 2A254e41 4.8935e41 271 3A869e-03 6A911e43 1.lll6e-01 I 272 23960e41 1.9294e41 2A582e41 273 4.4476e44 -4.574k-04 1.1927e-02 I

4 1

l 57 o____--_-______--______---_______- - . . _ - - _ _ - - - - .- - - _ . - _ - _ _ - -

Table B.3 Effective Modal Masses in X, Y, and Z Directions 1

Mode MmX MmY MmZ Male MmX MmY MmZ l 5.981k+01 1.8161e+03 9.2019c43 33 3.2033e41 4.9367e42 1.4589e41 2 1.1931e+04 2.8h02 53860e+00 34 3.13h42 7.8179ee 1

3.1768e41 3 2.6970e+02 6.40kd2 5.9843e42 35 4.6025c-01 1.1017e+00 6.2423e 01 4 1.1479e41 83916e+03 2.4725e+00 36 4.07E43 1.1937e+00 53510e41 5 53447e+02 2.4212e+03 43615e42 37 339h 03 74280e4 1.0497e42

6. 13876e+00 23489e+01 93105e41 38 6.4485ee 4.8045e42 82029e44 7 9.4680e41 4.4613e41 33194e42 39 2.8023e42 13537e 01 3.6007e6 8 5.902e+00 9.8h00 53919e41 40 2.6006e44 1.1685e41 83370e42 9 4.6516e41 13657ed0 6.6616e42 41 13148e42 13567e41 1.6640e+00 10 2.6762e+01 3.2608e+00 6.1366e-01 42 4.4426e+00 23n2ne+00 3.1220edi 11 23104e+02 13800e+01 3.2658e+00 43 5.1481e 01 33392e41 2.1095e+00 12 1.99%+01 9.4805e@ 2.8259e 01 44 23758e42 1.0317c-01 1.8091e 01 i

13 2.1583e+00 3.6402e41 7.15 % 43 45 3.1547e42 43032e 04 1.1917e+00 14 3.9216e+00 21791ed0 83228e 01 46 3.9456e+00 4.8455e+00 1.900ledi 15 3.0210e+01 1.0717e+01 13789c41 47 1.5929e41 5.4659e43 6.6672e41 j

16 2.9412e+00 83850e-01 1.9545e 04 48 7.07E42 4.4608e+00 3.9794e+01 17 4.4550eG 2.4655e+00 6.0451e42 49 5.5329e43 4.0155e41 31546e+00  !

18 23338e+00 13090e41 1.4114e 01 50 13215e41 1.5993e41 1.8059e41 19 1]473e+01 3.9664e41 1.6109e+00 51 33988e+00 1.9074e+00 83814e+00 20 3.1431e41 3.1541e 01 32148e-01 52 2.9246e42 93223e41 1.9358e+00 21 4]911e+00 4.4%3e-0) 1.4019e+00 53 1.8899e42 33948e42 2300le+00 22 82480c01 1.8 h 00 1.8799e41 , 54 4.43h+00 2.0609e41 1.1008e 01 1 23 5.3513e+00 8.4164e-01 33466e 01 55 3.0008e+00 33823e41 7.1790e 01 24 3.8170e-02 4.9745e.03 '

2.2424e 03 56 5.8572e41 6.9191e41 4.3112e42 25 6]589e41 1.6940e42 2.4020e 01 57 5.4339e-01 5.0732e+00 63232e+01 26 2.4139e43 2.4484e43 7.4248e45 58 3.0120c-01 43540c41 1.0848e43 27 13841e41 4.6967e42 3.9864e-02 59 1.1867e41 3.4781e+00 2.4169e+02 28 33054e+00 4.8945e4 23870e+00 60 1.2789e 04 13411e42 13952e+01 29 93095e01 43330e42 9.1484e 01 61 4.4977e42 5.2725e 02 3.2090e+00 30 1.8485e42 43393e42 33510e-02 62 5.7829e 02 3.1941e41 3.5220e+00 31 1.1281e41 2]l64eG 4.6372c 02 63 1.55 & 42 2.6274e 02 1.2292e+00 32 4.6470t42 1.9660e 03 4.1676e 02 64 1.4067e 02 3.4433e41 3.Il42e 01 58 I

l

i l

Table B.3 Effective Modal Masses in X, Y, and Z Directions ( Continued )

Mode MmX Massy Mas Z Mode MassX Massy MassZ 65 2.013ke 6.9063e42 4.6250e41 97 6.8320e42 8.5616e-01 63973e@

66 4.D45c 03 4.1580c02 2.9241c+00 98 23090e 02 13378e42 83572c42 67 2.6490e43 1.1389e41 2.5889e+00 99 2.197/c-02 2.2043e42 43296e 01 68 2.4979e42 4.832k@ 2.2437e+00 100 63291e42 2.5244e43 23203e42 69 5.0888e41 2.9035e 01 3.9U6e41 101 1A651e+00 1.1639e41 13252e+01 s 70 7.1809c45 1.9435e-03 13305e+00 102 4.0263e42 8.8007e-03 43416e41 71 2.9970e43 2.8355e42 2A90le+00 103 1A502e+00 3.1048e42 73048e+00 '

72 43919e41 1.1218e+00 1.1060e+0! 104 3.6324e+00 13237e41 3.%13e+01 73 2.6114e41 13648e 01 4A074e+00 105 23892c+00 13663e+00 3.1061e+01 74 33137e42 1.0389e41 3.9151e+00 106 5.2564e+00 7.6385e 01 7.6707e+01 3

75 6.3663e 02 1.2113e+00 63193c 01 107 33905e41 2.06k41 2.6542e@

76 2.0414e42 23703e01 7.9315e+00 108 4.6683e+00 6340le42 46950e+01 77 1.8299e41 4A090e-01 3.9431e+00 109 2A777e+02 1.1005e+00 5.5328e+03 78 1.5882e42 3.2117e-02 3.1546e43 110 13440e+01 5.2123e+00 23943e+02 79

{

2.6959e41 1A846e 02 7A263e+00 111 1.5822e+00 43274e41 ID7%+01 I 80 1.550le41 1.1761e42 7.1322e41 112 5A935e41 1.5338e41 1.0712c+01 81 4.6823e42 4A047e-01 1.0440e@ 113 1.2156e+00 8.6522e41 IJil2c+01 82 1.8247e41 1.1526e-07 13662e+00 114 1.8863e+00 1A755e+00 1.6163e+01 83 83828e02 53950e@ 5A682c Ol 115 1.5127e+00 1.8369e+00 1.9749e+01 84 61721e04 33862e41 9A767e42 116 1A855e42 13691e-01 3.2188e41 85 8.8139e42 1.9066e-01 2.6920e41 117 7.3735e.01 23477e42 2.8010c+00 86 8.8887e41 i' 5A702e41 1.0369e+01 118 1.6949e41 93579e 02 43357e+00 87 7.5819e43 5.2684e-01 93681e 01 119 1.9336e 01 1.1386e41 1.2838e+00 88 3.1622e 01 8.0152e-02 13817e+00 120 3.2979e41 5.06R42 2.0772e41 89 51342e41 3.2120c42 13958e+01 121 13669e+00 23235e+00 2.0281e+00 90 4.6604e41 23027e41 7.6991e@ 122 8.6176e41 SA105e41 1.6227e@

91 4.0493e+00 4.0239e@ 64281c+01 123 1.6143e41 2.1767e41 1.2318e43 92 2.9229e+00 1.0480e+00 63725e+0) 124 2.0126e41 13059e42 8.6117c+00 93 23258e@ 1.9160e 01 3.9007e+00 125 1A523e+00 1.1500L-01 2.0163e+01 M 2.89%+00 3.2388e 02 6.6245e+01 126 13132e42 13284e+00 1.2891e+0!

95 3.8511e42 23831e41 1.0349e@ 127 33213e+00 7.2122e-01 13552e+01

% 3.0063e41 63363e 02 63277e41 128 83006e41 53474e-01 3.0976e+01 59

Table B.3 Effective Modal Masses in X, Y, and Z Directions ( Continued )

Mode MmX MmY MmZ Mode MmX MmY MmZ 129 2.2525c+00 1.0728e-01 33694e42 161 12459e@ 1.9271c+00 9.09&+01 130 1.2130c41 2.4687e42 2.6117e41 162 1.1448c+00 3.1917e+00 3.9885e+00 131 23476e42 5.9563e-04 4.4064e41 163 2.093k41 24749e-01 13439e42 132 23118e+00 4.9243e 01 33305e41 164 1.1926e42 13119c42 13209e41

.133 3.4009e41 Il672e+00 5.47k+01 165 1.178k01 54127e Ol 4.7N01 134 13242e42 1.2822e-01 8.2759e+00 166 12 % 01 2.1033e+00 24661e41 135 5.1559e 01 If452e41 22367c41 167 14318e41 2.1930e41 2.980le41 136 53906e+00 4]I24e-01 5.0902c42 168 3.8077c42 64759e-03 35403e01 137 3.8640e+00 34939c 01 4.0ll7e+02 169 3.1267c41 7.4994e41 4.4294e41 138 8.243k43 93064e42 5.0831e+00 170 2.2792c+00 2.1983e+00 1380le+00 139 1.0740e43 3.4261e44 1.0513e-01 171 13933e 01 43937e41 2.1476e+00 140 1.1727e42 22968e44 2.1847e-01 172 33871e+00 94069e 02 23608e+01 141 2.1326e41 7.4827e41 7.4855e@ 173 2.2640e+00 2.0862e41 1.0942e+01 142 1393k41 2.0880e 02 4.4241e+01 174 43444e+00 9.9332e41 9.0796e+00 143 13626e42 2.1735e 01 22997e+00 175 54225e+00 1.0308e+00 1.9615ed!

144 1.0531e42 3]l88e-03 1.2414e+01 176 42591c+00 4.0900c4i 63992e+00 145 2332k41 1.1045e41 1.6810edl 177 7.2197e41 33992e43 7350$e+00 146 1.41lle41 1.9774e41 1.427k42 178 1.1992c+01 2.4775c.02 12903e+02 l 147 2.0427e 02 3.8276e41 2.5023e+01 179 13340e+02 5.9157e40 2.0212e+03 148 2.9730e42 3.886k42 3.9498e+00 180 13395e+02 317kd0 1.6322e+03 149 23894c42 2.9065e41 3.6926e+00 181 2.6841e+00 3.0852e-04 2.3502e+01 150 43199e42 138W-01 i 9.5746e 01 182 3.1330e+01 1.0671e44 2.8304c+02 151 5.877k42 1.020le-01 53068e41 183 5.4887e+00 1.0985e 01 2.3556e+01 152 7.264k42 1.9593c42 13138e41 184 13545e+00 3.0748e41 2.1784e+01 153 32935e42 2.0188e+00 1.1464e+00 185 53932e-02 1.0840e41 5.2533e41 154 3.8822e42 2.4846e42 5.0200e-01 186 4.9058e+00 8.2989e41 43074e+0!

155 If535e43 334k 04 1.0956e 02 187 2.1950c41 12530e42 2.2487e@

156 13721e43 2.1985e 03 33740e43 188 2.0158e+00 3.9945e41 14009e+00 157 2.4765e41 1.4315e+00 4]4 h 00 189 8.8385e40 1.1740c+01 5.2544e+01 158 3.1090e41 1.4260e+00 4.8805e-01 190 1.6149e+00 13663e42 1.0049e+0) 159 5.4393e+00 1.0223e+01 32680e+00 191 3.0824e+00 6.2052e+00 3.9406e+01 160 3.2763e42 2.8030e41 92508e42 192 4.6370e-02 23053e+00 2.8503e+01 e

1

+

Table B.3 Effective Modal Masses in X, Y, and Z Directions ( Continued )

Mode MassX Massy MassZ i

Mode MassX Massy MassZ 193 5364k+00 3]09k43 8.4047e+01 225 1.886~ +00 23090e42 4.66Me41 19A 13800c+00 33160e+00 2.9251e+01 226 4.6823e42 6.8820e42 2.6959e01 195 9.271kG 1339k41 740k42 227 1.4502e+00 1.8247c41 1.5127e+00 1% 1.400le+00 1314k+00 2.1260e+00 228 2.9970e43 2.89k+00 4.0263e42 197 5.1821cG 4.2165e45 2.6649e-01 229 53829e42 2.9229e+00 1.4067e42 198 If913e+00 ' l.2441e+00 53287e+01 230 1.4651c+00 1.8247c41 83228c42 199 1341k+00 6.0342e41 7.2542e+00 231 4.4977e42 6366k42 1.8247e41 200 2]545e+00 3.1723e41 1.0914e+01 232 1.4855e42 4.682k 02 6.2721e44 201 2.4533e41 63444e42 '

23322e+00 233 1.4855e42 4.682k 02 44604c41 202 1.9706e+00 1.5210e41 1.14E+0! 234 4.0493e+00 44823e42 2.9970e43  ;

203 2.020k+00 9.0822e41 6.125k+00 235 5.2342e41 8.8139e42 4.4977e42 204 1.0515e+00 23563e+00 6.5743e42 236 2.9970eG 2.8934e+00 3.006k41 205 4.8987e41 7.ll87ee 1.0935e+00 237 13882e42 1.5586e42 1.8247e41 2% If25k+00 1.4330e 01 5.6803e+00 238 1.550le41 23258e4 43919e41 207 2.8331e+00 1.6179e+00 1.2246e+01 239 6.2721e44 1.4855e-02 1.4502e+00 208 13603e# 6.0559e42 2.1210e+00 240 1.6577e+00 2.5043e+01 7.2429e41 209 3.8571e-02 2.9229e+00 7.5819e43 241 3.1451e+01 1.4316e+01 3]773e+01 210 1350le41 3.8571e42 3]I37e42 242 7.950le+00 9.1065e+0! 3.1278e41 211 44604e 01 2.8934e+00 3]905e41 243 1.6091e+02 94693e+01 2.7807e+01 212 2.6490e4 2.6114e41 43919eal 244 1.9882e+02 1.8523e+02 33915e+01 213 44683c+00 1.8247c 01 1.55E42 245 3.20lle+02 2.4273e+02 2.4785e+01 l 214 4.4977e42 8.8139e42 3.1622e41 246 2.4662e+01 1.876+01 7.0333e43 215 1.4855e02 2.0414e 02 1.4502e+00 247 2.2517e+01 1.2805e+0) 34398e41 216 13822e+00 4.4977c 02 23892e+00 248 3.6059e+02 3.4017e+00 13091e+01 217 3]905e41 1.886k+00 3]905e41 249 3.4032e+02 9.0776e+02 1.2170e41 218 1350le41 5]829e-02 13127e+00 250 7.8117e 04 1.0617e+02 J.2360e 01 219 5.088k-01 5.0888e 01 8.8887e41 251 1.1083e+02 4.0272e+02 13649e41 220 3.1622e41 3.8571e 02 135E42 252 13810e+00 7.0225e+01 4.0850e42 221 4468k+00 6366k42 1350lc41 253 5.ll29c44 2.1779e+01 1.4369e41 222 43919e41 4.4977e 02 2.4979e42 254 3.0092e+00 33450e 01 43752e41 213 . 2.013kG 4.6P23e-02 1.8247e41 255 1.4102e+00 7.640le 01 2.2879e42 224 5.0888e41 8.8139e42 4]919c41 256 4.8005e+00 3.55h41 2.6417e 01 61

Table B.3 Effective Modal Masses in X, Y, and Z Directions ( Continued )

Mode MassX Mass Y MassZ 257 4.7729e40 13855e+01 7.2726e41 258 1.8997e+01 33337e4 1.ll64c+00 259 6.2791e+00 8.735ie+00 2.9839e41 260 5.4935e-01 5.4702e-01 2.490le40 261 7.1809e45 83616e41 3.9431e40 262 53829e42 1.0389e-01 2.2437e+00 263 4]919e41 1.1218e+00 3.9431e40 264 4.7919e-01 1.1761e42 3.9776e41 265 1.8247e41 1.2113e+00 3.ll42e-01 266 1.8299e41 4.0239e+00 1.0369e+01 267 63291e42 2.9035e41 33220e+00 268 4.0263e-02 1.1218e@0 3.ll42e41 269 8.8887e-01 8.8007e-03 13305e+00 270 2325&e-03 8.6522e41 33220e+00 271 73819e-03 2.6274e42 1]D48e+00 272 2.9970e-03 1.9435e43 3.1546e43 273 63291e42 6.9063e-02 4.6950e+01 62 i

L_-_____________ _ _ _ _. _ l

Table 1.1 Buckling Factors of Safety Obtained from Axisymmetric Analysis SRP Designation Service Reference Load ofLoad Factor of Limit Number Combination Combination Safety Buckling Location A (iii)(a)(3) Dead load + live DBAl 3.10 General tensile yield load + internal in the cylinder pressure,45 psig +

rise in temperature to 280 F C (iii)(c)(1) Dead load + live DBA4 2.02 Between base and load + extemal lower stiffeners pressure,3 psig +

rise in temperature to 120 F Table 1.2 Weight of the AP600 Attachments Location Weight From To Attachment (Ibs) Elev. Elev.

Air baffle 410,845 142' 241' Walkway 25,000 162' -

HVAC Duct 44,761 190' ---

HVAC Duct 13,239 205'4" ---

HVAC 51,000 155' 192' Cable Trays 72,000 152' 160' Containment Recirculation Unit 82,300 162'1" Concrete on External Stiffener 78,500 132'3" l

63 l

Table 1.3 Factors of Safety for ASME Service Limits Factor of Safety Regulatory i Service Limit NE 3222.2 Guide 1.57

• Case N-284 Design Condition 3.0 --

2.0 Levels A & B 3.0 --

2.0 j Ixvel C 2.5 2 1.67 Level D 2.0 2 1.3

• Does not explicitly identify the service limit except as being associated with the loading causing the largest compressive stress.

4 Table 2.1 Buckling Modes of a Perfect Circular Cylinder Buckling Stress, o,(psi) Buckling Mode F.E.M. l Theory (Eq. 3.3)

Mode F.E.M. Theory (Eq. 3.2) m n m n 1 36491 36566 16 18 16 17.96 2 36540 36566 21 9 21 9.481

~

3 36563 36566 22 0 22.34 0 4 36621 36566 24 0 22.34 0 64 I

Table 2.2 Five Mesh Parameters Parameter A B C D E Number of elements 616 308 156 78 38 Hoop 0.55 1.0 5 1.0 5 2.0 5 2.0 5 Element Direction Size Vertical 0.5 5 0.5 5 1.0 5 1.05 2.05 Direction Aspect Ratio 1.0 2.0 1.0 2.0 1.0 Number of Variables 15810 8370 4266 2370 1170 R.M.S. Wavefront. . 137 90 90 66 66 Storage / Analysis

• 36.4 12.6 4.4 1.7 0.8 Storage / Increment

• 7.1 3.6 1.8 0.9 0.5

• Units are in megabytes.

Table 2.3 Results of Mesh Sensitivity Study Critical Load Mesh Proportionality Factor  % Error I A 0.3263 3.6%

B 0.3264 3.6%

C 0.3285 4.3%

D 0.3288 4.4%

E 0.3963 25.8 %

i 65

1 l

Table 4.1 Analytical Solution for Natural Frequencies and Mode Shapes of a Clamped-Clamped Cylinder Mode Shape Frequency m n (Hz) 1 9 4.7429 1 10 4.2994 1 11 4.2094 1 12 4.3918 i 1 13 4.7757 1 14 5.3067 1 15 5.9479 1 16 6.6765 2 12 7.6152 2 13 7.1369 l 2 14 7.0064 2 15 7.1602 2 16 7.5404 1

Table 4.2 Frequencies of First Horizontal and Vertical Modes Obtained by ABAQUS and BOSOR4 l Mode ABAQUS BOSOR4  % '

Description (F.E.M.) (F.D.M.) Difference Horizontal Mode 5.910 6.14 3.74 ,

Vertical Mode 12.398 13.69 9.44 .

Table 4.3 Percentage of Total Effective Modal Masses From Total Mass l  % of Effective Mass  %

Direction ABAQUS BOSOR Difference 4 X (North-South) 86.74 91.12 4.38 Y (East-West) 88.91 91.12 2.21 Z (Venical) 73.77 83.58 9.81 4

66 f

i Table 6.1 Element Sizes in the Three-Dimensional Model l

Elements Size

• Jit Aspect Region Hoop Direction Meridional Direction Ratio 1 1.00 1.69 1.69 2 0.32 0.52 1.63 3 0.34 0.65 1.92 4 0.34 0.67 1.98 5 0.34 0.56 1.64 6 0.34 0.55 1.63 7 0.34 0.53 1.55 8 0.34 0.36 1.05 67

4Z EL.256' ,

i

! Spring Line

! EL218'-8.5" i

b~ ~'

j , --~\ EL.208'-4" i s. . . Top of Crane l Girder l

i y EL. l?O'.0-i

! , 1.625"

! Equipment Hatch e ,

e i

9 .10 -

ya ,

__ . .2.2 <. . . . ._ EL.144*-6" e

EL 138'-7_"(_  !

d EL.132'-3" g'ig *i Personnel Air Locks

- - l EL.110*-6" .

EL.112'-6" ' .

Spring Line l- .  !

EL.108

• EL.104'-0" Equipmelit Hatch

.100'4

h\N '

\N i

l Fig.1.1 Elevation of AP600 Steel Containment (Penetrations Shown Distoned).

68

l JLy

!270 North a

s s

1 4

i I

i 180..

- - 0 X

-.7.-.-......... ......._._._._._ _ ,

i 54' s

67*

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.+

i ,

s ',

• CL.16' /

• f _ CL. 22'-0" 1.D.

Equipment Hatch / ;900 Equipment liatch Az.126 Az. 67 CL. 9'-10" I.D.

Personnel Air 1.ocks Az.10f Fig.1.2 Orientation of Major Penetrations.

69

n r

4 CONT. I. R. 65' 0' , y=

23' 0' x 4' THK INSERT PLATE

%. /

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. ll l ll l ll 11/2' l *l s . ..

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u l ll G. EQUIPMENT HATCH

~ AZ. 6f l ll l ll l ll l ll l ll l ll l ll 5' 313/16' LONG X 4 3/4' ll THK BARREL U \  !!

b 61' 0' .

t 1, 66' 13/16' _

J, Fig.l.3 Details of the 22 ft. Diameter Equipment Hatch.

l l

70 I l

I

_ . _ - . _ _ . _ _ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _______ ___o

l l

l 1

CONT.1. R. 65' 0" 4 _

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& 6" -

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/

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l AZ 126 i l l  !!

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. ss j 63' 6 5/16" '

r ELLIPSOIDAL BOTTOM HEAD i

.g 66' O13/16" _

f

~

Fig.1.4 Details of t'le 16 ft. Diameter Equipment Hatch 71

Load h

Limit Load (Perfect structure)

P b X C Fundamental path 1c +

Bifurcation -

'L N k -E s ' -

\

,/ N Post-Buckling path i x-(

F o

/

/ Limit Load (Irnperfect structure)

/

O >

Displacement Fig. 2.1 General Load Displacement Behavior of a Shell.

72

t 0 t a

Geometric configuration:

r = 780" t = 1.625" 0 = 20 e

1 = 1375" a) Geometric Configuration.- )

i I

l I

b) Finite Element Model.

Fig. 2.2 Finite Element Model of Cylindrical Shell.

73

l l

E I

see 4EE E I

Fig. 2.3 Buckling Modgof a Perfect Cylinder.

i

i I

70000 60000 -

50000 -

40000 -

.c E

w 30000 -

20000 -

1 10000 -

0 '

O.000 0.001 0.002 0.003 0.004 0.005 0.006 Strain (in/in)

Fig. 2.4 Stress-Strain Curve Used in the Analysis.

75

7 9

E v h _ _ _ 0 s _ 2 0 6 e _ _ _ Rx 1 6 M V 0

2 R

D V 0

h ____ _

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x

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BOSOR '

---

N2 8 4 0.4 -

o e

O A

C W e 0.3 -

o e4 M

U 0.2 -

0.1 I I I I I Mesh A Mesh B Mesh C Mesh D Mesh E Fig. 2.8 Re.sults of the Mesh Sensitivity Study.

I 79

+

i e '

, ______ 256*-4" l

_ Upper Ellipsoidal _

~

Head ' ,

' l 218'-8.5" Crane Girder i .

.- - T 209'M

-8 L-I PART A2 Upper Stiffener '  :

N i

. .. 170'-0" I Transition Zone  ;

I w ut 3

E.u; G_N;pwev+ s ,s

- e ; PAR;T;MsJ s ..

9 A 48s.

w u.x-. 4g y ~ -- nws.g;f5

. y.

3s c - - g &'.K:

Lower Stiffener ssDF t g g !:in jj?. l,l;Fg@

_ _ w=w; g3Wuysna;;pr 7,,ji

.. ,g .l 1

$w,.(- _gp(j;:{r si:M ,' 132'-3"

^

am

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Lower Ellipsoidal , ,'

• W i: -

4 6.-n;4*. '

l Head >

100'-0"

,\

Penetrations

Area s

. Fig. 3.1 Descritization of AP600 Containment. i i

l.

80 I

Fig. 3.2 Three-Dimensional Model of AP600 Containment Vessel 1

81 4 1

1

p7,=, , /

7 I /

N

/ N

/

~ i

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\\

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//

\\(s~c //

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- f Fig. 3.5 Finite Element Model of the Upper Ellipsoidal Head.

I t /\

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-52.i

---]

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==m

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._-4

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--4

__4

---4

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--q ha-Hc Fig. 3.4 Portion of the Cylindrical Pan Showing the Transition Zone.

82 l

L--_----------------_---- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - _ - - --- _----

/

e  !!

. n

-r Fig. 3.5 Finite Element Model of the Crane Girder.

~

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%-s me Nge paned connymen l ,

p -

o

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Fig. 3.6 Crane Bridge and Wheel System.

t 83

1. Upper Stiffener ( Elevation 170'-0")

Inside Shell 1.625" Shell Elements "

r- -+ ~

'o 1.75"  ?

I '-2' g e e qm, y

~ ~ -

Beam O g 2 '-6" _

Elements y r -

i t I' Sectional Elevation F.E. Model Elevation Top View

2. Lower Stiffener ( Elevation 132'-3") o , Beam Elements O

_ Outside

' Shell 60 Ard-3/4"

- o h

• + 1.625" O Shell 0.75" lr lilements 9- O y 2,l tl F.E. Model Elevation s

E.

=

S ]

Sectional Elevation

. .-- rw ,

J Ton View

Fig. 3.7 Finite Element Idealization of Stiffeners.

1 84

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l Fig.3*8 ANSyg OdefinE in penet:ngjon 85

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Fig. 3.9 Deformed Shape of the Containment due to Internal Pressure.

i l

i- 86 l l l

i 52.5?

165'-9"

\  ; Air Locks Lower Equip. '

h^

l Hatch

• 100'-0" l

l I

l hh

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t 87 1

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/

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=~- .--

I Fig. 3.11 Deformed Shape due to Axial Compression.

0.40 Besor: 0.315 P, N284 :0.252 P, ,

0.30 Limit load = 0.292 P.

5 u.

.N 3

8 0.20 8

& l E

3 3

0.10 s

0.00 0.00 0.20 0 40 0 60 h

Vertical Denectum (m.)

Fig. 3.12 Load-Deflection Curve at the top of the Tested Portion.

88 1

i 1

80 i

0 0.5 Axial Wave (m=1)

-- 1.0 Axial Wave (m=2)

-- 1.5 Axial Wave (m=3) 60 -

i 50 -I 9

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y 40 - I

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5 10 15 20 25 30 35 40 45 50 Number of Circumferential Waves, n Fig. 4.1 Fundamental Frequency of a Freely Vibrating Cylinder with Clamped Ends.

I 89 I

[ . _ _ _ - . _ _ _ _ _ _ _ _ _ _ _ _ - _ _ _ - - - - - _ - - - - - - -- -

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Fig. 4.2 Three-Dimensional Model Classification According to Mass Density l

l 90

ABAQUS ABAQUS t

?

lll' ]

> ,s lis!Es' __ --

lj' == j!  !

_r,-- D -

ss as =ip! s l

3 as --

E Y

I  ! !g j~ lR X U i E-- -

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1. lllE F i kW Afode ll) f= 3.6895 ffz Afode (2) f = S.9103 ffz ABAQUS ABAQUS 5

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Afode (4) (= 6.3589 fit Afode (5) f = 6.6476111 l

Fig. 4.3 First Five Free Vibration Modes.

91 j

ABAQUS ABAQUS z ) Z Y

/ l Afode (51) f= 9.5350 Hz Afode (56) f u 9.6158 Hz ABAQUS ABAQUS i --

. 5. .

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i Afode (58) f= 9.7083 Hz Mode (60) f = 9.7408 Hz l

Fig. 4.4 Selected Four Modes from the Firri sixty Modes of Vibration.

92

20000 in X Direction 17500 -

• " - - In Y Direction

- ~ In 2 Direction 15000 -

A

... - ---a -a 12500 -

j C"~~

.E s I E i b goooo a' l V e l.

3 l- I e .

4 7500 -

l

I i 1 5000 - 1 l

e i e' I 4 l 2500 - 4 d l j g

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I O '-

n s" 00 50 10 0 I5 O 20 0 25 0 i Frequency (liz) 1 Fig. 4.5 Accumulative Effective Modal Mass Versus Frequency.

i 93 l- l h

.I

ABAQUS _

ABAQUS i

3 j e ::'

$! isss y ==::

i l! h Y

X  ; 7 m . ,##

Mode (109) f=12.398 Hz Mode (179) f=15.384 Hz ABAQUS ABAQUS j m

l q'7; , ,; s's

,{li'i Ny

'. II I I 1

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Mode (180) f=15.396 Hz Mode (249) f=18.816 liz Fig. 4.6 Modes 109,179,180,249.

l 94 w__-_________________-_-___-_-_ - - - - - -

r l

l Fig. 4.7 Contour Plot of N max i Stress Resultants  !

l I

6000 l

l 5000 l

4000

)

E 3000 2000 1000 0 40 to 820 160 200 140 2sn 120 M0 Angle (degreest Fig. 4.8 Distribution of N m i x at the Base l

l 95

3250 l Theta = 286.875

--- Theta = 293 5000 ny I

l l

2750 l

i s North

~+

S4 1 47 2500 /

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l LOVER

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\

l l

1250

\ %

1000 0 2500 5000 7500 10000 12500 15000 Ni max (ibhn) l Fig. 4.9 Distribution of Ni max at Azimuth 670.and 73.1250 96

3250 Deta = 234 7

--- Heta = 253 3000 ar 27.;0 i a North i s 2500 g " ~ s

\

I LOVER l' (O HAT H  %

\ UPP[R

\ AjR EO HAICH p 2250 'g '

.9

's L

% /

g / \

~ /

ua 2000  %

N

)  %

s 1750 .-

I500

/

i250 l(XO O 2500 5000 7500 10000 12500 150(0 Ni max (ib/in)

Fig. 4.10 Distribution of N Imax at Azimuth 1070and 1260 l 97

- - - - - - . - . - - - - - - - - - - - - _ _ ------_____-.__--_--___-----___---J

I l

i i

l i

I 1

l i

l Fig. 4.11 Contour Plot of N ,2iia, Stress Resultants o

4 t'

.4 i

4 g.

Fig. 4.12 Potential Buciding Regions 98

l l

l 1

3250 Equivalent Static Leeds

, --- Ni max 1 8

i,F g

i t

2750 g i

I I

. I l

2sw

,j s'o

/

'.j t

i l

8

? 2250 towte 8 0 to ut = ~ s 8 unece I

'= aie to mits i I LOCEE 4

-" s W 2000 ,# .

\

• ' o

• j

=*

• ,* 3 1750

)

N.

%g 1500

\

t t

1250 1000 15000 t2500 10000 -7500 5000 2500 0 2500 vuo Ni (Ibbn)

Fig. 4.13 Comparison of N1 max Stress Resultants at 0=286.8750, t

i l

l l

u__.____.________________.____..__.____.__ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ . _ _

. .J.*J I

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ll i f

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i

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t I

3 3 E= E E

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I i

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I i

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it .

.!y == -f

%illi..!i?'{.M{td, Q3. ..i.ih..d..i..M.Siin *2 w~ = = = =g 2'"'

y .

.i.l..l.i.!.E.EEEEE Fig. 4.14 Contour Plot of N iDue to Equivalen'. Static Loads 100

1 s

UPPER UPPER I AIR EQUIPMENT seoo '6Aa - LOCK HATCH sos I6Aa FLOWER I 3, #6An EQUIPMEN,T ,,no a/to HATCH l LOWER

/ I AIR

,/

./

\

i Fig. 4.15 Equivalent Static Loads of Potential Buckling Region 2.

I i

l i

l

< 101 e

. _ _ _ _. _ _ _ _ _ _ _ m _

l l

l sta w.ws 1

= -2 $1s.04 l'

= -3.35s.04

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.- mmamemmassammmmmmesumaniassammedE El II,ist.amii esamism .4. ,, f<, ll;W simumm I11 areammm . , em.wiwi.c4

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= = = ====:= ======ss = = =U 8um = mamm t am an um Y

i .

== ======:===mnemme x

Fig. 4.16 Contour Plot of N Due to Equivalent Static Loads at Region 2 102

80000

- Equnalent stars Lea <i i

--- Ni man 5000

)

E g # #% g I

= ~ \ ,, ,, ,' '

l z ~ ~. , ,- l s , , , . , ~- ,

s *

, ,i s 10000 'g ,# ( ,#

's

\ I l

\ l 1$000 %t

-20000 0 40 30 120 160 200 240 280 320 40 Angle (degrees)

Fig.4.17 Comparison of Nimax and N1due to Equivalent Static Loads at the Upper i Equipment Hatch. '

10000 I

- Equwaleni Staic Leads

--- Ni man ]

VJ00 0

I \ #

/ \g t

! m,

\ .

I i, <

/

s

,/

s

'g' s

,, N.- - .

- -.. ~,j ,

. 50.

20000 0 40 to 120 len 200 240 280 320 M0 Angle (degrees)

Fig. 4.18 Comparison of N1max and N idue to Equivalent Static Loads at the Lower Equipment Hatch.

i 103 L

10000

- Equivakna State loads

--- m ,

$000 0

s

• e~ ', ,

I< ~ s, g l

N, $000 , # n i s ',

, o

/

r l ', ,'

~'

/ s, ,-

~~ - s ,-

., ' ~s#

t5000 20000 0 40 80 120 too 200 240 280 320 M0 Angle (degrees)

Fig.4.19 Comparison of Nimax and Ng due to Equivalent Static Loads at the Upper Air Lock. .

k

- Equwalen Stais toads

-- - m mu O

g $, ,,

8 x

== '

s. ~ -

/

~, ,

/

.i0000 13000 20000 0 40 30 s20 160 200 240 220 32n w Angle (detreell i Fig. 4.2.0 Comparison of Nimax and Nj due to Equivalent Static Loads at the Lower Air Lock.

104

! 0 ,

i l - = = Ni snam Equivalent Suir toed s i I o a

-2500 -t ' s '

' i t t s e I

\, - ,

g 8,

i e t

's, ,

~

~..'  : ,'  : 's l

{

-5000 8 s

1. 8

\ e' '

g l j n

C t

g f %e*%

i i

-7500 8#

E i ,

l 1

te i i

• lM l

12500

-15000 0 40 80 120 160 200 240 280 320 360 Angle (degrees)

Fig.4.21 Comparison of Nlmax and N idue to Equivalent Static Loads in the Thin Shell Around the Upper Equipment Hatch Reinforcing Collar.

1 i

105

i l

I l 70000 l.

I 60000 -

l 50000 -

l' .

40000 -

i c E

m l

30000 -

20000 -

l n

i 10000 -

0 ' ' '

O.000 0.001 0.002 0.003 0.004 0.005 0.006 Strain (in/in)

Fig. 5.1 Stress Strain Relation at T=1200 F.

]

I 106 1

l

i b

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i W- - / l .

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i l

l I 107

II O

Oo .

o 3.0 2.5 f lj ' ,

,a ,r'

,s'

• 2.0 ,f' I

1 8 I G I 1.5 ,#

3 l l

l 10 f l

r Point A I

--- Point B 0.5 f 00 0 2 4 6 8 io Deflection (in)

Fig. 5.3 lead Deflection Curves at Points A and B (Load Case 1).

l 108

_ . _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ - _ - - - - - - - - - - - - - - - - - - - - - - - - -- -~ --

83CTItsi Potert

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Fig. 5.4 Contour Lines of Von Mises Stresses at A = 2.54 (bad Case 1) 109

O Oo

• s 3.0 2.5 * ~~~~~~~~~~~~ s

,o#  %

r  %

8 I

i 2.0 g i

i i

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1 l

I I

I l0 I I

I I

I I

I I

05 I

--- Poirt B

,# point A I

I I

O 10000 20000 30000 40000 50000 Von Mises Stresses (Ibhn)

Fig. 5.5 Von Mises Stresses at Points A and B (Load Case 1).

I10

l t ,

i

'\ /\, /\ '

/\ /\ /\

f'l l' i

?

a

' ?l

/

}lU

l

l \\r- \", ~ ;A,,i1 5} .: YI y Fig. 5,5 kiled vi'* Of the Dero A = 2*62 04ad case 2). a r i 111 N - O C-M. Oo _ . _ . - 3.0 l 8 2.5 20 1 I b s ( i.5 -l .5  : I l .0 8 l i I romi c 03 l - - - ro ni n 0 2 4 6 3 10 Deflection (in) Fig. 5.7 Load Deflection Curves at Points C and D (Load Case 2). I12 l 1 l ' l SECTICu polart 3 j . . tor.o: l milestrillumiinunum su!sunami surm ate Fm

• 82s.o*

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• 203,' -

Region 5  ! h r 170' , v! - ey: Region 6 i i i < 132'-3" Region 7 i i Region 8 104'-1.5" L - 100' Fig 6.1 Partitioning of the Wedge Modelinto Eight Regions For Meshing z il '% >: C5!!! Fig 6.2 Finite Element Model of the Top Ellipsoidal Head 114 l _ _ _ _ _ _ _ _ _ _ _ . _ _ _ . . . _ _ _ _ _ _ _ ] o Mid side Node e CornerNode Sheti Elements Shell Elements 2'-6" l: t> { t35" l '.2" . 1 -- --O----H > A - f < > Beam Beam M }+- 135" q g J Elements Elements inside  ; Shell

a. Sectional Elevation Elevation Plan

b. Finite Element Model Fig 6.3 Modeling of Upper Stiffener (Elev.170'-0")

) l I o Mid side Node stet e Corner Node Elements Outside Shell O Shell Elements 3 '-8 " _i o } 0.75" 9, , -i, r f 12" .1 ,, Beam y __ 0 Elements Beam > Elements

a. Sectional Elevation Elevation Plan

b. Finite Element Model i

Fig 6.4 Modeling of Lower Stiffener (Elev.132'.3") i l 115 i L_ _ _ _ _ _ ._ _ _ _ _ . - _ _ _ _ - - _ _ __ . - _ . _ - - _ _ _ - - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ r-l o Mid side Node l Inside 3g e CornerNode Beam Elements 1.5" 4 - [ . l a e -

1 l i, t  :

" y o h i l 88: l ll ll- 6*-0" o ,  : e  ; es q  :  ; ! t II 88 , t- .s- . l r'r .. l 8" 3'-0" - Shell j Eiements 7

a. Sectional Elevation Elevation Plan .

b. Finite Element Model Fig 6.5 Modeling of Crane Girder i

I I I 4 fl 8 l l l d ! I i l Fig 6.6 Assembled Finite Element Model 116 i l 3M BOSOR INPUT --- SMOOTHED DISTRIBU110N 2M s \

\

\ t-7g \ \ \ i \ l 8 \ a \ l g Dry \ Wet

8. 2e ,

7, \ E s x

1. 30 % 's 70 %

's s i 220 ',- ! s

s. .

I 2W t l l-l I 180 l 0 10 20 30 40 Circumferential Arc Length (in) Fig 6.7 Case 2 Temperature Distribution Above Elev.132'-3" 1 l 117 , l i ____._._.._________._____m l I i 3000 Mendaonal Stress Resultants ) I ! --- Hoop Stress Resultants _ _ , y _w s j

• I I

i i g \ s* \ 2500 i s I t i n i i .5 I i E I s> s N E 2000 ,' t i i i 1 i 8 l t 1 I 8 l l / / l J l 1500 i i i i l l 1000  ! -60000 -45000 30000 -15000 0 15000 Stress Resultants (Ib/in) l l . Fig 6.8 Meridional (N1) and Cicumferential (N2) Stress Resultants 1 l Due to Case 2 Temperature Loading (Dry Strip) 1 l ( f 118 I i r 3000 _ Mendsonal Stress Resultants --- Hoop Stress Resukants p I I I I I I / o A 2500 - N; , 's l a f> < I ^ ] l bc I 1 -S I !l' > N  % iu 2000 - ,# 4 / I I I I I I I i ..,==---

• 1500 't .

1 i 's l i ,__________________ -------- -- ) ) i i 1000 ' N -45000 30000 -15000 0 15000 Stress Resultants (Ib/m) Fig 6.9 Meridional (NI) and Circumferential (N2) Stress Resultants Due to Case 2 Temperature Loading (Wet Strip) 6 119 Rigidly / Connected -- f, v v F T1 T2 L U llOIfl J L O F Stress Di bution on Sec. A-A ' bl ' b2 ' Theoretical Model Fig 6.10 Approximate Theoretical Model kl. P Mode (1) A = 11.213 Mode (2) A = 14.235 Fig 6.11 Elastic Buckling Modes Due to Striping 120 ~

y 6

pk W dlli Mode (1) A = 6.790 Mode (2) A = 6.832 Fig 6.12 Elastic Buckling Modes Using Worst Meridian Approach 121

-100000 - w.a w i -- wonc-I 0000 - 4 0000 - g ___ --- a - E a , / m m . ,/ / / / .. / 20000 -# 04 5A 10 4 154 20.0 25.0 30.0 Lead Factor Fig 6.13 Variation of Stress Resultants with Load Factor (Theoretical Model Results) -lu1AJU l - w.=c- i t --

• M aimum Seems Realians (Imperfect) l

-83100 - y ,' .g a _ ,e - - i* s' 1 b # m o "E # 8 < 1 * ~ l 5 l I I I -20m0 -l a 0 ' ' OD Sh 10D 154 20.0 23D 30.0 Lead Factor Fig 6.14 Variation of Stress Resultants with Load Factor (Finite Element Results) 122 10J 10.0 1 E 5 a ,s r D W m 9.0 g _. O I 2 3 m Fig 6.15 Effect of Three Dimensional Imperfections on Buciding Safety Factor 280 ,. 's \. M0 - \g \. g - 20% Dry seip 30% Iky Saip j 'g -- 50% Dry Sair s

• 260 'g .

s \ s s 5 \ 250 - s 'g \ s E \ B  % \. 7A0 s \, 5 p \s \. 230 Ns \ 's . \s \ = 220 - N \ 's \, 210 's *., \* 's s \, g . . . . . . . . s _ 0 5 10 15 20 25 30 35 40 45 50 Circumferential AstIssth(in) Fig 6.16 Variation of Temperature Distribution with d 123 le 11 - ) 82 13 1 . so - E 2, . 8 - 7 - a n. a , 6 " =

• 4  ; A e Fig 6.17 Effect of Stri Ping waves Number on Buckling Safety Factor l

i i Fig 6.18 Deformed Shape ( DB A1. Temperature Case 1) 124 ) i4 L. I l Fig 6.19 Deformed Shape ( DB Al, Tempera: re Case 2 ) Perfect Case j$ Fig 6.20 Deformed Shape ( DBA1, Temperature Case 2 ) Imperfect Case, k=4.0 125 - _ _ _ _ _ _ _ _ .