ML20044G020
| ML20044G020 | |
| Person / Time | |
|---|---|
| Site: | 05200002 |
| Issue date: | 05/31/1993 |
| From: | Bluhm D, Challa R, Fanous F, Greimann L IOWA STATE UNIV., AMES, IA |
| To: | Office of Nuclear Reactor Regulation |
| References | |
| CON-FIN-L-1898 IS-5083, NUREG-CR-5957, NUDOCS 9306010330 | |
| Download: ML20044G020 (112) | |
Text
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i NUREG/CR-5957 IS-5083 l
l System 80 Containment --
Struc~ ural Design Review s
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Prepared by L Greimann, F. Fanous, R. Challa, D. Illuhm Ames Laboratory l
Iowa State University Prepared for U.S. Nuclear Regulatory Commission I
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9306010330 930531 PDR ADOCK 05200002 A
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1.
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3.
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NUREG/CR-5957 IS-5083 l
i System 80 +
Containment --
Structural Design Review 1
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Manuscript Completed: April 1993 l
Date Published: May 1993 j
I Prepared by L Greimann, F. Fanous, R. Challa, D. Bluhm l
5 Ames Laboratory i
Iowa State University
.l Ames,IA 50011 j
Prepared for Division of Engineering Office of Nuclear Reactor Regulation i
i U.S. Nuclear Regulatory Commission Washington, DC 20555 NRC FIN L1898
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ABSTRACT
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I A review of the structural design of the Comb.sstion Engineering (CE) System 80+* steel containment was completed. The stress analysis and the evaluation of the structure against i
buckling were performed by using BOSOR4 and BOSORS finite difference sc'tware, respectively.
The CE System 80+* containment was modelled as an axisymmetric shell consisting of different segments and mesh points with the additional mass of the penetrations and appurtenance being j
smeared around the circumference. The transition region was modelled using elastic springs with l
a foundation modulus of 180 lbs/in'. The stresses due to the individual loads (dead loads, l
intemal and extemal pressures and temperatures) were computed using the stress analysis option in the BOSOR4 program. The stresses from individualloads were combined according to ASME Code into stress intensities. Service Level B loadings produced a 20 percent over-stress in a small zone just above the transition region. All other stress intensities were within allowable limits. For the System 80+*, the perfect shell with an elastic material was initially analyzed. The l
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calculated factor of safety values were 2.3 (Level B) and 1.59 (Levels C and D). Finally, i
sensitivity studies were conducted to investigate the effects of mesh size and transition zone stiffness on the controlling buckling load.
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TABLE OF CONTENTS s
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EXECUTIVE
SUMMARY
1 1.
I NTR O D UCTI O N................................................
2 s
2 1.1 Background..........
2 l
1.2 Objective...........
1.3 Methodology............
2 2.
STRESS ANALYSIS 3
3 2.1 Axisymmetric Modelling of the System 80+* Conta;nment..
3' 2.2 individual load Cases..................
3 i
L 2.2.1 Dead Loads............
3 2.2.2 Temperature....
3 2.2.3 Pressure 4
2.2.4 Seismic Loading.......
4 i
2.3 Combinat;on of Stresses...
5 2.4 References..........
5 l
I 3.
COMPLETE SPHERE WITH EXTERNAL PRESSURE.................
7 3.1 Section NE3133...
7 3.2 Analysis Using Code Case N-284...............................
7 5
3.3 BOSOR Analysis........................
7 3.4 References.
8 4.
BUCKLING ANALYSIS OF SYSTEM 80+
CO NTAINM ENT................
9:
4.1 Loading and Solution Process.......................
9 4.2 Level C S ervice Limits....................................... 10 4.3 Level B Service Limits.................
10 4.4 Controlling Case...........................
10_
4.5 Foundation Modulus and Mesh Sensitivity Studies..................
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4.6 Di scu ssion................................................ 11 4.7 R eferences..........................
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5.
SUMMARY
, CONCLUSIONS AND RECOMf#
3ATIO N S.................. 12
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5.1 S u m m a ry...........................................
12 5.2 Conclu sions....................................
13 5.3 Recommendations.........................
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APPENDIX A. BUCKLING ASSESSMENT TECHNIQUES............
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A.1 The Shell Buckling Problem..
15 A.2 The ASME Boiler and Pressure Vessel Code and Regulatory Guide 1.57 16 A.2.1 Section NE3133...
17 A.2.2 Code Case N-284 17 A.2.3 Regulatory Guide 1.57.
. 18 A.3 Numerical Approach......
18 A.3.1 BOSOR4 and BOSORS Programs Capabilities 19 A.3.2 Nonaxisymmetric Loads.......
19 A.3.3 Elastic Buckling of Nonaxisymmetrically Loaded Perfect Shells....
21 A.3.3.1 Longitudinal Stress Resultant 21 A.3.3.2 Circumferential Stress Resultants........
. 22 A.3.3.3 Shear Stress Resultants....
22 A.3.4 Imoerfect Shells.............
23 A.3.5 Openings in Shells 24 A.3.6 BOSOR Model Guidelines.....
25 A.3.6.1 Mesh Guidelines.............
25 A.3.6.2 Modelling of Material Nonlinearities.........
25 A.3.6.3 Convergence Criteria..
27 A.3.7 BOSOR5 Calibration 27 A.3.8 Summary 27 A.4 References........
. 28 APPENDIX B. BOSOR4 AND BOSORS MODIFICATIONS......................
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i B.1 Modal Participation Factor (BOSOR4).
31 B.2 Modal Stress Resultants end Stresses Output....
32 i
l B.3 Effective Uniaxial Strain and Plastic Strain Output.....
32 B.4 R efe re n ce s.............................................
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APPENDIX C. EQUIVALENT AXISYI(METRIC PRESSURES 33
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C.1 Equivalent Axisymmetric Pressures...........................
33 C.2 References......................
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I LIST OF TABLES i
i Table 2.1 Smeared Mass Densities for the Different Segments of Axisymmetric Model in Fig. 2.2....................
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Table 2.2 Design Conditions..........
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t Table 2.3 Level A Service Limits.....
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Table 2.4 Level B Service Limits....................................
39 Table 2.5 Level C Service Limits 40 I
Table 2.6 Level D Service Limits..
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Table 2.7 Nomenclature.......
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Table A.1 Factors of Safety for ASME Service Limits..
44 Table A.2 BOSORS Comparison to Experimental Results.....
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i Table B.1 Modal Participation Factor for the First l
Mode of Vibration of the Cantilevered Beam i
in Fig. B.1..
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i Table B.2 Modified Libraries and Subroutines of BOSOR4
.t Source Code for Creating FILE 14.DAT.............................
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Table B.3 Modified source code of LOCAL, MODE and OUT2 subroutines of BOSOR4 48 L
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l LIST OF FIGURES Figure 2.1 Cross Sectional Elevation of the System 80+* Containment 65 Figure i.2 BOSOR Axisymmetric Model of the System 80+T" Containment........ 66 i
Figure 2.3 Extreme Fiber Stresses Due to a Temperature Differential of 220 F (indicating only the first two segments)...
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Figure 2.4 Extreme Fiber Stresses Due to a Temperature Differential of 40 F (indicating only the first two segments)...............
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Figure 2.5 Response Spectra for Operating Basis Earthquake for Soil Condition B4'..
69 Figure 2.6 Response Spectra for Safe Shutdown Earthquake for Soil Condition B4 70 l
Figure 2.7 Response Spectra for Safe Shutdown Earthquake-for Soil Condition C1 71 Figure 2.8 Response Spectra for Safe Shutdown Earthquake for Fixed Base Condition..
...................72 Figure 2.9 Comparison of SRSS Meridional Giress Resultants at 0 deg.,45 deg. and 90 deg. Meridians.............
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Figure 2.10 Comparison of SRSS Circumferential Stress Resu!tants at 0 deg.,45 deg.,
74 and 90 deg. Meridians.
Figure 3.1 Complete Sphere with Extemal Pressure - Variation of Imperfection Wavelength.............................................
75.
Figure 3.2 Complete Sphere with Extemal Pressure - Variation of Imperfection Amplitude..............................................
76 Figure 3.3 Buckled Wave Shape and Imperfection Wave Shape for Segments 2 and i
3...........
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Figure 4.1 Impeifection Sensitivity Study for Levnt C Service Limits (Meridional Stress Resultants in Compression) 78 Figure 4.2 Imperfection Sensitivity Study for Level C Service Limits (Circumferential Stress Resultants in Compression)............
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Figure 4.3 Imperfection Sensitivity Study for Level B Service Limits (Meridional Stress Resultants in Compression)........
79 Figure 4.4 Buckled Mode for the Imperfect Shell with an Imperfection Amplitude of.
0.875 in. and Imperfection Wavelength of 114.5..
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Figure 4.5 Variation of Effective Uniaxial Strain at Middle Surface of the Shell at Elevation 95.4 ft...
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Figure 4.6 Variation of Radial Displacement at Elevation 95.4 ft.................. 82 i
Figure A.1 Fundamental and Postbuckling Load. Displacement Paths.............
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Figure A.2 Coordinate System used in BOSOR Computer Programs Indicating the Circumferential Angle 0....
................................84 Figure A.3 Ground Accelerations in the X, Y, and Z Directions................
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Figure A.4 Ground Motion Corresponding to Symmetric and Asymmetric Terms.....
85 Figure A.5 Effect of Nonaxisymmetric Longitudinal Distribution on Critical Stress 86 Figure A.6 Torus Model for Buckling of Axisymmetric Shell with Nonaxisymmetric Stress Resultant..
87 Figure A.7 Effect of Circumferentially Varying Longitudinal Strass on the Critical Load.
for Stiffened Cylinders....
88 Figure A.8 Buckling Pattems of Axisymmetrically and Nonaxisymmetrically Loaded Cylinder 89 Figure A.9 Effect of Circumferential Pressure Variation....
90 Figure A.10 Modetting vf Imperfection as per ASME Specified Tolerances..........
91 Figure A.11 Configuration of Axisymmetric imperfection in BOSORS Buckling i
Analysis 92 t
i Figure A.12 Effect of Reinforcement Percentage on the Penetrated Shell Critical Load...................................................
93 Figure A.13 Effect of Penetration Size on Shell Critical Load....................
94 Figure A.14 Stress-Strain Curve Derived from Equations for Plasticity Reduction Factor in Code Case N-284........................................
95 Figure 8.1 Sample Problem Used in Calibration of BOSOR4 Program............
96 Figure B.2 Format of FILE 14.DAT Created During Vibration Analysis (INDIC=2) Ootion of the BOSOR4 Program...
97 Figure C.1 Geometry and Membrane Stresses for a Surface of Revolution.........
98 Figure C.2 Meridional and Circumferential Stress isultants for the First Horizontal M ode of Vibration.......................................... 99 Figure C.3 Comparison of SRSS Stress Resultants with Stresses Induced Due to Axisymmetric Pressures........................
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7 ACKNOWLEDGMENT The authors would like to express their appreciation for the members of the U.S. Nuclear Regulatory Commission, Mr. Harold Polk (Project Manager), Mr. Goutam Bagchi (Branch Chief, i'
Structural and Geosciences), Mr. P.T. Kuo (Section Chief), Mr. Tom Cheng (technical staff), Mr.
David Tang (Lead Engineer), and Mr. Yong Kim (Lead Engineer) for their help throughout the course of this work. The authors would also like to acknowledge the able assistance of the project Program Assistant, Ms. Connie Bates and the project Secretaries, Ms. Jeanine Crosman and Ms. Denise Wood for word processor operations and secretarial services associated with this project.
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EXECUTIVE
SUMMARY
j The objective of the present workis to review the structural design of the Combustion Engineering I
(CE) System 80+
steel containment. The stress analysis and the evaluation of the structure against buckling were performed by using BOSOR4 and BOSORS finite difference software, respectively.
t The Nuclear Regulatory Commission (NRC) Standard Review Plan (SRP) 3.8.2 stipulates that the design and analysis procedures be in compliance with the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code and Regulatory Guide 1.57. SRP 3.8.2 i
further prescribes the load combinations pertaining to Design Conditions and Service Limits i
classified by the ASME Code. Section NE3222.1 of the Code establishes the admissible factors of safety against buckling.
The CE System 80+ containment is a spherical steel shell structure with a 1200 in. radius and a 1.75 in. thickness. It is embedded in a concrete base with flexible support in a transition region.
The major penetrations are the equipment hatch and tw; personnel locks.
The major appurtenance is the spray header system. The containment was modelled as an axisymmetric shell consisting of different segments and mesh points with the additional mass of the i
penetrations and appurtenance being smeared around the circumference. The transition region was modelled using elastic springs with a foundation modulus of 180 lbs/in'.
The stresses due to the individual loads (dead loads, intemal and extemal pressures and temperatures) were computed using the stress analysis option in the BOSOR4 program. The seismic loading to the structure was provided by the plant owner through NRC channels in the l
form of earthquake response spectra for different soil conditions. The modal stress quantities were combined by the Square Root of the Sum of Squares (SRSS) method. Seismic stresses for those soil conditions and several meridians were compared to select the controlling seismic case. The stresses from individual loads were combined according to ASME Code into stress j
intensities. Service Level B loadings produced a 20 percent over-stress in a small zone just I
above the transition region. All other stress intensities were within allowable limits.
As a prelude to the buckling analysis of the CE System 80+* containment, a complete sphere with uniform extemal pressure was analyzed using the BOSOR4 program to study buckling and imperfection sensitivity. The loading combination, for the buckling evaluation of the CE System 80+
containment, was chosen on the basis of the stress analysis results as that loading which
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resulted in the largest compressive stress. The buckling analysis was accomplished using the BOSORS program. The buckling analysis was performed on the basis of the worst meridian assumption, that is the maximum stresses are assumed to exist uniformly around the i
circumference. The material nonlinearities and residual stresses were incorporated using a stress-strain constitutive relationship derived from the ASME Code Case N-284.
For the System 80+*, the perfect shell with an elastic material was initially analyzed. A sinusoidal imperfection was introduced and the imperfection wavelength was varied to identify the critical imperfection configuration. The calculated factor of safety values were 2.3 (Level B) and 1.59 (Levels C and D). Finally, sensitivity studies were conducted to investigate the effects of mesh size and transition zone stiffness on the controlling buckling load.
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INTRODUCTION 1.1 Backcround The CE System 80+* containment is a spherical structure of thin steel plates with large radii of curvature. It is embedded in concrete at its base with a flexible transition zone. As the containment is subjected to various loading conditions, regions of compressive membrane forces develop in the steel containment which may cause the shell to fait due to compression instability.
In order for the containment to perform its intended safety function to sustain these loads, a sufficient margin of safety against buckling should exist.
1.2 Obiective The objective of the present work was to assess the adequacy of the structural design of the System 80+* containment.
1.3 Methodoloov The current structural acceptance criteria as per SRP 3.8.2 was reviewed with regard to buckling.
Investigation of different buckling analysis techniques was made with regard to applicability of analysis software, treatment of axisymmetric and nonaxisymmetric features, effects of penetrations, imperfections, and material behavior. On the basis of the investigation, the BOSOR4 and BOSOR5 finite difference programs for axisymmetric shells were selected for the analysis. The BOSOR programs were verified for stress analysis and buckling by analyzing several test cases for which closed form solutions are available and by reference to usage by others.
I initially, a stress analysis of the containment shell was made for the different loading combinations as specified in SRP 3.8.2. The different loads that constitute the loading combinations are the dead load, uniform intemal or extemal pressure and temperature increment, which give rise to axisymmetric stresses and earthquake events which gives rise to nonaxisymmetric stresses. The nonaxisymmetric seismic stresses were incorporated into the axisymmetric analysis by utilizing equivalent axisymmetric pressures. The equivalent axisymmetric pressures, which were derived on the basis of the shell theory, involved the extraction of modal quantities and the calculation of inertial pressures. Supplemental programs were developed to perform this process. The results of the stress analysis were checked against the allowable stress intensity limits for different Service Limits as per the ASillE Code.
Prior to the buckling analysis, a complete sphere under uniform extemal pressure was analyzed to investigate the sensitivity of the buckling load to geometric imperfections and the finite difference mesh size. The criticalload combinations, for buckling analysis of the System 80+*,
were identified on the basis of the stress analysis results as those that cause the largest compressive stress. In addition to the buckling analysis of the perfect shell, sensitivity studies were conducted with regard to variation of buckling load with imperfection wavelength, stiffness modulus of the transition region, the stress-strain const;tutive law of the containment material, and mesh size.
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STRESS ANALYSIS This section summarizes the stress intensities induced in the System 80+
containment shell considering the load combinations specified for Design Conditions and Levels A, B, C, and D as in the SRP 3.8.2 [2.1). The structural analysis of the containment shell was first performed for each individualload case utilizing the BOSOR4 finite difference program with the INDIC=0 option.
(The INDIC=0 option is an elastic axisymmetric stress analysis incorporating geometric nonlinear effects). The results were combined to calculate the stress intensities for the Design Conditions and Levels A, B, C, and D, which were compared to allowable stress intensities from the ASME Code [2.2).
2.1 Axisymmetric Modellino of the System 80+
Containment A cross-sectional elevation of the System 80+" containment is shown in Fig. 2.1.
The containment consists of a spherical steel (SA 537 - Class 2) shell with a radius of 1,200 in. and a wall thickness of 1.75 in. The major appurtenances are the spray header system, the equipment hatch, and two personnel locks. The components contribute weights of 120,000 lb.,
50,000 lb., and 120,000 lb. (for each of the two personnel locks), respectively. The portion of the containment building up to Elev. 86.3 ft. is embedded in a concrete foundation. Between Elev.
66.3 ft. and 90.1 ft., i.e., from the concrete embedment to the top of the concrete surface, the l
containment is embedded in a cork material (along the outside surface only) which provides a transition region.
The axisymmetric model of the System 80+
containment is shown in Fig. 2.2.
Maximum element length was limited to 20.94 in, which is less than (rt)v2/2 (see Sec. A.3.6.1). It is comprised of 12 BOSOR segments. The choice of segment size is dictated by the element size, the level of detail and the input requirements of BOSOR4. For example, the BOSOR4 program l
allows only 20 elements in a segment that is subjected to nonuniform pressure. In the axisymmetric model, the masses of the major penetrations and appurtenances were smeared around the circumference. The different shell segments and the corresponding mass densities are shown in Table 2.1.
The containment portion between Elev. 86.3 ft. and 90.1 ft. was modelled as a shell supported by an elastic foundation with a subgrade modulus of 180 lblin*,
i which is the same value as that for the cork in the design of the Cherokee containment [2.3]. The shell was assumed to be fixed at Elev. 86.3 ft. where the steel shell is embedded in the concrete.
2.2 Individual Load Cases 2.2.1 Dead Loads l
The weight of the structure is input as the normal and meridional components of pressure on the different elements and a stress analysis is performed using the INDIC=0 option of BOSOR4. The maximum meridional membrane stress of -1,123 psi occurs at the base of the structure (Elev.
86.3 ft.). The maximum circumferential membrane stress of 1,196 psi occurs at Elev. 94.1 ft.
The maximum extreme fiber stress in meridional direction of-2,120 psi occurs at the base of the structure. The maximum extreme fiber stress in the circumferential direction of 1,282 psi occurs at Elev. 94.1 ft.
2.2.2 Temperature Accident and operating temperatures of 290*F and 110*F, respectively, were considered. The 3
structure was assumed to be constructed at 70*F. The stress analyses was performed using the INDIC=0 option cf the BOSOR4 code. For a temperature differential of 220*F, the maximum stresses occurred at Elev. 86.3 ft. The membrane stresses with the maximum absolute value in i
the meridional and circumferential directions were 2,130 psi and -41,690 psi, respectively. Plots I
of extreme fiber stresses are shown in Fig. 2.3, with the abscissa indicating the elevation starting from Elev. 86.3 ft., at the base. The extreme fiber stresses with the maximum absolute value in the meridional and circumferential directions were 25,490 psi and -48,700 psi, respectively. For i
2 a temperature differential of 40*F, the maximum stresses also occurred at base. Plots of extreme
.l fiber stresses are shown in Fig. 2.4.
Their maximum absolute value in the meridional and circumferential diractions were 4,636 psi and -8,855 psi, respectively. The membrane stresses i
with the maximum absolute value in the meridional and circumferential directions were 387 psi i
and -7,580 psi, respectively.
2.2.3 Pressure The two uniform pressure loadings for System 80+
are intemal and extemal pressures of 49 and 2 psi, respectively. The stress analysis performed using the INDIC=0 option of BOSOR4 l
resulted in equal meridional and circumferential membrane stresses at points remote from the base with the stress values of 16,800 psi and -685 psi for the intemal and extemal pressures, respectively. These stress values are similar to those obtained by shell theory, j
2.2.4 Seismic Loading f
Seismic loading is a special case of nonaxisymmetric loading as discussed in Sec. A.3.
A response spectrum analysis of the structure for a base excitation is performed by combining the j
modal responses to estimate the system maximum response. One approach to estimate the l
maximum response or stress quantity, R,,, for a particular response quantity is to combine the l
modal response utilizing the SRSS method [2.4,2.5,2.6). For example, the maximum response, l
i R,.,,, caused by seismic excitations in the X, Y, Z directions can be calculated summarizing the modal responses as:
'*#2 3
m i
[ { R[y (2'I)
R
=
m k=1 y=1 j
where m is the number of modes and k corresponds to the directions X, Y, and Z. Equations i
similar to Eq. 2.1 can be used to calculate maximum displacements, stresses, and stress resultants.
i Despite the simplicity and practical advantages of the SRSS method, this method fails to take into account the effects of closely spaced frequencies on the overall response of a multi-degree of freedom system [2.7, 2.8]. Reference [2.8) suggests that modes should be considered to be closely spaced if their frequencies differ by less than 10 percent. If the modes are closely I
spaced, Eq. 2.1 should be modified as follows:
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3 m
p p
Rnl (2 2)
\\
R, =
[ Rl3 + 2 [ [ lRu
=1 g ]=1 l=1 r=1 n
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where p is the number of closely spaced modes.
Response spectra for Operating Basis Earthquakes (OBE) and Safe Shutdown Earthquakes j
(SSE) were considered. Figure 2.5 illustrates the OBE response spectrum for a soil condition -
designated as B4. Figures 2.6 through 2.8 show the response spectra for a SSE associated with three different soil conditions. Two horizontal (X and Y) and one vertical (Z) spectra are shown for each soil. The SSE spectra associated with the three soils were used to predict the seismic stress resultants at several different circumferential locations. The modal quantities were determined by performing a vibration analysis of the containment using the INDIC=2 option of modified BOSOR4 program (see Sec. B.2). An error in calculation of the modal participation factor by BOSOR4 was discovered during the preliminary check runs and corrected prior to performing the vibration analysis of the containment (see Sec. B.1). The SSE corresponding to soil condition B4 produced the largest seismic response.
The SRSS meridional and circumferential stress resultants, N and Nw, for mendians at 0, 45 and 90 degrees, are o
shown in Figs. 2.9 and 2.10 respectively. The meridian with the highest stress resultants, or the worst meridian, was in the YZ plane (or 0 equal zero in Fig. A.2).
2.3 Combination of Stresses Section 1.3 of SRP 3.8.2 [2.1) stipulates that the design loading combinations be in compliance with Subsection NE, Section 111, Division 1 of the ASME Code and Regulatory Guide 1.57 [2.9].
l The pertinent load combinations are listed in Tables 2.2 to 2.6 for the Design Condition and Service Levels A, B, C, and D. Table 2.7 defines the nomenclature used in Tables 2.2 to 2.6.
In addition, Tables 2.2 to 2.6 refer to the specific SRP 3.8.2 loading combination numbers.
r Subsection NE of the ASME Code [2.2 (Table NE3221.1)] has established allowable stress intensP.ies for the Design Conditions and Service Levels A, B, C, and D as listed in Tables 2.2 i
to 2.6. Allowable stresses depend upon the stress classification, i.e., primary or secondary (see Table 2.7).
The stress intensities were evaluated for each of the load combinations and the maximum stresses are listed in Tables 2.2 to 2.6.
3 One violation of the allowable membrane stress intensity limit occurred for the Level B Service Limit with an intemal pressure of 49 psi and the OBE (Load Combination (iii)(b)(1)). The maximum stress intensity in the vicinity of Elevation 95 ft. was approximately 20 percent over the allowable value. The stress intensities were 10 percent beyond the allowable between Elevation 91.5 ft. and 101.0 ft. The conservative nature of the axisymmetric analysis may have contributed i
to this slight over-stress.
2.4 References 2.1 U.S. NRC Standard Review Plan (SRP) Section 3.8.2, Rev.1 - July 1981, pp. 3.8.2.9 -
3.8.2.11.
2.2 American Society of Mechanical Engine.ers, Boiler and Pressure Vessel Code, Section NE3220,1989.
1 2.3 NUREG CR-3127, L. Greimann, et al., "Probabilistic Seismic Resistance of Steel Containments," prepared for NRC, Washington, January,1984.
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2.4 Clough, R W. and Penzien, T., Dynamics of Structures. New York, McGraw Hill,1964.
2.5 Biggs, J. M., Introduction to Structural Dynamics. New York, McGraw Hill,1964.
.l 2.6 Venmarcke, E. H., " Structural Response to Earthquakes," Chapter 8, Seismic Risk and Enoineerino Decisions, B. C. Lominitz and E. Rosenbluth, Editors, Elsevier Fublishing Co.,
1976.
2.7 Wilson, E. L., Kiureghuian, A., and Bayo, E. P., "A Replacement for the SRSS Method in Seismic Analysis," Earthouake Encineerino and Structural Dynamics. University of Califomia. Berkeley, Vol. 9, No. 2, March 1981.
2.8 American Society of Civil Engineers, Manuals and Reports on Engineering Practice No.
58, Structural Analysis and Desion of Nuclear Plant Facilities.1980.
2.9 U.S. NRC Regulatory Guide 1.57, " Design Units and Loading Combinatio1s for Metal Primary Reactor Containment System Components."
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l 3.
COMPLETE SPHERE WITH EXTERNAL PRESSURE Appendix A outlines shell buckling assessment techniques and guidelines as per the ASME Boiler -
and Pressure Vessel Code and the SRP 3.8.2.
To demonstrate the procedures outlined in Appendix A and prior to the analysis of the System 80+ containment, a complete spherical shell was analyzed. The complete sphere has a radius, r, of 1200 in. and a thickness, t, of 1.75 in.
A modulus of elasticity of 29,000 ksi was used for the material. The loading is uniform extemal pressure. The analysis was accomplished using ASME Section NE3133, ASME Code Case N-284, and the BOSOR finite difference program.
{
3.1 Section NE3133 The complete sphere was analyzed using Section NE3133.4 of the Code. The allowable extemal pressure was calculated by the equation in step 5 as 3.9 psi.
F 3.2 Analysis Usino Code Case N-284 i
The complete sphere was also ans.lyzed using the Code Case N-284 dated August 25,1980.
The steps involved in N-284 are: determination tef. tresses in the shell wall under extemal pressure; determination of the capacity reduction factor, n, and plasticity reduction factor, n; and, i
enhancement of the calculated stresses by the reduction factors. The enhanced stresces are compared to the calculated theoretical buckling values. Sirre the shell is under equal biaxial compressive stress, the capacity reduction factor, a, was calculated by Article 1512(b) of N-284 as 0.124. The plasticity reduction factor q was calculated according to the Code Case as 1.0, i.e., elastic buckling controls (see Article 1620(a) of Code Case N-284). The theoretical buckling values were calculated as per Article 1712.1.3(a) of Code Case N-284 as 74.63 psi. The predicted buckling pressure, which is at times y times the theoretical buckling pressure, is 9.25 psi. Utilizing the factor of safety of two in N-284 for design conditions, the allowable extemal pressure is 4.6 psi, somewhat larger than that calculated using Section NE3133.
3.3 BOSOR Analysis The axisymmetric BOSOR4 analysis of the complete sphere was accomplished following Sec.
i A.3.G. The vessel was modelled using three shell segments with a total number of 183 mesh points (1 degree central angle). The finite difference point spacing of 20.94 in.,is some what less than (rt)u2/2 (see Sec. A.3.6.2). The BOSOR4 buckling pressure of 74.6 psi compares well to the classical theoretical value of 74.63 psi.
Sensitivity studies on the complete sphere were conducted with regard to mesh size, imperfection amplitude and imperfection wavelength. The study was performed considering three different element sizes of 0.5,1, and 2 degree central angles, or lengths of 10.47 in.,20.94 in. and 41.89 in., respectively. There was no significant difference in the buckling load values. An element of 20.94 in. was selected for all subsequent analyses.
An axisymmetric imperfection for the sphere is mode!!ed as a sinusoidal wave of the form shown t
in Eq. A.7 in Appendix A. The imperfection wavelength,1,, was taken as:
Li = K (r c) 1/2 (3.1)
I where K is the wavelength parameter. To perform an imperfection sensitivity analysis based on l
7 i
._ q wavelength, the imperfection peak to trough amplitude was fixed at 3.5 in., while varying the wavelength from 92 in. to 628 in., which corresponds to K values of 2 to 13.7, respectively. This imperfection amplitude is twice the allowable ASME value of 1.75 in. The larger imperfection amplitude was used to approach the N-284 value of the buckling pressure, which is equal to the theoretical value modified by a capacity reduction factor. The results of the analysis are presented in Fig. 3.1.
Note that the imperfection wavelength corresponding to the minimum buckling load is 3.92 (rt)'8 which is about equal to the 3.5 (rt)'8 value recommended by Koiter, as presented in [3.1] (see Sec. A.3.4).
The imperfection wavelength was then fix2d at a K value of 3.92 and the imperfection amplitude, W,, varied. Figure 3.2 illustrates that the buckling load continues to decrease with an increasing imperfection amplitude. The BOSOR buckling load approaches the N-284 predicted buckling value of 9.2 psi for a peak to trough imperfection amplitude equal to twice the thickness. This is twice the ASME recommended maximum allowable deviation of one thickness (see Sec. A.3.4).
7 i
The buckled wave shape and imperfection shape are shown in Fig. 3.3. The latitude is plotted i
as the abscissa. The buckled wavelength is about twice the wavelength of the imperfection. This is consistent with Koiter's observation for cylinders as reported in [3.2].
3.4 References 3.1 Brush, D. O. and Almroth, B. O., Bucklino of Bars. Plates and Shells. Chapter 7, New l
York, McGraw Hill,1975.
3.2 Teng, J. G. and Rotter, J. M., " Buckling of Pressurized Axisymmetrically Imperfect f
Cylinders Under Axial Loads," ASCE Engineering Mechanics Joumal, Vol.118, No. 2, i
Feb.1992.
I i
i 8
i 4.
BUCKLING ANALYSIS OF SYSTEM 80+
CONTAINMENT This section summarizes the buckling analysis results for the System 80+
containment. The
[
analysis was performed to determine the buckling factors of safety for Service Levels A, B, C, and i
D.
4.1 Loadino and Solution Process 1
The process of determining the seismic loads for the buckling analysis of the containment l
consisted of performing a vibration analysis of the containment, combining the modal stress
[
quantities by the SRSS me' hod, and determining the equivalent axisymmetric pressures corresponding to the modal stress quantities (see Appendix C). The equivalent axisymmetric pressures approach converts the nonsymmetric seismic loading into an equivalent axisymmetric loading which gives the same maximum stress resultants on the most highly stressed meridian, which, as shown in Sec. 2.2.4 was in the YZ plane or is at a 0 value of zero degrees (see Fig.
l A.2). As summarized in Appendix C, the equivalent axisymmetric pressures are determined with meridional and circumferential stress resultants in eithec (1) compression and tension respectively, or (2) tension and compression respectively.
(Both the meridional and circumferential stresses can not be in compression at the same time since this would be incompatible with the modal quantities). This process is performed by the LOADS program (Appendix C). The input to the LOADS program consists of the vibration modes, modal stress quantities, the pertinent earthquake response spectrum. The LOADS program also adds the extemal pressure and dead loads. Thermal loads are input directly into BOSOR 5. The output l
corisists of equivalent axisymmetric pressures in the meridional and normal directions as j
segment-wise input files for buckling analysis.
The capabilities of the BOSOR programs are summarized in Appendix A. The stress-strain curve, f
derived from equations in ASME Code Case N-284, is summarized in Sec. A.3.6.2. The factor 3
of safety against buckling, A,is defined as a load multiplier which is equal to the ratio of the loads j
at which buckling occurs to the input loads. In addition to the input loads, the initial load multiplier and its increment are specified. The buckling analysis phase consists of prebuckling and eigen
{
buckling stages. During the prebuckling stage, the analysis is performed for the current load multiplier and the stability detenninant value for the specified starting number of circumferential waves is evaluated. The loads are then incremented and the process is repeated. If there is a change in the sign of the de'erminant, the solution moves to the eigen buckling stage. Within a j
specified range of circumferer.tial wave numbers, the wave number associated with the minimum i
A is determined. This minimun A represents the predicted buckling factor of safety.
l The stress analysis of the System 80+" containment was performed for the load combinations f
in Tables 2.2 through 2.6. On the basis of the stress analysis results, the controlling load case i
for buckling (largest compressive stress) will be inadvertent actuation of the spray system (extemal pressure of 2 psi and temperature of 110 F) and an earthquake corresponding to soit j
condition B4. Service Level A does not have seismic loading. Level B, which has OBE, will therefore, have control over Level A since both have the same factor of safety (see Table A.1).
j The loadings for Levels C and D are the same (compare Tables 2.5 and 2.6). As Level C has a larger factor of safety requirement, it will have control over Level D.
Level C results are i
~
I presented first in the following discussion.
i f
1 4
4.2 Level C Service Limits i
The contro!!ing load combination for buckling in Service Level C is combination (iii)(c)(1) of SRP 3.8.2 (see Table 2.5) which includes the SSE corresponding to soil condition 84. As a point of l
reference, an elastic analysis of the perfect containment shell was performed. The A value was determined to be 3.92. Material nonlinearities (see Fig. A.14) were next introduced into the perfect shell model and a A value of 2.4 was obtained.
4 The predicted buckling load was determined by introducing geometric imperfections in addition to the material nonlinearities. The imperfection sensitivity analysis was performed by varying the imperfection wavelength parameter, K (see Sec. 3.3).
The imperfection is modelled as a sinusoidal wave with a radial imperfection amplitude of 0.875 in. which corresponds to the Code specified maximum deviation of one shell thickness (1.75 in.) (see Sec. A.3.4). The starting wavelength corresponded to a K value of 3.92, which was the critical wavelength obtained from the analysis of the complete sphere under extemal pressure (see Sec. 3.3). For this imperfection, the material model in Fig. A.14 was used. The A value obtained was 1.9. The imperfection l
wavelength parsmeter was next varied from 2 to 4.5 in increments of 0.5, corresponding to i
wavelengths of 92 in. to 205 in., respectively. The variation of the predicted value of A is shown in Fig. 4.1. The minimum buckling load multiplier of 1.59 corresponds to a K value of 2.5 or an imperfection wavelength of 114.5 in. As an observation, the buckling load multiplier is not as sensitive to imperfections as a shell subjected to uniform extemal pressure alone (compare Fig.
3.1 and Fig. 4.1).
l 4
The above imperfection sensitivity analyses were performed with the meridional stress resultants in compression and the circumferential stress resultants in tension (see Sec. 4.1). The process was raeated assuming the circumferential stress resultants to be in compression and the meridional stress resultants in tension. The results of the latter analysis, which are shown in Fig.
4.2, give a minimum buckling load multiplier of 3.6. Hence, this case does not control.
4.3 Level B Service Limits The controlling load combination for Level B is combination (iii)(b)(1) of SRP 3.8.2 (see Table 2.4), which includes the OBE corresponding to soil condition B4. Following the steps summarized above, the variation of buckling load multiplier with imperfection wavelength parameter is presented in Fig. 4.3. The minimum value of A is 2.3.
l 1
4.4 Controllino Case As a summary of the above results, the controlling buckling load multiplier, A, has a value of 1.59 for Level C Service Limit. The imperfection wavelength is 114.5 in. with a Code deviation of 1.75 in. The buckled shape is shown in Fig. 4.4. The buckled shape was axisymmetric, i.e., the number of circumferential waves is zero. As described earlier (see Sec. A.3.3 and Sec. A.3.4),
the analysis assumes that the stress resultants and the imperfections are uniform over the entire circumference and equal to the largest values from nonaxisymmetric analysis.
From the prebuckling analysis, the variation of the effective uniaxial strain, of the middle surface and at the elevation of 95.4 ft., is shown in Fig. 4.5. Note that, corresponding to the maximum effective strain of 0.23%, the maximum middle surface effective stress is about 45,000 psi (see Fig. A.14), which is above the proportionallimit of the N-284 derived stress-strain curve. Hence, the buckling of the containment is not elastic and is associated with significant plastic straining, 10
l as shown Fig. 4.5. As an additional observation, the maximum surface strain of 0.46% occurs on the inner surface of the containment at the A value of 1.59.
Figure 4.6 shows the variation of radial displacement with load multiplier at the elevation 95.4 ft.
The prebuckling displacements are small and are probably within the constraining limits such as bellows expansion joint limits. The post buckling displacements are not predicted.
4.5 Foundation Modulus and Mesh Sensitivity Studies 8
The foundation modulus for the controlling case was 180 lblin. The effect oithe variation of the foundation modulus was studied for two different foundation moduli: 90 lblin' and 360 lb/in'. The i
variation in the A value appears to be insignificant.
The maximum element size for the above analyses was 20.94 in. As a final mesh size study, the mesh size was doubled to a maximum element size of 41.9 in. The A value was 1.61 indicating that the mesh size variation had very little effect on the controlling load multiplier.
4.6 Discussion The minimum factor of safety against buckling prescribed in Section NE3222.1 of the ASME Code is 3.0 for Design Conditions and Level A and B Service Limits,2.5 for Level C Service Limits, and 2.0 for Level D Service Limits (see Table A.1). On the basis of the above analysis, the factor of safety for Level B is 2.3 and for Level C is 1.59, which are less than the recommended design values. Code Case N-284 prescribes safety factors of 2.0 (Level A and B),
1.67 (Level C) and 1.34 (Level D) (see Sec. A.2.2 and Table A.1). The predicted factors of safety satisfy these criteria for Levels A, B, and D and very nearty satisfy Level C. Regulatory Guide 1.57 (Appendix A) prescribes a factor of safety for buckling of 2.0 without identifying the relevant Service Levels or load combinations. However, it does mention that the factor should be applied to the case causing maximum compressive stress which, in this situation, is Levels C and D.
Hence, Regulatory Guide 1.57 is not satisfied.
The analysis is conservative because of the following assumptions and procedures:
(1)
The stresses are assumed to be uniform around the circumference and equal to their maximum values from a nonaxisymmetric analysis (see Sec. A.3.3).
(2)
The imperfection is assumed to be axisymmetric, which is conservative [4.1].
(3)
The imperfection is assumed to have the largest deviation permitted by the ASME Boiler and Pressure Vessel Code and to have a wavelength which causes minimum buckling load (see Sec. A.3.4).
(4)
The material nonlinear stress-strain curve is assumed to have a proportional limit occurring at 0.55 times the yield stress (see Sec. A.3.6.2).
4.7 References 4.1 Teng, J. G. and Rotter, J. M., " Buckling of Pressurized Axisymmetrically Imperfect Cylinders under Axial Loads," ASCE Engineering Mechanics Joumal, Vol.118, No. 2, Feb.
1992.
l I
11 m
5.
SUMMARY
, CONCLUSIONS AND RECOMMENDATIONS 5.1 Summary The objective of the present work is to review the structural design of the CE System 80+*
containment. An analysis was performed to check stress levels against the ASME Code
{
requirements. The adequacy of the structure against buckling was reviewed by finite difference i
analysis software.
f The System 80+
containment is a spherical steel shell stmeture with 1,200 in. radius and 1.75 in. thickness. It is embedded in a concrete base with flexible support in a transition region. The j
containment cross section inscribes a central angle of 270 degrees. The steelis Type SA - 537 class 2. The major penetrations are the equipment hatch and two personnellocks. The major j
appurtenance was the spray header system.
[
SRP 3.8.2 stipulates that the design and analysis procedures be in compliance with the ASME Boiler and Pressure Vessel Code. Loadings on the structure include dead load, operating I
conditions, design basis accident, inadvertent spray actuation and seismic loading. SRP 3.8.2 further prescribes the load combinations pertaining to Design Conditions and Service Limits A, B, C, and D as classified ' y the ASME Code. The Code permits buckling evaluation by design j
o type equations or numerical approaches. The different buckling analysis techniques were summarized in Appendix A of this report. Section NE3222.1 of the Code establishes the l
admissible factors of safety against buckling for Design Conditions and Service Levels A, B, C, j
and D.
j
?
The numerical method of analysis was performed with the BOSOR4 and BOSOR5 finite j
difference software. The containment was modelled as an axisymmetric shell consisting of
[
different segments and mesh points.
The choice of the segments was dictated by the l
penetrations and appurtenance. The additional mass of the penetrations and the appurtenance was smeared around the circumference. The transition regiun was modelled using elastic springs
[
8 with a foundation modulus of 180 lbs/in.
The stresses due to the individual loads (dead loads, intemal and extemal pressures and temperatures) were computed using the stress analysis option in BOSOR4. The seismic loading i
I to the structure was considered in the form of earthquake response spectra for different soil conditions. Modifications were made to the source code of the vibration analysis part of BOSOR4 l
to obtain the modal stress quantities and correct the modal participation factor calculation. The modal stress quantities were combined by the SRSS method. Seismic stresses for several soil j
conditions and several meridians were compared to select the controlling seismic case. The stresses from individual loads and seismic event were combined according to SRP 3.8.2 into j
stress intensities and were found to satisfy the allowable limits, except in a small zone just above I
the transition region.
l As a prelude to the buckling analysis of the System 80+ containment, a complete sphere with uniform extemal pressure was analyzed using the BOSOR4 program. The buckling pressure calculated by BOSOR4 for the perfed shell agreed well with the classical theoretical value. Next, an axisymmetric, sinusoidal imperfection was incorporated into the BOSOR model. The variation of the buckling load values with respect to the imperfection wavelength and the imperfection amplitude were studied and compared to the ASME Code Case N-284. The BOSOR buckling load, for an imperfection amplitude of 1.75 in. (twice ASME tolerance) and an imperfection l
12
i i
l I
wavelength of 179.7 in., approached the predicted N-284 buckling value with a capacity reduction factor of 0.124.
l 5
The loading combination, for the Suckling evaluation of System 80+*, was chosen on the basis cf the stress analysis results as that loading which resulted in the largest compressive stress.
11 was inadver'ent spray actuation (extemal pressure of 2 psi and a temperature of 110*F) in l
addition to seismic events (OBE or SSE). The buckling analysis was accomplished using the i
BOSORS program. Since BOSORS does not accept stress quantities for input, the SRSS stress
[
quantities were transformed into equivalent axisymmetric pressures. The buckling analysis was i
performed on the basis of the worst meridian assumption; that is, the maximum stresses are assumed to exist uniformly around the circumference.
The material nonlinearities were j
incorporated using a stress-strain constitutive relationship derived from the equations for the i
plasticity reduction factor given in Code Case N-284. The effects of the residual stresses were incorporated using a reduced proportional limit.
l t
in general, the predicted buckling load for a structure, is evaluated as a load multiplier or a factor of safety times the applied loads. For the System 80+, the perfect shell with an elastic material i
was initially analyzed.
The sinusoidal imperfection was introduced and the imperfection wavelength was varied to identify the critical imperfection configuration. The factors of safety values were 2.3 (Level B) and 1.59 (Levels C and D), respectively. Finally, sensitivity studies which were conducted to investigate the effects of mesh size and transition zone stiffness showed little effect on the predicted buckling load.
5.2 Conclusions On the basis of the analyses performed herein, the following can be concluded:
t (1)
Based on the stress analysis, all stress intensities were below the allowable limit, as specified in Section NE3221 of the code, except for a zone above the transition i
region, which was slightly over-stressed for Service Level B. The stress analysis was conservative.
(2)
The predicted buckling factor of safety is 2.3 for Service Level B and 1.59 for Service Levels C and D. These values are conservative because: (a) the analysis was performed using a two-dimensional axisymmetric code, with the stresses assumed to be uniform around the circumference and equal to their maximum value; (b) the imperfection, based on the Code recommended tolerances, was also assumed to be axisymmetric; and, (c) the material model was assumed to have a reduced proportional limit at 0.55 times the yield stress to account for residual stresses. The calculated factors of safety do not satisfy the requirements of l
NE3222.1 or Regulatory Guide 1.57. However, they do very neariy satisfy Code Case N-284.
(3)
The buckling load, as govemed by the combined loading, is not as sensitive to
[
imperfections as a shell subjected to uniform extemal pressure alone.
(4)
The buckling load is not much affected by the variation in the foundation modulus of the material in the transition region.
(5)
The material model incorporating residual stresses predicts a larger zone of 13
i plasticity and a lower buckling load than the elasto-plastic one.
5.3 Recommendations in order to gain further understanding of the buckling of the System 80+*, the following recommendations are made:
f (1)
The behavior of the structure needs to be examined using a three-dimensional finite element code. Most of the conservative assumptions involved in the solution 3
(see Sec. 4.4 and Sec. 5.2) can be relaxed only if a complete three-dimensional j
model is studied. However, a three-dimensional analysis would require more computational effort.
l (2) in a three-dimensional code, nonaxisymmetric imperfections depicting a state nearer to true fabricated she!!s can be represented. However, a representative imperfection shape must be established using Code specified tolerances, measured insitu imperfections or randomly introduced imperfections.
(3)
To ascertain actual shell failure, i.e., containment leakage, a study of the post
{
buckling behaviorwould be necessary. The extent and magnitude of post buckling strains could be compared to strain failure criteria to predict shell failure. The extent of the buckling mode (local versus general) could be predicted.
i (4)
The design-type equations approach, e.g., N-284, and the finite element work i
should be calibrated with test data on spherical shells, if possible, for both static and dynamic tests. The evaluation should be reliability based and utilize statistical j
principles (see Sec. A.2.2).
i f
l i
i i
J 14 i
1
t i
APPENDIX A BUCKLING ASSESSMENT TECHNIQUES A.1 The Shell Bucklina Problem Shell buckling is not a well understood phenomenon. The analysis of the elastic (and inelastic) buckling strength of thin shells has long frustrated structural engineers. Two classic problems -
that of an axially compressed thin circular cylinder and that of a sphere under extemal pressure -
- which appear on the surface to be simple, have proven to be extremely complex. For these cases, the buckling load predicted by classic, small displacement elastic theory is well above observed experimental results (by 3 to 5 times). It is beyond the scope of the current work to provide a complete review of the state-of-the-art in this area, but the reader is referred to [A.1, A.2, A.3, A.4, A.5, A.6, A.7] for good summaries of this topic.
1 Many explanations and theories have been advanced to reconcile the difference between prediction and reality. There is now general agreement that the discrepancy is due to imperfections in the shell, i.e., deviations of the shell from the ideal. These imperfections include geometric imperfections (deviations from a perfect shape), material imperfections (residual stress),
and boundary condition imperfections (geometric constraint conditions and applied load eccentricities). Undoubtedly these parameters are of random nature and are not well defined at this time of design. Remedies for the discrepancy between theory and experiment involve varied approaches ranging from theoretical models that include the imperfection (as-built shapes and materials, weld and fabrication residual stresses) to classical analysis reduced by margins or knockdown factors which reflect the difference between theoretical and experimental load capacities. Typically the former are too expensive and the lat'er too crude. However, a theoretical model enables the analyst to include the imperfection parameters (if known). Most current approaches represent a combination of these two extremes.
Before proceeding to ccasideration of various shell analysis techniques, a conceptual description of the instability behavior of shells is appropriate. (This follows [A.1 and A.2].) Figure A.1 schematically illustrates the possible load-displacement behavior of a shell, e.g., a cylindrical shell under axial load. Loadirig may progress along a fundamental path OBA. The deformed shape remains essentially the same along this path with only the magnitude of the displacements increasing, e.g., axisymmetric shape for axial tension or compression, first Fourier component for pure bending. Eventually, an instability occurs along this path, point A, at a point of maximum load. This point is variously referred to as the limit point buckling or" snap-through" or the plastic collapse load. For this behavior, the " failed" deflected shape differs from the initial deflected shape only in magnitude.
For many shell configurations and loadings, e.g., axial compession of cylindrical shell, an attemative equilibrium path is available. That is, a bifurcation point B exists at which two equilibrium shapes are possible. At this point, the shell will follow the path of the least energy, path _ BC. This behavior is termed bifurcation buckling. The postbuckling path is called a secondary path. Deformations along this path differ completely from the prebifurcation path, e.g.,
symmet.ic buckling of an axially compressed cylinder in which wrinkles or lobes form around the circumference. Elastic buckling ocews if the shell is elastic at the bifurcation point, i.e., point B is below point Y. If B is above Y, inelastic (or plastic) buckling occurs.
The purpose of shell analysis is to predict the load displacement behavior, i.e., Path OBA or OBC in Fig. A.1. No postbuckling enalysis will be performed in the current work. For almost all shell 15
problems, the postbuckling path is unstable (as shown) and the structure has reached its maximum loads. For certain structures, e.g., flat plates with in-plane compression and ellipsoidai heads with intemal pressure, the postbuckling path is stable. It will here be assumed that the postbuckling path is unstable and carries the shell well into the inelastic range, i.e., to strains several times the yield value. This is a conservative assumption. For example, for the case of elastic buckling, strains may remain below yield and the structure could retum to its undeformed I
l state upon removal of the load.
Current shell buckling analysis techniques can be (loosely) subdivided into two types:
Design type equations in an analytical form with appropriate corrections for imperfections [A.5 (Papers No. 10,10,17,18), A.8]. Typically these equations are for special, but common, cases (e.g., stiffened and unstiffened cylinders, cones, spheres).
Numerical (finite element) type approaches which can handle more general shell shapes.
Unfortunately, these solutions must also be empirically corrected for imperfections and are very expensive [A.3, A.4, A.5 (Papers No. 5,19), A.6, A.9].
The ASME Code Subsection NE, Section lil, Division I [A.11] permits these two analysis techniques. SRP 3.8.2 [A.10] stipulates that the design and analysis procedures for steel containment structures be in compliance with the Code as augmented by Regulatory Guide 1.57
[A.15].
A.2 The ASME Boiler and Pressure Vessel Code and Reculatory Guide 1.57 Section NE3222.1 of the Code specifies the basic allowable compressive stress for the stability of structures as:
"The maximum buckling stress values to be used for the evaluation of instability 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, nonlinearities, large deformations, and inertial 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 of restraint and loading the same as those to which the confiamtion is expected to be subjected.
(b)
The value determined by the applicable rules of NE3133."
As per Section NE3222.2, the stability stress limits for various loading conditions, such as the Design Conditions and Service Limits A, B, C, and D, have the factors of safety listed in Table A.1.
16
r Method (a)(2) refers to the prediction of buckling strength by classical methods as reduced by specific margins, often referred to as knockdown factors. This method corresponds to the design type equation approach described in Sec. A.1. ASME Section NE3133 (Method (b))and Code Case N-284 utilize this method as summarized in Sec. A.2.1 and Sec. A.2.2, respectively.
Method (a)(1) corresponds to the numerical approach described in Sec. A.1.
The BOSOR analysis summarized in Sec. A.3 satisfies thesc criteria. Method (a)(3) involves physical testing and will not be discussed further.
A.2.1 Section NE3133 As referred to in Section NE3222(b), Section NE3133 of the Code establishes general rules for determining the thickness under extemal pressure loading for different structural geometries such as cylindrical she!!s and spherical shells. It outlines design procedures which establish the thickness of the component under investigation as a function of shell parameters (the radius and shell thickness in case of as spherical shell) and material type. The design procedure involves i
j the evaluation of allowable extemal pressure using two factors, A and B, which are calculated using the shell parameters and material and temperature charts in Appendix Vil of the Code.
A.2.2 Code Case N-284 ASME Nde Case N-284 (A.12) provides well-defined stability criteria for detarmining the structural adequacy against buck!ing of shells with more complex geometries and loading conditions than those covered by Section NE3133. The rules are based on linear elastic bifurcation buckling theory which has been reduced by knockdown factors to account for the effect of imperfections, boundary conditions, material nonlinearities and residual stresses. It is based on the work collected by Miller [A.5 (Paper No.16)]. The design equations used in the Code Case N-284 are established by comparison of theoretical buckling values and test results. The theoretical values are predicted by classical linear elastic buckling theory considering a shell with perfect initial shape and perfect elastic behavior. Two knockdown factors or margins (capacity and plasticity reduction) are applied. The capacity reduction factor, et, is geometry dependent and is evaluated for various shell geometries and buckling modes (local and general). The plasticity reduction factor, q, is stress dependent and is also evaluated for various shell geometries and buckling modes (local and general). The design equations for the capacity and plasticity reduction factor are a result of a lower bound fit to the test data [ Article 1400 of A.12). In addition to the Code Case N-284, Miller's equations are used in an American Petroleum Institute bulletin for the stability design of cylindrical shells [A.13).
Code Case N-2B4 also outlines buckling design procedures for use with computer analyses. The computed theoretical buckling stress is reduced by the same two factors, et and q. In addition, interaction equations must be satisfied for combined loading.
As per Code Case N-284, the stability stress limits for various loading conditions, such as the Design Conditions and Service Limits A, B, C, and D, correspond to the factors of safety shown in Table A.1. The factors of safety are lower than those specified by NE3222.2 of the Code, but l
are consistent with other ASME factors of safety for other failure criteria, e.g., yielding due to intemal pressure [A.14). One could justify use of the lower factors of safety of Ccds Case N-284, l
since the knockdown factors fit a lower bound of the test data. Although difficult to prove, the factors of safety established in Section NE3222.2 of the code may have been developed as an extra precarson against the typical wide spread in shell buckling test data. The spread would dictate a Egher factor of safety if the knockdown factors are derived from the mean of the test t
17 i
data. Consistent rules for factor of safety selection could be developed only with a thorough statistical study of the test data. Buckling factors of safety should oe the same as other factors of safety if the analysis techniques consistently fit the same statistical property of the test data, e.g., mean, mean mirius one standard deviation,90% lower confidence interval.
Code Case N-284 recommends that circumferential variations of geometry and material properties be smeared around the circumference for an axisymmetric stress analysis. For nonaxisymmetric loading conditions such as in a seismic event, the Code Case N-284 recommends that a concurrent maximum value set (longitudinal compression, circumfereritial compression, in-plane shear) be assumed to act uniformly over the entire circumference. This analysis is termed as the
" worst meridian" approach (see Sec. A.3.2 and Sec. A.3.3). Code Case N-284 also recognizes that locally high stress states may exist near discontinuities. Such stresses may be averaged over a distance of (rt)v2 on either side of the peak stress.
A.2.3 Regulatory Guide 1.57 The U.S. NRC Regulatory Guide 1.57 [A.15] delineates the acceptable design limits and appropriate loading combinations associated with normal operation, postulated accidents and r
I specified seismic events for the design of components of metal primary reactor containment systems. The regulatory guide recognizes the design limits as specified in Code Section NE3222 i
in regard to the containment function and the combination of design loadings in the regulatory l
position. However, the guide states that if a detailed analysis is performed, note 7 to the l
l i
regulatory position applies. Note 7 explicitly states that:
i
If a detailed rigorous analysis of shells that contain the maximum a!!owabic deviation from
[
inae theoretical form is performed for instability (buckling) due to loadings that induce j
compressive stresses, such analyses, considering inelastic behavior, should demonstrate
[
that a factor of at least two exists, between the critical buckling stress and the applied i
stress."
The factor of safety of two against buckling is not explicitly associated with a specific Service Limit such as Design Conditions or Level A. However, Regulatory Guide 1.57 mentions that the loading l
combinations should encompass that loading which produces the greatest potential for shell l
instability. Hence, this factor can be associated with Level C and D service limits, which usually i
produce the greatest compressive stress in the shell since they are associated with the SSE
]
event.
1 A.3 Numerical Accroach 1
Within the numericsi approaches, two attematives are available [A.1, A.3, A.6, A.9, A.16):
i l
A complete three-dimensional model of the shell using two-dimensional shell elements.
This model can be used to model both meridionally and circumferentially varying imperfections and loads. Penetrations can also be included in the model.
An axisymmetric model of the shell with one-dimensional axisymmetric shel! elements.
Only axisymmetric imperfections and loads may be modelled.
Code Case N-284 also identifies these two methds.
1 18 l
l
.. -. ~.
t 1
i From an analytical point of view, the first choice is preferable. This method has, to date, been i
used primarily in a research mode for analyzing experimental models with carefully measured l
(known) imperfections for localized analyses, e.g., openings and penetrations [A.9, A.16]. Results have been interesting and useful, but not necessarily more accurate than those for the simpir axisymmetric model. Froni a practical viewpoint, the first approach is quite expensive, and the authors agree with [A.16), that the second is, at this time, a better choice.
In the remainder of this section, guidelines for buckling analysis of nominally axisymmetric shells with an axisymmetric model will be presented. Approximate incorporation of nonaxisymmetric features will be emphasized.
A.3.1 BOSOR4 and BOSORS Programs Capabilities The BOSOR programs are finite difference codes for the analysis of complex branched shells of revolution [A.17, A.18). BOSOR4 can handle both axisymmetric and nonaxisymmetric loading; however, it is limled to linear material properties. BOSORS was developed from BOSOR4 and is designed for axisymmetric and nonaxisymmetric buckling analyses resulting from axisymmetric loading only, but it does handle nonlinear material behavior. Both programs are based on the finite difference energy method.
BOSOR4 performs stress, buckling and modal vibration analyses of smooth or ring-stiffened, branched, segmented shells of revolution with compax wall constructions, loaded either axisymmetrica!!y or nonsymmetrically. Progtam branches include large-deflection axisymmetric stress analysis, small-deflection nonsymmetric stress analysis, modal vibration analysis with axisymmetric nonlinear prestress, and buckling analysis with axisymmetric or nonsymmetric i
prestress. Realistic engineering details such as interior supports may also be modelled. The i
program has restart and graphics capability.
BOSOR5 has also been widely used. However, it does not supersede BOSOR4, since it does t
not perform modal vibration or linear nonsymmetric stress analysis. The program can handle i
segmented and branched axisymmetric shells with material nonlinearities. BOSORS also has restart and graphics capability.
The BOSOR programs were chosen over other programs for several reasons. _ They are, i
probably, the most widely recognized buckling programs for axisymmetric shells. In addition, the programs are not general purpose, but rather are designed only for she!!s of revolution and, as l
such, the input p meters are relatively few. Additionally, interactive versions of the BOSOR programs are edu.! e on workstations with plots of the undeformed and deformed structures.
For the present analysis, the BOSOR4 program is used for vibration and elastic buckling analyses and BOSOR5 for the inelastic buckling analysis.
A.3.2 Nonaxisymmetric Loads Many loadings for shells of revolution are not axisymmetnc, i.e., the shell stress resultants vary around the shell circumference as in the case of overall bending, in this case, the meridional, circumferential, and normal displacements, i.e., u, v, and w can be expressed by Fourier series in terms of the circumferential angle, 0, (see Fig. A.2): in which n represents the symmetric i
Fourier terms,ii,,, % and R, and m represents the asymmetric termsli,( and %. In the above i
m representation n and m correspond to the circumferential wave numbers in the prebuckled shape.
19
eo ao u = { II, sin n0 + { 17, cos m0 (A.1) n=1 m=1 eo ao v = { V, cos n0 + { F, sin m0 j
n=1 m=1 i
ao eo v = { F, sin n0 + { F, cos m0 n=1 m=1 Equations similar to Eq. A.1 can be used to express stress resultants and stresses. For a vioration analysis, the expansion is also used for the modal quantities.
The seismic loading on the containment structure is considered as a special case of nonaxisymmetric loading. The dynamic equilibrium equation for horizontal and vertical ground excitation for a multidegree of freedom system can be written as [A.22, A.23].
[M}{0}+[C]{0}+[K){u} = -R,[M]{d,},[M]{d,},[M){d,}
(A.2) where [M] is the mass matrix, [C] is the camping matrix, [K) is the stiffness matrix, and X,, ?,, and I, are the ground accelerations in the X, Y, Z direction, respectively (see Fig. A.3).
The vectors {d,}, {d,} and {d,} relate a unit static displacement at the base in the X, Y, Z direction, respectively, to the relative displacement at the degrees of freedom. For example, for a structure with three degrees of freedom at each of its nodes, i.e., translation in the X, Y, and Z directions, the vectors {d,}, {d,}, {d,} are:
{d,} = {1 0 0 1 0 0.. 1 0 0}'
{d,} = {010 010... O d 0}'
(A.3)
{d,} = {0 010 01.. 0 01}'
Using modal analysis techniques [A.23], Eq. A.2 can be uncoupled into the modal coordinates.
The modal participation factors are defined by:
{r,} = [Mf[o)'[M){d,}
{r,} = [Mf[o)'[M}{d,}
(A.4)
{r.} = [Mf[o)'[M){d,)
where:
[MJ = [o)'[M][0]
(A.5) is the generalized mass matrix. The matrix [o] is a rectangular matrix containing the eigen vectors.
If the ground displacement is expanded into a Fourier series similar to Eq. A.1, a motion in the directica corresponds to the symmetric (n=1) terms, a motion in Y-direction corresponds to the asymmetric (m=1) terms and a motion in direction Z conesponds to the asymmetric (m=0) 20 e
I t
terms. Hence, one tmds the modal participation factors to be zero for all n and m except for n=1 with X motion, m=1 with Y motion and n=0 with Z motion as illustrated in Fig. A.4.
A.3.3 Elastic Buckling of Nonaxisymmetrically Loaded Perfect Shells i
T When BOSOR4 or BOSORS is used for the buckling analysis, the circumferential va'iation of the stress resulants is not permitted. Buckling of nonsymmetrically loaded shells can be handled by the worst meridian approach [A.4, A.19).
Code Case N-284 [ Article 1720 of A.12] also recommends this approach. The nonsymmetric stress resultants are first calculated for a typical i
meridian by the Fourier series. A buckling analysis is conducted by assuming that these stress resultants are uniformly distributed around the entire shell circumference. Several meridians were examined to locate the " worst" case, i.e., the case with the lowest buckling load.
Several authors agree that if the predominate stress resultants vary slowly in the circumferential i
directir with respect to the buckling wave, the buckling stress is not sensitive to the nonsymmetry of the loading [A.3, A.4, A.5, A.6, A.9, A.16, A.20, A.21). They also agree that if it does vary rapidly, the " worst" meridian is at least, conservative. From an intuitive view, it appears reasonable that uniform compression around the entire circumference is a " worse" case than a circumferentially varying compression with the same maximum value. The following sections summarize some previous studies which considered the stability of axisymmetric shells under nonaxisymmetric stress states, as related to the " worse" meridian approach.
t A.3.3.1 Longitudinal Stress Resultant The elastic buckling of a long cylindrical shell with an axial compressive stress resultant over a small circumferential arc length (see Fig. A.5) is reported in [A.21). The results are summarized l
in Fig. A.S. The figure illustrates the ratio of critical stress of the nonaxisymmetric case, N,, to the critical stress for uniform axial compression, N,:
y N
critical stress, nonaxisymmetric case cr _
gg_gy N
critical Stress, axisymmetric case cr plotted versus a nondimensional arc length. The critical stress, N,,, is always equal to or greater than the critical stress of the same shell under axisymmetric axial compression. The increase in N,, is noticeable only if the loaded arc length is small.
The elastic stability of circular cylindrical shells under bending (n=1 for prebuckled shape) has been investigated by Seide and Weingarten [A.24] as a bifurcation buckling problem. They concluded that the critical buckling stress for bending of a shell with an r/t ratio of 100 is about 1.5 percent higher than the criticai buckling stress under uniform axial load. For thinner shells this difference is even smaller. This result is well above the critical stress found by Brazier [A.25) who included the effect of the shell cross-section ovalization on the stability of long cylinders
[
under pure bending. If a long cylinder is subjected to bending, its cross-section flattens and, consequently, the bending stiffness deteriorates with increasing moment. However, it appears from experiments on the bending of thin shells that the shells fail through buckling in a wavy pattem, similar to shells under axial compression, rather than through collapse by flattening of the cross-section. The restraint at the ends of the tested shells is a contributing factor. Hence, a j
finite length she!! will buckle at load levels higher than these predicted by Brazier.
21
BOSOR4 has been used to study the variation of the elastic buckling stress of stiffened she!!s with circumferentially varying longitudinal stress resultants [A.9]. To study a nonaxisymmetric effect with an axisymmetric model, the authors of [A.9] modelled a cylinder of infinite length as a portion of a torus with a large radius R, (see Fig. A.6). As R, approaches infinity, the torus approaches a straight cylinder [A.9, A.20]. The cross-section need not be circular nor the thickness constant. Loading can vary around the circumference. Since the torus is a shell of revolution, the BOSOR programs can be used without any special afterations. The loading j
considered in [A.9) is similar to that in Fig. A.5, i.e., a uniform longitudinal stress resultant is applied over the arc length, b. The results are summarized in Fig. A.7 where the ratio N,/N,,is plotted against the ratio of the loaded are length b to 'he arc length of the buckled wave for the uniform compression case, b (see Fig. A.8). These results are shown for the case of a ring and l
stringer stiffened shell (smeared stiffeners, general buckling), a she!! with stringers only (discrete stringers, general buckling), and a stringer stiffened shell with a restricted axial buckled wavelength (discrete st.inger and rings, local buckling between rings).
These results illustrate that the elastic buckling stress of a shell subjected to circumferentially varying longitudinal stress does not differ significantly from that of uniformly loaded shell if the variation is not rapid with rc pect to the buckled shape. Hence, if the longitudinal stress resultant i
is approximately uniform c= 3r one-half a buckled wavelength, the circumferential variation of i
stress can be ignored. In this regard, a " worst" meridian axisymmetric analysis will produce acceptable resuits.
If the circumferential variation is rapid, the results will, at least, be l
conservative.
i A.3.3.2 Circumferential Stress Resultants Circumferential variations may also have an effect for situations in which the circumferential stress dominates. The effect of a circumferential!y varying radial pressure on the critical elastic buckling stress is shown in Fig. A.9 [A.26]. By comparing the dashed curves (uniform pressure case) to the solid (cosine variation of pressure) one can see that the uniform case always gives a slightly l
lower critical stress.
l Several cylindrical and conical shells heated along an axial strip of varying width have been i
analyzed [A.19). It was found that a cylinder or a cone heated along a narrow strip will buckle at approximately the same stress as a uniformly heated shell. This result applies even if the half-i wavelength of the buckled shape is approximately the same as the circumferential extent of the i
l heated region. Although this project is not concemed with localized temperature loading, this j
l example helps to demonstrate the validity of the " worst" meridian approach.
The conclusion, then, for nonaxisymmetric circumferer31 stress resultants is similar to that obtained for shells with nonaxisymmetric lor,;itudinal stress resultants, i.e., the circumferential variation of stress resultants does not significantly affect the elastic buckling stress if the variation i
is skms with respect to the buckled shape. In general, the elastic buckling stress for a nonaxisymmetrically loaded shellis conservatively approximated by an axisymmetric model. This j
is the basis of the " worst" meridian approach.
l A.3.3.3 Shear Stress Resultants BOSOR5, as currently configured, does not include shear stress resultants even though such cases may be axisymmetric, e.g., torsion. Hence, the program cannot be used to detect the elastic buckling strength for cases with shear stress resultants or for cases with combined normal 22 l
l
and shear stress resultants. This does not appear to be a severe limitation because, in many cases, the shear stress is not significant, e.g., planes of symmetry. The user is wamed, however, that, in the most general case, shear stress, in combination with other stress resultants, reduces the buckling capacity [A.27).
A.3.4 Imperfect She!!s As previously mentioned, differences between the first-order theoretical and the measured buckling strengths are usually ascribed to imperfections. In the case of real structures which contain unavoidable imperfections, there is no such thing as the true bifurcation buckling illustrated in Fig. A.1. Instead, the structure will follow a path below OBC with a maximum load below Point B. How much below B depends on the imperfection sensitivity of the shall[A.1, A.2].
For the sphere under extemal pressure and a cylinder under axial compression, the reduction is -
targe. For the imperfect shell case, the distinction between plastic collapse and bifurcation buckling is less clear. The deformed shape at maximum load can be similar to the prebuckled shape, the perfect bifurcation buckled shape, the dominant imperfection mode, or a combination-i of the three.
Ifit were possible to predict all of the imperfection pattems resulting from the fabrication process, one could conceivably evaluate the critical load of such a structure with much greater certainty.
A three-dimensional finite element model in which the nonsymmetric imperfections can be explicitly included would be appropriate [A.9). An attemative is to utilize an axisymmetric model t
with an approximate axisymmetric imperfection, as would be appropriate for a BOSORS analysis
[A.1, A.3). Studies indicate that an axisymmetric imperfection is more critical than nonsymmetric imperfections [A.28].
A nominally axisymmetric shell has circumferential and longitudinal imperfections. Imperfections sympathetic to the various buckling modes are considered to be the most important [A.1, A.4, A.5 (Paper No.19), A.6). Thus, longitudinalimperfections tend to have the most effect on longitudinal buckling strength and circumferential imperfections on circumferential strength [A.3, A.5 (Paper Nos. 7,9,15)). As suggested by several researchers [A.1, A.3, A.4, A.5 (Paper Nos. 3,5,6,19)],
the longitudinal imperfection can be modelled as a sine wave in the meridional direction, s, i.e.:
2 Wg = W, sin (A.7) t i
where W, is the radialimperfection amplitude Fig. A.10 (one half the peak to trough amplitude of the imperfection, e). In the design case where e is not known, it can be estimated in terms of ASME fabrication tolerances [A.29) as-W, = 0. 0 02 5 y D,,
(A.8) f where:
"d" y = 0. 01 D (A.9) om and in v,tiich D and D are the maximum and minimum diameters of the imperfect shape, and i
23
l l
l I
l D
is the average diameter. Another restriction is:
l 0.315 e 51.0t (A.10) where t is the thickness of the shell.
-l For axial compression of unstiffened shells which buckle nonsymmetrically, Koiter [A.1, A.7] has suggested that the imperfection wavelength ( be taken as equal to the wavelength for axisymmetric buckling of a perfect cylinder, (see Fig. A.11) which is:
ly = 3.5 frt)sr2 (A.11) l Gellin [A.30) gave a similar expression to evaluate the critical waveler.gth for cylindrical shells stressed beyond the proportional limit.
i Circumferential imperfections cannot be included directly in the axisymmetric model. Reference
[A.2] suggests that cylinders subject to radial pressure are imperfection insensitive. However, i
cylinders with hydrostatic pressure are imperfection sensitive because the pressure on the ends produces axial loads to which cylinders are notoriously imperfection sensitive. If this is true, circumferentially varying imperfections need not be included in the model since they are not generally sympathetic to the axial compression buckling mode. Bushnell[A.1) suggests that one can account for the effect of nonaxisymmetric imperfections on longitudinal buckling by dividing the critical load (based upon an axisymmetric imperfection pattem) by the ratio of the local radius of curvature at a " flat spot" to the nominal radius. He observes, however, that the standard deviation of the ratio of theory to test is not noticeably reduced when this is done.
A.3.5 Openings in Shells A cylinder with a penetration cannot be analyzed with a BOSORS axisymmetric model. Itis known that if the penetrations are reinforced by the ASME area replacement rule [A.31), the-strength of a penetrated shell under uniform intemal pressure will exceed that of an unpenetrated shell. In general, the ASME rules require the area removed for the opening should be replaced by an equal area of reinforcernent around the opening circumference. In this situation yielding is the usual failure mode and, as the stresses are redistributed, the reinforcement participates fully in the plastic behavior.
Numerous investigations have been conducted to determine the effects of a penetration and its j
reinforcement on the elastic and inelastic buckling resistance of the shell [A.32, A.33, A.34). In i
all of these studies, experimental cylindrical models with various ratios of reinforcement area to -
cutout area were tested under axial compression. Figure A.12, which summarizes the results, i!!ustrates that the area replacement rule does not necessarily restore the full unpenetrated buckling strength of the shell. In [A.3) it is concluded, however, that small holes (diameter less than 2 (rt)2) have less effect than other imperfections and, for this case, the area replacement j
reinforcement adequately restores the strength.
An analytical study using the STAGSC [A.35] has been conducted by Me!!er and Bushnell [A.36) l to calculate the strength of the models tested in [A.34). The unpenetrated specimen was analyzed with and without the measured initial imperfections. The analytical collapse load for the j
perfect shell was nine percent greater than the analytical collapse load for the imperfect shell.
-i The average experimental collapse load for two specimens was one percent higher than the 24
calculated load for the imperfect cylinder. No measured imperfections were reported in [A.34] for the shells with openings; hence, perfect analytical models were used in the analysis of the penetrated shells. Figure A.13 is a plot of the ratio of the analytical calculated buckling load of the perfect penetrated shell to that of the unpenetrated perfect shell as a function of the opening radius, a, divided by (rt)'8 The discrepancy in the results reported in [A.02] and [A.33] makes it difficult to establish specific conclusions conceming the buckling resistance of penetrated shells in relation to unpenetrated shells. The results given in [A.34] show that openings in stiffened shells will reduce the buckling load, even though the penetrations are reinforced in accordance with the ASME rules. However, the buckling strength of an axisymmetric shell is not affected by reinforced openings with diameters less than 2(rt)v2 Larger penetrations, such as equipment hatch openings, could reduce the shell buckling resistance, even if they are reinforced as required by ASME rules. However, a good first approximation of the effect of the reinforced opening on the buckling strength can be obtained by an axisymmetric analysis. The penetration and its reinforcement would be assumed to be axisymmetric about the axis of the penetrating cylinder and the containment shell would be approximated by a segment of an equivalent spherical shell. The relative effect of the reinforced opening may be studied by comparing the buckling strength of this approximation to that of the unpenetrated equivalent spherical shell.
(Note, this approximation should not be used to determine absolute buckling strength but only to determine relative buckling strength of the penetrated vs. unpenetrated shells.) In many cases, a three-dimensional finite element analysis is recommended to evaluate the shell critical load, similar to that in [A.36). Each large penetration must be handled on an individual basis. Code Case N-284 recommends that the circumferential variation of geometry and material properties be smeared around the circumference for an axisymmetric analysis. It also recommends that articles 1710 and 1720 can be used in analysis, without consideration of penetrations if the diameter of the penetration is less than 10% of the vessel diameter.
A.3.6 BOSOR Model Guidelines A.3.6.1 Mesh Guidelines The BOSOR programs are based on the finite difference energy method to analyze smooth or stiffened shell structures. The shell and the ring stiffeners can be modelled for the finite difference analysis using shell segments. A discrete ring option is available in the BOSOR programs for the idealization of the circumferential stiffeners. A maximu,n of 95 shell segments can be used to model a structure. Each segment is divided into a number of finite difference elements using at least five mesh (noda!) points with a maximum of 95 points. In calculating buckling loads, the BOSOR manual recommends using more than four mesh points per half wavelength of the buckling mode. The number of these mesh points in the segment is one of the most important variables in the analysis. This govems, to a large extent, the accuracy of the solution. It is recommended that the distance between the mesh points be restricted to (rt)v2/2.
This also satisfies the recommended mesh point spacing for the buckling analysis.
A.3.6.2 Modelling of Material Nonlinearities in the fabrication process, flat steel plates are deformed to fit the singly and doubly curved surfaces of the containment. Even though the plates and the subsequent welding may be stress-relieved, the proportionallimit of the material is seldom restored to its virgin value near the yield point. This has a significant effect on the buckling strength of steel compone'.ts because the 25
material tangerst modulus is reduced, in lieu of knowing the actual residual stress pattem, this effect can be approximated by using an effective stress-strain curve with a proportionallimit below the yield strength [A.37, A.38). Several choices exist for incorporation of the material nonlinearities in the analytical model. For this study, the approximate stress strain curve was derived from the equations for the plasticity reduction factor, y, in the Code Case N-284, which expresses the predicted buckling pressure as (A.43):
Pc
- 9 P.
(A.12) in which p, = inelastic buckling pressure and p, = elastic buckling pressure l
l Miller established the plasticity reduction factor, y, in Code Case N-284 as a lower bound to test
'l' data to account for material nonlinearites and residual stresses. Since the data resulted from experimental tests on fabricated steel shells with residual stresses and no stress relief, it is representative of actualin-situ conditions.
j in several theoretical inelastic buckling developments, the elasticity reduction factor is approximated by [A.43 (Eq.18.3))
y = E /E (A.13) r I
in which d
E7=
(A.14) is the slope of the stress-strain curve. The effective stress-strain curve is obtained by combining Eq. [A.13) and Eq. [A.14) and integrating (A15) i da, q g de with y given by Millers equation [A.11 (Article 1600)). As an exce!Ient approximation, this effective stress-strain curve incorporates the material nonlinearity effects observed in the testing of actual fabricated steel shells. The proportional limit of the effective stress-strain relation, indicated in the Opde Case as the dividing line between elastic and inelastic behavior, occurs at 0.55 times the yield stress. This approximates residual stresses of the order of 0.45 times the l
yield stress which is consistent with the observed residual stresses in fabricated structures of about half yield. For purposes of the BOSORS analysis, this curve is approximated by a piece-
?
wise linear curve as shown in Fig. A.14.
26 1
BOSORS [A.18] includes options for both the deformation and flow theories of plasticity. Even though tre flow theory is analytically more attractive, buckling test results tend to agree better with deformation theory analysis [A.3, A.4, A.18, A.39). Deformation theory will be used here.
A.3.6.3 Convergence Criteria To accomplish the nonlinear (material and geometry) solution, an iterative approach is used with the BOSOR programs. The procedure is to increase the applied load by small increments and allow the program to iterate until it converges to a final solution. The margin of safety against buckling is defined as a load multiplier which is equal to the ratio of applied loads to the loads at which buckling occurs. Hence, to find the margin at safety, the total load i.e., (D + L + seismic i
+ P + T), is incremented. In BOSOR4 and BOSORS, the geometric nonlinearity solution, is said to te converged if the ratio of the change in the displacement vector to the current total displacement vector is less than a specified value of 0.001 [A.18). The nonlinear material problem is handled in BOSORS using a similar technique. In this case, the solution is said to be converged when the plastic strain increment becomes less than 0.1 percent [A.18).
A.3.7 BOSORS Calibration One advantage of the BOSORS program is its widespread use. Hence, results have been compared to experimental results in many cases [A.39, A.40, A.41, A.42). Much of the applicable work has been summarized in Table A.2 and can be used in calibrating BOSCRS results. In Table A.2, the shell and loading type for the experimental work are identified. Deformation plasticity theory was used for all analyses.
The specimens were aluminum and steel.
i Imperfections were not included in the torispherical and ellipsoidal head analysis since these heads were not considered sensitive to imperfections [A.40, A.41). The stiffened cylinders with extemal pressure were machined and had small imperfections, which were not included in the BOSORS model [A.42). Axisymmetric imperfections were included in the analysis in [A.1, A.3]
and [A.5 (Paper No. 5)] as discussed in Sec. A.3.4.
The other cylinders from [A.18) were relatively thick (r/t < 100) and imperfection insensitive. The mean and standard deviations of the ratio of experimental result to BOSOR5 result are listed. As one would expect, the mean and standard deviation of this ratio vary noticeably with experimentation. The results shown in Table A.2 yield a mean and coefficient of variation of the ratio of the experimental to BOSOR5 results of 1.000 and 0.135, respectively.
A.3.8 Summary Many shells of revolution structures experience loadings (static and/or dynamic) that produce nonaxisymmetric stress resultants which may cause instability. Guidelines for using BOSOR4 and BOSORS programs to calculate the buckling stress resultants for non-axisymmetrically loaded axisymmetric shells are presented.
r A circumferential variation in the stress resultants is not permitted in the BOSOR5 axisymmetric model. Several investigations have concluded that the buckling Stress is not sensitive to variations which are small with respect to the buckling wave. Therefore, the buckling strength can be conservatively evaluated assuming that the membrane stresses along the " worst" meridian are axisymmetric if shear stresses are not significant.
Geometric imperfections have a significant effect on the shell buckling load. In the BOSORS 27 i
i i
axisymmetric model, an effective imperfection can be modelled as a sine wave in the meridional direction.
Other nonaxisymmetric factors such as penetrations cannot be included in the BOSORS axisymmetric model.
i A.4 References A.1 Bushnell, D., " Elastic-Plastic Buckling of Axia!!y Compressed Ring Stiffened Cylinders--
Test Versus Theory," Weldina Research Council. Bulletin 282, November 1982.
A.2 Chajes, A., " Post Buckling Behavior," ASCE Joumal of the Structural Division.109 (10),
pp. 2450-2462, Oct.1983.
A.3
- Bushnell, D., " Plastic Buckling of Various Shells," Joint ASME/ASCE Mechanics i
Conference. June 21-29, 1981, Boulder, CO.
A.4 Bushnell, D., " Buckling of Shells-Pitfall for Designers," AIAA Joumal.19 (9), pp.1183-t 1226, Sept.1981.
A.5 Bucklina of Shells in Offshore Structures. Ed. by Harding, J. E., Dowling, P. J., and -
I Agelidis, N., London: Granada Publishing,1982, selected papers.
A.6 Seide, P., Weingarten, V., and Masri, S., " Buckling Criteria and Application of Criteria to Design of Steel Containment Shells," Report to NRC, NUREG/CR-0793, May 1979.
A.7 Brush, D. O. and Almroth, B. O., 3ucklina of Bars. Plates and Shells. Chapter 7, New York: McGraw Hill,1975.
A.8
" Shells (Buckling)," Chapter 8, Enoineerina Joumal, American Institute of Steel Construction, Second Quarter, pp. 102-111, 1982.
A.9 Almroth, B., Rankin, N., and Bushnell, D., " Buckling of Steel Con *ainment Shells, Task 3:
Parameter Studies," Report to the U.S. NRC, NUREG/CR-2836, Vol. 3, December 1982.
A.10 U.S. NRC Standard Review Plan (SRP), Section 3.8.2, Rev.1 - July 1981. pp. 3.8.2.4 and 3.8.3.11.
'I A.11 American Society of Mechanical Engineers, Boiler and Pressure Vessel Code, Section NE3222,1989.
A.12 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,1980.
A.13 American Petroleum institute," Bulletin on Stability Design of Cylindrical Shells."
l A.14 ASME Boiler and Pressure Vessel Code Appendix 111, paragraphs 2100 and 3100.
i i
f 28
A.15 U.S. Nuclear Regulatory Commission " Regulatory guide 1.57, Design Limits and Loading Combinations for Metal Primary Reactor Containment Systems Components," NRC, Washington, June 1973.
A.16 " Chicago Bridge and Iron's Comments on NUREG CR-0793, ' Buckling Criteria and Application of Criteria to Design of Steel Containment Shells,' May 1979," available from i
NRC, Washington, November 1979.
j l
A.17 Bushnell, D.,"BOSOR4 Program for Stress, Buckling and Vibration of Complex Shells of A
Revolution," Structural Mechanics Laboratory, Lockheed Missiles and Space Co., Inc.,
Palo Alto, CA,1975.
A.18 Bushnell, D., "BOSORS Program for Buckling of Elastic-Plastic Complex Shells of Revolution including Large Deflections and Creep," Stmetural Mechanics Laboratory, Lockheed Missiles and Space Co., Inc., Palo Alto, CA,1974.
A.19 Bushnell, D. and Smith, S., " Stress and Buckling of Nonuniformly Heated Cylindrical and Conical Shells," AIAA Joumal. 9 (12), pp. 2314-2321, Dec.1971.
A.20 Bushnell, D.," Stress Buckling and Vibration of Prismatic Shells," AIAA Joumal. 9 (10), pp.
2004-2013 Oct.1971.
A.21 Hoffs, N. J., Chao, C., and Madsen, W. A.,
- Buckling of a Thin-Walled Circular Cylindrical Shell Heated Along an Axial Strip," Joumal of Applied Mechanics. pp. 253-258, June 1964.
A.22 Berg, Glen V., " Elements of Structural Dynamics," ist Edition, Published by Prentice Hall, inc., Englewood Cliffs, CA,1989.
A.23 Clough, R. W. and Penzien, J., " Dynamics of Structures," Published by McGraw Hill Book Company, New York,1985.
A.24 Seide, P. and Weingarten, V. I., "On the Buckling of Circular Cylindrical Shells Under Pure Bending," Joumal of Applied Mechanics. Vol. 28, pp.112-116,1961.
A.25 Brazier, L G., "On the Flexure of Thin Cylindrical Shells and Other Thin Sections," Proc.
Roy. Soc.. Series A., Vol.116, pp.106-115,1926.
A.26 Almroth, B. O., ' Buckling of a Cylindrical Shell Subjected to Nonuniform Extemal i
Pressure," Joumal of Applied Mechanics. 29, pp. 675-682 Dec.1962.
A.27 Baker, W. E. and Bennet, J. G., Experimental investigation of the Buckling of Nuclear Containment-like Cylindrical Geometrics Under Combined Shear and Bending," distributed at Structural Mechanics in Reactor Technoloov. Chicago, August,1983, and submitted to Nuclear Enoineerino and Desian Jouma!.
A.28 Teng, J. G. and Rotter, J. M., " Buckling of Pressurized Axisymmetrically Imperfect Cylinders Under Axial Loads," ASCE Engineering Mechanics Joumal, Vol.118, No. 2, Feb.1992.
i 29 s
L
A.29 ASME Boiler and Pressure Vessel Code, Sec.111, Class MC Components, Subsection NE, paragraphs 4221.1 and 4221.2, and Section Vill, Division 2, Subsection AF, paragraphs AF130.1, AF130.2,1989.
A.30 Gellin, S., "Effect of an Axisymmetric imperfection on the Plastic Buckling of Axially Compressed Cylindrical Shell," Joumal of Aeolied Mechanics. 46, pp.125-131, March l
1979.
i A.31 ASME Boiler and Pressure Vessel Code, Sec.111, Class MC Components, Subsection NE, Paragraphs 3331-3335,1989.
A.32 Miller, C. D., " Experimental Study of Buckling of Cylindrical Shells with Reinforced Openings," ASME/ANS Joint Conference. Portland, July 26-28,1982.
A.33 Bennett, J. G., Dove, R. C. and Butler, T. A., "An investigation of Buckling of Steel Cylinders with Circular Cutout Reinforced in Accordance with ASME Rules," Report to the U.S. NRC, NUREG/CR-2165, June 1982.
A.34 Bennett, J. G. and Baker, W., " Buckling Investigation of Ring-Stiffened Cylindrical Shell l
with Reinforced Openings Under Unsymmetrical Axial Loads," Report to the U.S. NRC, NUREG/CR-2966, Oct.1982.
A.35 Almroth, O. B. and Brogan, F. A., 'The STAGS Computer Code," NASA CR-2950, Feb, 1978.
A.36 Meller, E. and Bushnell, D., " Buckling of Steel Containment Shells, Task 2: Elastic-Plastic Collapse of Nonuniformly Axially Compressed Ring-Stiffened Cylindrical Shell with Reinforced Openings," Report to the U.S. NRC, NUREG/CR-2836,2 December 1982.
A.37 Brockenbrough, R. L. and Johnston, B. G., Steel Desian Manual. U.S. Steel, Pittsburgh, PA,1974, pg. 47.
A.38 Salmon, C. G. and Johnson, J. E., Steel Structures Desian and Behavior,3rd edition, Harper and tiow, New York,1989, p. 308.
A.39 Bushnell, D.," Elastic-Plastic Buckling ofIntemally Pressurized Ellipsoidal Pressure Vessel Heads," Weldino Research Council. Bulletin 267, May 1981.
A.40 Lagae, G. and Bushnell, D.," Elastic-Plastic Buckling ofIntemally Pressurized Torispherical Vessel Heads," Nuclear Enoineerina and Desion. 48, pp. 405-414,1978.
A.41 - Bushnell, D. and Gallety, G. D., " Stress and Buckling of intemally Pressurized, Elastic-Plastic Torispherical Vessel Heads Comparison of Test and Theory," Joumal of Pressurized Vessel Techniaues. 99, pp. 39-53, Feb.1977.
i A.42 Bushnell, D., " Buckling of Elastic-Plastic Shells of Revolution with Discrete Elastic-Plastic -
Rings," Intemational Joumat of Solids and Structures.12, pp. 51-66,1976.
A.43 Galambos, T. V., Ed., " Guide to Stability Design Criteria for Metal Structures," 4Q Edition, published by John Wiley & Sons, New York,1988.
30 5
APPENDIX B BOSOR4 AND BOSORS MODIFICATIONS B.1 Modal Particioation Fsetor (BOSOR4)
BOSOR4 software calculates eigenvalues, eigenvectors, generalized mass and modal participation factors for each vibration mode for a modal analysis performed with INDIC = 2. In BOSOR4 output, values of the modal participation factor were incorrect. This was discovered during the solution of the sample problem shown in Fig. B.1. The sample problem consists of an axisymmetric cylinder of length 80 in. and diameter 8 in. The material data are as follows:
modulus of elasticity, E of 3,000 ksi; Poissons ratio, v of 0.3; and, mass density, p of 0.000728 lb-sec'/in'. The sample problem was analyzed with BOSOR4 software using a total of 5 equally spaced mesh points. Hand calculations were also performed to evaluate the modal participation factors using the mode shapes from BOSOR4 software. The problem was also run on the finite -
element software ANSYS [8.1). The ANSYS model consisted of 5 equally spaced nodes connected by axisymmetric straight shell elements (STIFF 61). The results of the modal participation factor calculation were compared as shown in Table B.1. In this table, the value of
}
circumferential wave number, N, represents the direction of the ground motion (see Appendix A).
The horizontal ground motion in X and Y directions is represented by N = 1 and -1, while N = 0, represents vertical ground motion in Z direction (see Fig. A.4). For N = 1, it is clear that the modal participation factor from BOSOR4 is about 1/2 of the ANCYS and hand calculation values.
Moreover, for all three values of N, (i.e., N = 1, O and -1) the values of the modal participation factors from BOSOR4 are quite different than the corresponding values obtained from ANSYS or l
hand calculations. The sample problem was repeated with different boundary conditions at the two ends of the cylinder and the above observations were confirmed.
When the BOSOR4 source code was studied, two errors were discovered in the calculation of the f
modal participation factor-1.
For N = 0 and -1, BOSOR4 calculates incorrect values of the modal i
participation factor (variable AMPLTD in subroutine LOCAL of MODE library).
2.
The BOSOR4 software introduces a factor of 1/2 in the modal participation factor calculation (variable GAMP in subroutine MODE of MODE library).
The errors in subroutines LOCAL and MODE of MODE library were corrected. In addition, FORTRAN lines were incorporated into subroutine OUT2 of B4 POST library to inform the user that the values of the modal participation factors are based on the modified version of BOSOR4 software. The above changes are documented in detail within the BOSOR4 source code in subroutines LOCAL and MODE of MODE library, and OUT2 of B4 POST library. The modified i
source code of the above subroutines is shown in Table B.3.
The sample problem shown in Fig. B.1 was rerun on the modified version of the BOSOR4 software and the correct values of the modal participation factor (see Table B.1, hand calculations) were obtained.
31
B.2 Modal Stress Resultants and Stresses Output For a modal analysis (INDIC=2), BOSOR4 software does not calculate the Fourier coefficients for the modal stress resultants and modal stresses. These quantities are required for a seismic analysis. BOSOR4 was modified to calculate these coefficients for each of the modes of vibration
-j and at each mesh point of the computer model. When the modified BOSOR4 software is run, it
~
creates an additional output file named as FILE 14.DAT which contains the modal stress resultants and stresses as well as the geometrical properties of the structure.
Figure B.2 shows a chart explaining the basic format of FILE 14.DAT along with the names of a!!
i the quantities in it. Table B.2 lists the names of the subroutines and libraries of BOSOR4 source j
code where modifications were made for creating FILE 14.DAT.
B.3 Effective Uniaxial Strain and Plastic Strain Outout BOSORS software has an option with which the user can specify if strains should be written to the output file. If this option is chosen, BOSORS writes the strain quantities at all the mesh points of all the segments.
The BOSORS source code was modified (FLOW subroutine of PLAST library) so that, in addition to the above output, a separate output file called RESULT.DAT is created during the modified BOSOR5 run. The RESULT.DAT file contains the strain quantities (plastic meridional and circumferential strain and effective uniaxial strain) at a!! the mesh points where plastic straining
?
occurs. The effective plastic strain E
is defined as E,=
cyj cyj, where cyj is the p
plastic component of the total Ejy. These quantities are printed out at all the loadsteps requested by the user during the POST-SETUP input session of BOSOR5.
l B.4 References B.1 De Salvo, G. J. and Swanson, J. A., "ANSYS Engineering Analysis System - User Manuals I and 11, Theoretical Manual, Examples Manual," Swanson Analysis System inc.
1985.
B.2 Chen, W. F. and Han, D. J., " Plasticity for Stmetural Engineers," Springer-Verlag,1988, New York.
i f
3 i-i i.
32
i APPENDIX C EQUIVALENT AXISYMMETRIC PRESSURES I
C.1 Ecuivalent Axisymmetric Pressures Seismic loading implies time varying stresses. Ideally, the seismic problem should be analyzed i
as a dynamic buckling problem in which inertial forces interact with buckling forces. Since, for example, peak stresses exist for only a short period of time, it is reasonable to expect that, in some cases, buckling may not occur because the inertial forces of the shell material may prevent it. In this work, the seismic stress resu!tants will be applied as static loads (" quasi static" analysis), which several have recognized as conservative [C.1, C.2, C.3]. The SRSS stresses and stress resultants cannot be combined with other stresses and input directly into the BOSOR codes to check buckling. BOSOR and most other codes accept only extemal forces as loading.
1 Therefore, it is necessary to convert the SRSS stress resultants into equivalent pressure loadings.
The maximum SRSS stress resultants on a selected meridian are first decomposed into a linear combination of the modal components:
{1 6 } = [N ] {$}
(C.1) 4 where {NJ = maximum SRSS stress resultants (Eq. 2.1, Figs. 2.9 and 2.10) and [N,] are the modai stress resukants. The values of {y} can be interpreted as the combination of the individual modes which superimpose to produce the maximum SRSS seismic response. Next, the modal stress resultants [N,] must be converted to modat inertia pressures which would be nonaxisymmetric. They must further be converted to equivalent axisymmetric pressures for the axisymmetric BOSOR5 analysis (see Sec. A.3.2). For an axisymmetric shell, two of the goveming differential equations for nonsymmetric dynamic loading [C.4, C.5] are:
aN,
i
, a(rM )
a(42)
(C.2)
. Tc'i5, N cos$,
B {rn )
4 1
2 1
i x
rr, as ao ras a
, a(M cos$), 202 (IM ),N N SiD@
6:M,
, a'(rM),
3 803 r as x3 a6Bs r1 x
2 12 1,
2 r as*
1 I
in which N,, N, and N,2 are the meridional, circumferential and shear stress resultants, 2
respectively, and M,, M, and M are the corresponding moment resultants. The displacements, 2
i2 u and w, are in the meridional and normal directions, respectively, and m is the mass per unit l
area. These and other quantities are defined in Fig. C.1. The acceleration terms on the right l
represent inertial pressures.
The applied pressures are p, and p,.
For an equivalent axisymmetric response N,, and M,,, derivatives with respect to O must be zero. For mode j, an j
axisymmetric analysis will reproduce the modal stress resultants on a specific meridian at 0 if the following equivalent axisymmetric modal pressures are applied for the nth harmonic (Eq. A.1):
i 33
p1 oj m i*1 - " #**' - m _
- =
sin no (C.3) rk I
g i
1 -
(r E,)
sin n0 P,' = 0) m Y.,, +
22 in which ej is the modal frequency, 4 and 4, are the meridional and normal modal displacements, respectively, and N f
and,M are modal stress resultants. The 12 -
12 2
super bar refers to the amplitude of the# Fourierharmonic, a,s in Eq. A.1.
l l
The total equivalent pressures are obtained by superposition of the modal pressures as follows:
i
?
I r
pl = { P,,* $
(CA) 3 J
P$= { Pa,'43 J
'i where y, are the modal coordinates. Equation C.3 shows only the asymmetric terms of the Fourier series. A complete series involves the symmetric terms also (see Eq. A.1). Several possibilities exist for detc.nining the value of {y}. The peak stress resultants in {N_} represent maximum values which. annot all occur at the same time since this would be inconsistent with the m mode shapes. To assume, for example, that both the meridional, N,, and circumferential,-
N, stress resultants reach their maximum compressive values at the'same time is conservative 2
and modally inconsistent. See, for example, Fig. C.2 which indicates the stress resultants for the first horizontal mode of vibration. For mode one, which dominates the behavior, compressive j
values of N do not occur with compressive values of N. The method of least squares will be i
2 i
used to select the values of {y} which best" fit the SRSS resultants {N } for both the meridional and circumferential directions, that is:
i
((N ]r[y )
[y 3#IN )) {9}
(C.5) 41 1
2 42
=W
[N 3 r {y
}+y IN 3 IN 3
2 41 2
42 2
o 34
r-t where:
modal meridional stress resultants
[N,,)
=
i modal circumferential stress resultants
[N ]
=
o SRSS meridional stress resultants (positive)
{N,J
=
SRSS circumferential stress resultants (positive)
{N Q
=
weight parameters which take values of +1 or-1.
l
=
w,w2 i
These m equations are solved for {y}. The weight parameters w, and w are -1 and +1 to check 2
buckling due to meridional compression and +1 and -1 to check circumferential compression (see Fig. C.1).
The program LOADS was written to convert the SRSS stresses into equivalent axisymmetric pressures by the equations in this appendix. It accepts the mode shapes, frequencies, and participation factors from BOSOR4 and calculates the SRSS values by Eq. 2.1. It then calculates the equivaleat axisymmetric pressures and prepares the input file to BOSORS for the buckling analysis. As a check, an imperfect System 80+* shell with a double amplitude of 1.75 in. and l
a wavelength parameter of 3.92 was considered (see Sec A.3.4). The SRSS stress resultants and the BOSOR5 stress resultants, which result from the equivalent axisymmetric pressures, are i
compared in Fig. C.3. The resulting comparison validates the approach.
C.2 References C.1 Bucklino of Shetis in Offshore Structures. Ed. by J. E. Harding, P. J. Dowling, and N.
Agelidis, London, Granada Publishing,1982, selected papers.
C.2 NUREGICR-0793, " Buckling Criteria and Application of Criteria to Design of Steel Containment Shell," Report to NRC, May 1979.
C.3 Chicago Bridge and Iron's Comments on NUREG/CR-0793, May 1979, available from NRC, Washington, November 1979.
1 C.4 Brush, D. O. and Almroth, B. O., Bucklina of Bars. Plates and Shells. New York, McGraw Hill,1975, Chapter 6.
C.5 Kraus, H. Thin Elastic Shells. New York, John Wiley and Sons,1967, Chapter 6.
l t
35 l
~
i i
'i Table 2.1 Smeared Mass Densities for the Different Segments of Axisymmetric Model in Fig. 2.2 i
Segment Description Mass Density
~ '
lb-s*/in' 1
Compressive surface 0.000735 2
1.75 in. shell 0.000735~
i 3
Personnellock 2 0.0008192 4
1.75 in, shell -
0.000735 i
5 Equipment hatch +
0.0008936 Personnellock 1 6
Equipment hatch 0.0008153 7,8,9 1.75 in. shell 0.000735-10 Sprayheader system 0.0002986 11,12 1.75 in. shell-0.000735 s
P k
t r
l 36 I
l l
l l
f:
Table 2.2 Design Conditions SRP Design Allowable Stress Maximum Reference Description
- Intensity Umit Calculated Stress Number Type
- Limit Value Value Elev.
(psi)
(psi)
(ii)
D+L+P,+T,+R, P,
1.0 S.
22,000
- 2.154 94.1 (P,=-2 psi, T,=70*F)
D+L+P,+T,+R, P,,,
1.0 S.
22,000 17,852 98.2 (ii)
(P,=49 psi, T,=70*F)
{
- Definition of variables in the column is given in Table 2.7.
l L
1 1
37 1
i
Table 2.3 Level A Service Umits t
SRP Design Allowable Stress Maximum Value Reference Description
- Intensity Umit as per Stress Number Analysis (psi) j Type
- Umit Value '
Value Elev.
(psi)
(iii)(a)(1)
D+L+P,+T,+R, P,
1.0 S.
22,000 2,160-94.1 (P,=0,-T,=70* F)
(iii)(a)(1)
D+L+ P,+T,+ R, P +PpO
' 3.0 S,,
80,100 10,852 86.3-t (P,=0, T,=110*F)
(iii)(a)(2)
Not applicable (iii)(a)(3)
D+ L+ P,+T,+R, P,
1.0 S.
22,000 2,154 94.1 (P,=-2, T,=70*F)
(iii)(a)(3)
D + L+ P,+T,+ R, PgP,+O 3.0 S.,
80,100 10,170 86.3
-L (P
-2, T,=110*F)
(iii)(a)(3)
D+L+P,+TpR, P,
1.0 S 22,000-17,852
'98.2 y
(P,=49, T,=70*F) j (iii)(a)(3)
D+ L+ P,+T,+ R, PgP,+O 3.0 S,
80,100 77,334 86.3 f
(P,=49,T,=290* F) i
- Definition of variables in this column is given in Table 2.7.
l r
r F
h i
38
i i
l Table 2.4 Level B Service Umits i
SRP Design Allowable Stress Maximum Value Reference Description
- Intensity Limit as per Stress
-j Number Analysis (psi) l Type
- Umit Value Value Elev.
l (psi)
(psi)
(iii)(b)(1)
D+L+ P,+T,+R,+E P,,,
1.0 S.
22,000 15,505 94.1 (P,= 2,T,=70*F,E=OBE)
. (iii)(b)(1)
D+ L+ P,+T,+R,+E P +P,+O 3.0 S.,
80,100 21,540 86.3 t
(P,=-2,T,=110*F,E=OBE)
- i r
I (iii)(b)(1)
D+L+ PgT,+ R,+E P,,,
1.0 S.
22,000 26,316 95.4
[
(P,=49,T,=70'F,E=OBE) t (iii)(b)(1)
D+ L+PgT,+R,+E P + P,+ O 3.0 S,,,,
80,100 79,986 86.3
[
o (P,=49,T,=290'F,E=OBE)
(iii)(b)(2)
D+ L+ P,+T,+ R,+ E P,-
1.0 S.
22,000 15,509 94.1 (P,=0,T,=70*F;E=OBE) ij (iii)(b)(2)
D+L+P,+T,+R,+ E
. P +P,+O 3.0 S,,,,
80,100 21,808 86.3 o
(P =0,T,=110'F E=OBE)
.l o
t (iii)(b)(3)
Not applicable M'
(iii)(b)(4)
Not applicable i
!-t t
- Definition of variables in this column is given in Table 2.7.
i 39 l
l l
l Table 2.5 Level C Service Umits l
SRP Design Allowable Stress -
Maximum Value l
Reference Description
- Intensity Limit (psi) as per Stress Number Analysis (psi)
Type
- Limit Value Value Elev.
(psi)
(psi)
(iii)(c)(1)
D+ L+P,+T,+ R,+ E' P,
1.0 S, 52,480 21,183 94.1 (P,=-2,T,=70*F,E'=SSE)
(iii)(c)(1)
D+L+P,+T,+R,+ E' P,,,
1.0 S, 52,480 30,269 94.1 (P,=49,T,=70*F,E'=SS E)
(iii)(c)(2)
D+ L+ P,+T,+R,+ E' P,,,
1.0S, 52,480 21,187 94.1 1
(P,=0,T,=70*F,E'=SSE)
(iii)(c)(3)
Not applicable
- Definition of variables in this column is given in Table 2.7.
I 3
40 l
1 I
i I
l l
Table 2.6 Level D Service Umits SRP Design Allowable Stress Maximum Value 1
Reference Description
- Intensity Umit (psi) as per Stress Number Analysis (psi)
)
Type
- Umit Value Value Elev.
(psi)
(psi) '
(iii)(d)(1)
D+L+ P,+T,+ R,+Y,+YgY,,,+ E' P,,,
S, 47,600 21,183 94.1
.l (P,=-2,T,=70*F,E'=SSE) i (iii)(d)(1)
D+L+P,+T,+R,+Y,+YpY,,,+E' P,,,
S, 47,600 30,269' 94.1 (P =49,T,=70*F.E'=SSE) i (iii)(d)(2)
Not applicable j
i
' Definition of variables in this column is given in Table 2.7.
I
+
t
[
i
-i l
41 t
Table 2.7 Nomenclature P
Stress Intensity (difference between the algebraically largest and smallest principal stresses, twice the maximum shear stress).
P,,,
General primary membrane stress intensity. Average stress across an entire section of a vessel. Not self limiting. Gross deformation occurs if this stress exceeds yield. An example is general membrane stress in a cylinder or sphere with intema! pressure. Temperature stresses are Dat included. Therefore, the temperature increment is set equal to zero in Tables 2.2 to 2.6 in those cases for which P,,, is checked. These stresses are checked at the shell middle surface.
O Secondary stress intensity. Self-limiting. An example is the stresses due to the bending stress resultants M,, M,, M,, for pressure or seismic loading. All thermal stresses are secondary. Hence, for those cases in Table 2.2 to 2.6 for which primary plus secondary stresses are checked, the temperature is at the operating or accident level. These stresses are checked at the shell surface.
l t
P Local primary membrane stress intensity. A stressed region may be considered j
t local if the distance over which the membrane stress intensity exceeds 1.1 S.
does not extend in meridional direction more than (rt)". Typically self-limiting like a secondary stress but redistribution takes place only after large deformations. An i
example is the local membrane stress near a gross structural discontinuity such as shell intersections at the springline or at a penetration. Membrane stresses near the base of the containment may be considered in this category.
P, Primary bending stress intensity. Same as P,,, except bending stress. An example is the center of a flat plate with lateral pressures).
S' ~
Yield stress, ASME Table 1.2.0: 60,000 psi @ T=0'F; 52,480 psi @ T=290"F; l
59,500 psi @ T=110*F.
S, Allowable stress intensity, ASME Table 1-10.0: 22,000 psi.
P S,
Allowable stress intensity, ASME Table 1-1.0: 26,700 psi.
I S,
Allowable stress intensity 85% of the allowable membrane stress intensity specified in Appendix F: 47,600 psi.
l D
Deadloads.
t L
Uve loads including all ioads resulting from platform flexibility and deformation, i
and crane loading if applicable, equal to zero for this containment.
T, Thermal effects and loads during startup, normal operating or shutdown conditions, based on the most critical transient or steady-state condition.
R.
Pipe reactions during startup, normal operating or shutdown conditions, based on the most critical transient or steady-state condition, equal to zero for this I
containment.
I 42 i
i Table 2.7 Nomenclature (continued) s P.
Extemal pressure loads resulting from pressure variation either inside or outside containment.
E Loads generated by the operating basis earthquake including sloshing effects, if appfighle, j
E' Loads generated by the safe shutdown earthquake including sloshing effects, if applicable.
P, Pressure load generated by the postulated pipe break accident including P,, pool j
swell and subsequent hydrodynamic loads. For this containment, accidental spray j
actuation is included in this category.
T, Thermal loads under thermal conditions generated by the postulated pipe break accident including T., pool swell, and subsequent hydrodynamic reaction loads.
l For this containment, accidental spray actuation is included in this category.
R.
Pipe reactions under thermal conditions generated by the postulated pipe break l
accdent including R., pool swell, and subsequent hydrodynamic reaction loads, i
equal to zero for this containment.
.j
?
P, All pressure loads which are caused by the actuation of safety relief valve discharge including pool swell and subsequent hydrodynamic loads, equal to zero l
for this containment.
T, All thermal loads which are generated by the actuation of safety relief valve j
discharge including pool swell and subsequent hydrodynamic thermal loads, equal to zero for this containment.
1 j
(
i i
I i
i t'
f' Table A.1 Factors of Safety for ASME Service Umits j
r
?
l Service Umits Factory of Safety NE-3??? 2 Case N-284 Design Conditions 3.0 2.00-s Level A & B 3.0 2.00 i
i Level C 2.5 -
1.67 I
Level D 2.0 1.33 I
.i t
.I t
t i
i 6
i l
L Table A.2 BOSOR5 Comparison to Experimental Results Experiment /BOSORS Result
{
Shell Type Load Type Number Mean C.O.V.
Reference i
Torispherical Head intemal Pressure 8
1.06 0.082
[A.40]
Torispherical Head intemal Pressure 6
0.97 0.120
[A.41]
Ellipsoidal Head Intemal F ressure 5
1.09' O.209
[A.39]
l Ellipsoidal Head intemal Pressure 10 1.09' O.061
[A.09]
Stiffened Cylinder Extemal Pressure 69 0.960 0.057
[A.4]
Cylinder / Cone External Pressure 3
1.008 0.007
[A.3]
Torisphere/ Cylinder Extemal Pressure 4
1.021 0.034
[A.3]
Cylinder Axial Compression 10 1.019 0.058
[A.3]
f Cylinder Axial Compression 24 0.968 0.089
[A.3]
l Stdfened & Unstiffened Axial Compression 40 0.995 0.117
{A.1,A.3]
l Staffened Cylinder Axial Compression 5
1.459 0.284
[A.5, No.5] -
Weighted 1.00 0.135 i
- Experimental pressures reduced by 20 percent as suggested in [A.29).
F f
i t
I i
[
45 t
i
I' l
I l
t Table B.1 Modal Participation Factor for the First Mode of Vibration of the Cantilevered Beam in Fg. B.1
- i Circumferential Modal Participation Factor.
Wave No.
BOSOR4 Hand ANSYS^
i N
- l Calculation 1
0.8125 1.625 1.558
' j
. e 0
0.6463 0
6.548E-05
-1 0.00105 1.625 1.558
. i t
1 l
l l
' )
+
1
[.
- i i
46
i l
i t
i v
i
'l h
}
i i
i I
Table B.2 Modified Ubraries and Subroutines of BOSOR4 -
Source Code for Creating FILE 14.DAT.
BOSOR4 Ubrary Subroutines i
1.
B4 READ READIT 2.
GEOM GEOMTY 3.
RFIVE RFIVE 4.
WALL '
CFB2' f
5.
MODE MODE, LOCAL t
t t
1 1
I 47 j
e i
Table B.3.
Modified source code of LOCAL, MODE and OUT2 subroutines of BOSOR4.
C-DECK LOCAL SUBROUTINE LOCAL (IW,lS,HC,15,FN,F.C.B,PSTS BMODE,NSEG, THERM, 1PFXD,1NDIC,TIO,lALL,lVEC.SMODE.TMODE,lPOINT,lDIST IANALY, i
/
1 1 LOOP.lPOS, PALL,FF.RHF,RHFIX,RVAR,RFIX XSINGL)
DOUBLE PRECISION H,HPAST HI,C1 C2.C3,W,U,V,R,RD,FK1.FK2, CURD
.l DOUBLE PRECISION UD,VD,WD WDD. CHI,PS
]
DOUBLE PRECISION F C
C CALLED FROM MODE, WHICH IS CALLED FROM MAIN.
f' C
CALCUMTES MODE SHAPE FOR BUCKLING AND VIBRATION MODES.
C ALSO CALCULATES STRESS RESULTANTS OR STRESSES FOR l
C NONSYMMETRIC 1
C UNEAR STRESS ANALYSIS (INDIC-3). LOCAL DOES THIS FOR CURRENT l
I C
SHELL SEGMENT AND FOR CURRENT HARMONIC.
C IN UNEAR STRESS ANALYSIS DISPLACEMENTS, STRESS RESULTANTS, C
AND STRESSES ARE SUPERPOSED. FORCES AND MOMENTS IN DISCRETE :
C RINGS ARE ALSO CALCUMTED, ALTHOUGH THESE ARE NOT SUPERPOSED ii C
UNTil MTER.
C C-MODE SHAPE IN EACH SHELL SEGMENT CALCULATED ALONG WITH MODAL C
[
STRES DIMENSION F(1),C(15,14),B(15,5),PSTS(15,3),BMODE(IALL,9)
DIMENSION SMODE(9000),TMODE(9000),lDIST(1000),XSINGL(3000)
DIMENSION THERM (15,8),TIO(15,4),Z(100),T(100) 1 DIMENSION PFXD(15,3),1W(1000),HC(1),FF(*),lPOS(*), PALL (*) -
DIMENSION RHF(*),RHFIX(*),RVAR(*),RFIX(*)
COMMON /NOHARM/NHARM COMMON'ALLRNG/NTOT,lPRE
[
COMMOi4/SHEL/ISHL(95),1WAL(95),lTHK(95),lARC(95),lLOAD(95)
COMMON /INTRVL/INTVAL(95),10UT,NDIST,NCIRC NTHETA COMMON /XTRSS/SIG11,SIG10,SIG21,SIG20 TAUI.TAUO,SIGEl,SIGEO' COMMONflDZREF/lZREF(95) -
]
COMMON /FICTP/ IFICT COMMON /SFLAGI MONOO COMMON /STRSEG/ISTRSS(95)
-[
REAL N10,N20 COMMON /ISTUFB/KK,15 TOT,lC,NUNK,KMP,1 CALL REAL N1,N2 N12,M1,M2,MT,K1,K2,K12
[
COMMON /INTRG/NSTATN(95),lRING(95,20)
COMMON /AMPFAC/OMEGDR,YMTRL,YAXIAL,BDAMP,AMPLTD GAMP(200) i COMMON /RESTRT/lREST COMMON /RGTEMP/TNR(98),TM R(98),TM RX(98)
COMMON /DRGTEM/TNRFIX(98),TMRFIX(98),TMXFIX(98)
C:
l!
N=FN
(
IK - 1 N10 = 0.0 N20 = 0.0 CHIO = 0.0 IF (ISTRSS(IS).EO.1) CALL GASP (T,15,3,lTHK(IS))
-j 48' l.
1
]
1 i
Tabte BJ (continued) l C
RETRIEVE REFERENCE SURFACE LOCATION I
IF (ISTRSS(IS).EO.1) CALL GASP (Z,15,3,1ZREF(IS))
ITOT - 15 TOT - 2*(IS-1) i LINDX - IPOS (3)
IF (ILOOP.EO2) UNDX - IPOS (6)
LINDX3 - UNDX + IVEC - 1 l
C INDX - IPOS (9) i IF (ILOOP.EO2) INDX - IPOS (12)
I INDX3 - INDX + IVEC - 1 C
TIN - O.
TOUT-O.
Hi - HC(1)
C START LOOP OVER CURRENT SEGMENT MESH POINTS.
DO 1101-1,15 IF (ISTRSS(IS).NE.1) GO TO 5 IF (INDX.EO.0) GO TO 5 TIN - TIO(l.1)* PALL (INDX3)
TOUT - TIO(1,2)* PALL (INDX3)
IF (ILOOP.NE.1) THEN TIN - TIO(1,3)* PALL (INDX3)
TOUT-TIO(1,4)* PALL (INDX3)
ENDIF C
5 CONTINUE H - HC(l)
HPAST - Hi Hi - 2.*H-HPAST ITOT - ITOT + 1 t
il - 15 TOT + 1 + 1 13M - IW(ll - 1) 10 - IW(ll) 13P - IW(ll + 1) 12M 2 11M 1 11P - 10+1 12P = 10+2 IF ((13P-10).EO.(IC+NUNK-KLAP)) 11P - 10 + IC + 1 IF ((13P-10).EO.(IC+NUNK-KLAP)) 12P - 10 + IC + 2 C
DISPLACEMENTS AND DERIVATIVES FOR VARIABLE SPACING MESH C
C1 - (HPAST-HI)*(HI + H)/(16.*HPAST*H)
C2 - (HI + H)*(HPAST + H)/(4.*HPAST*HI) -
C3 --(HPAST-HI)*(HPAST + H)/(16.*Hi*H) t W - C1*F(13M) + C2*F(10) + C3*F(13P)
U - (F(12M) + F(11P))/2.
V -'(F(11M) + F(12P))/2.
IF (IREST.EO2) V -- V R-B(1,1)
RD - B(1,2) 49
Table B.3 (continued) i>
FK1 - B(1,3)
FK2 - B(1,4) r CURD - B(1,5)
CURD - 0.0 RS - R RDS - RD CURV1 - FK1 t
CURV2 - FK2 C
C AMPUTUDE FACTOR FOR HARMONIC DRIVING AT THE RESONANT C
FREOQUENCY.
C C
FIND INTEGRAL OF Y M*PHl. M* PHI IS STORED IN TMODE.
C C**********************************************************************
C PLEASE NOTE: THIS SECTION OF SUBROUTINE
- LOCAL" WAS MODIFIED (1/92)
C BY THE RESEARCH TEAM OF STRUCTURAL ENGINEERING DEPARTMENT C (CIVIL ENGINEERING) AT IOWA STATE UNIVERSITY, SO THAT C THE PROGRAM CALCULATEE CORRECT VALUES OF
- MODAL PARTICIPATION C FACTORS" FOR CIRCUMFERENTIAL WAVE NUMBER (N)
-1, O AND 1.
C THESE MODIFICATIONS CHANGETHE PARTICIPATION FACTOR RESULTS C OR " VIBRATION PROBLEMS
- PERFORMED WITH INDIC - 2 ONLY.
C FOR BETTER UNDERSTANDING OF THE CHANGES MADE IN THE PROGRAM, C THE FORTRAN UNES OF ORIGINAL VERSION OF THE PROGRAM ARE I
C PRESENTED BELOW. THEN EACH OF THE MODIFICATIONS MADE IN THESE C UNES ARE EXPLAINED.
C C
C UNE FORTRAN I
C NO.
UNE C-C 1 IF (INDIC.NE2) GO TO !
C2 IF (IABS(N).GT.1) GO TO 7 C 3 IF (N.EO.0) GO TO 6 C 4 AMPLTD - AMPLTD +XSINGL(11M)*1. +XSINGL(10)*R*FK2 C
+XSINGL(11P)*RD C 5 IF (i.EO.1) AMPLTD-AMPLTD +XSINGL(13M)*R*FK2 +XSINGL(12M)*RD C 6 IF (i.EO.15)AMPLTD-AMPLTD +XSINGL(13P)*R*FK2 +XSINGL(12P)*1.
C C 7 NOTE...THE ABOVE FORMULA IS FOR LATERAL FORCED MOTIONS
~
C ONLY...
C C8 GO TO 7 C 9 6 CONTINUE C
C 10 AMPLTD - AMPLTD +XSINGL(11M)*1. -XSiNGL(10)*RD C
JSINGL(11P)*R*FK2 C 11 IF (1.EO.1) AMPLTD-AMPLTD -XSINGL(ISM)*RD +XSINGL(12M)*R*FK2 C 12 IF (1.EO.15)AMPLTD-AMPLTD -XSINGL(ISP)*RD *XSINGL(12P)*1.
C C 13 NOTE...THE ABOVE FORMULA IS FOR AXIAL FORCED VIBRATIONS 50
Table B.3 (continued)
C ONLY.
C C 14 7 CONTINUE C
C C
c C 1 FROM UNES 2 & 3, IT IS CLEAR THAT FOR *N*
-1 & 1, THE PROGRAM C
CALCULATES THE VARIABLE *AMPLTD* USING THE FORMULA GIVENIN C
UNES 4,5 & 6. FROM THEORY. FOR *N*
-1. THE FORMULA FOR C
CALCULATING THE VALUE OF *AMPLTD* SHOULD BE DIFFERENT.
C HENCE C
THE FOLLOWING FORTRAN UNE WAS ADDED BETWEEN LINES 3 & 4 OF C
THE ORIGINAL CODE, SO THAT THE PROGRAM DOES NOT USE THE 4
C SAME FORMULA FOR CALCULATING THE VALUE OF *AMPLTD*.
C C
IF (N.EO.-1) GO TO 999 C
C C
2 THE FOLLOWING FCRTRAN UNES WERE ADDED BETWEEN UNES 8 &
C 9 OF THE ORIGINAL CODE, SO THAT FOR *N"
-1, THE PROGRAM C
USES THE CORRECT FORMULA FOR CALCULATING THE VALUE OF r
C
- AMPLTD".
C l
C 999 AMPLTD - AMPLTD -XSINGL(11M)*1. +XSINGL(10)*R*FK2 C
+XSINGL(11P)*RD C
IF (i.EO.1) AMPLTD AMPLTD +XSINGL(13M)*R*FK2 +XSINGL(12M)*RD C
IF (i.EO.15)AMPLTD-AMPLTD +XSINGL(13P)*R*FK2 -XSINGL(12P)*1.
C GO TO 7 C
I C
C 3 FOR *N* - 0, i.e. FOR AXIAL FORCED VIBRATIONS, THE PROGRAM C
CALCULATES THE VALUE OF *AMPLTD* USING THE FORMULA GIVEN C
BY UNES 10,11 & 12 OF THE ORIGINAL CODE.
C C
- FOLLOWING ERROR WAS IDENTIFIED IN LINE 10 C
-l C
C
/
> should be *0* (ZERO)
C
/
C
/
C AMPLTD - AMPLTD +XSINGL(11M)*1. -XSINGL(10)*RD +XSINGL(11P)*R*FK2 C
C C
C
- FOLLOWING ERRORS WERE IDENTIFIED IN LINE 12 C
C L
C C
C C
51
Table B.3 (continued)
C C
should be *0"(ZERO) <
\\
C
\\
C
\\
C IF (1.EO.15)AMPLTD-AMPLTD -XSINGL(ISP)*RD *XSINGL(12P)*1.
C
/
C
/
C should be *+* <
/
C C
C C
THE ABOVE ERRORS IN LINES 10 & 12 WERE FIXED.
C C****************************************************************************
C END OF THE ABOVE EXPLANATION C""*"""""""""*"""""""""*""*"""""*"*"""""*"*
C IF (INDIC.NE.2) GO TO 7 IF (IABS(N).GT.1) GO TO 7 IF (N.EO.0) GO TO 6 IF (N.EO.-1) GO TO 999 AMPLTD - AMPLTD +XSINGL(11M)*1. +XSINGL(10)*R*FK2 +XSINGL(11P)*RD IF (1.EO.1) AMPLTD-AMPLTD +XSINGL(13M)*R*FK2 +XSINGL(12M)*RD IF (i.EO.15)AMPLTD-AMPLTD +XSINGL(13P)*R*FK2 +XSINGL(12P)*1.
C C
NOTE...THE ABOVE FORMULA IS FOR LATERAL FORCED MOTIONS ONLY..
C GO TO 7 999 AMPLTD - AMPLTD -XSINGL(11M)*1. +XSINGL(10)*R*FK2 +XSINGL(11P)*RD IF (l.EO.1) AMPLTD-AMPLTD +XSINGL(13M)*R*FK2 +XSit;3L(12M)*RD IF (1.EO.15)AMPLTD-AMPLTD +XSINGL(13P)*R*FK2 -XSINGL(12P)*1.
GO TO 7 6 CONTINUE C
AMPLTD - AMPLTD +XSINGL(11M)*0. -XSINGL(10)*RD +XSINGL(11P)*R*FK2 IF (1.EO.1) AMPLTD-AMPLTD -XSINGL(13M)*RD +XSINGL(12M)*R*FK2 IF (i.EO.15)AMPLTD-AMPLTD -XSINGL(ISP)*RD +XSINGL(12P)*0.
C C
NOTE...THE ABOVE FORMULA IS FOR AXIAL FORCED VIBRATIONS ONLY.
C 7 CONTINUE C
US - U VS - V WS - W UD - (F(11P) - F(12M))/H VD - (F(12P) - F(11M))/H 52 L
A4 f
Table B.3 (continued) i IF (IREST.EO2) VD - -VD
.j C1
.5'HPAST -
C3.5/HI C2 -C1 -C3 WD - C1*F(13M) + C2*F(10) + C3*F(13P)
C1 - 1J(HPAST*H)
C2 -21(HPAST*HI)
C3 - 11(Hi*H)
WDD - C1*F(13M) + C2*F(10) + C3*F(13P)
CHIO - O.
N10 - O.
N20 - O.
CHI - WD - FK1*U CHIS - CHI C
STRAINS AND CHANGES IN CURVATURE E1 - UD + W *FK1 + CHIO* CHI i
FMUL - 0.0 K1 -W D D-FK1 *UD-CURD *U-FM UL'E1
- FK1 IF (R.GT.O.0) GO TO 10 PS - - CHI E2 - E1 E12 - 0.0 K2 - K1 K12 - 0.0 GO TO 20
- 10 PS - W'FN/R - FK2*V E2 - V*FN/R + U*RD/R + W'FK2 E12 - VD - V*RD/R + U*FN/R + CHIO*PS K2-PS*FN/R+ chi *RD/R-FMUL*E2*FK2 K12 -- CHl*FN/R + PS*RD/R + VD*FK2 20 CONTINUE i
C C IN THE FOLLOWING THE CONDITION INDIC.NE.2 WAS ADDED TO GET STRESS C RESULTANTS FROM THE MODAL ANALYSIS. THIS WAS DONE BY FOUAD AND C RAMA C
' IF (INDIC.NE.3. AND. INDIC.NE2) GO TO 70 i
C C
STRESS AND MOMENT RESULTANTS.
N1
- C(1,1)*E1 + C(1,2)*E2 + C(1,3)*K1 + C(I,4)*K2 N2 - C(1,2)*E1 + C(1,5)*E2 + C(1,6)*K1 + C(I,7)*K2 M1
- C(1,3)*E1 + C(1,6)*E2 + C(1,8)*K1 + C(1,9)*K2 M2 - C(1,4)*E1 + C(1,7)*E2 + C(1,9)*K1 + C(1,10)*K2 N12 - C(1,11)*E12 + C(1,12)*K12*2.
MT - C(1,12)*E12 + C(1,13)*K12*2.
C IF (ILOOP.EO.1) THEN i
N1 - N1 + THERM (,1)* PALL (INDX3)
N2 - N2 + THERM (1,2)* PALL (INDX3)
M1 - M1 + THERM (1,3)* PALL (INDX3)
M2 - M2 + THERM (1,4)* PALL (INDX3)
ELSE t
b Table B.3 (continued)
N1 - N1 + THERM (1,5)* PALL (INDX3)
N2 - N2 + THERM (I,6)* PALL (INDX3)
M1 - M1 + THERM (1,7)* PALL (INDX3)
M2 - M2 + THERM (1,8)* PALL (INDX3)
ENDIF IF (1.EO.1.OR.I.EO.15) GO TO 30 GO TO 70 30 IF (R.GT.O.0) GO TO 40 GO TO 70 C
C CORRECTION ADDENDS FOR EDGE STRESS RESULTANTS CALCULATED.
C 40 F1 - 0.5*RD*H/R F2 - 0.5*FN*H/R CN1 - N2*F1 - N12*F2 + 0.5'N10*CHl'H'FK1 l
CN12 - N12*F1 + N2*F2 + 0.5*N20*PS*H*FK2 CM1
-M2*2.*F1 + MT*2 'F2 - N1*CH10*H - N10* CHI *H IF (1.EO.15) GO TO 50 GO TO 60 50 CN1 - CN1 CN12 - CN12 CM1 - CM1 60 N1 - N1 + CN1 N12 - N12 + CN12 M1 - M1 + CM1 N2 - N2 + C(1,2)*CN1/C(l.1)
M2 - M2 + C(1,9)*CM1/C(1,8) 70 CONTINUE C
C THESE WERE ADDED BY FOUAD AND RAMA C
USING - SNGL(U)
WSING - SNGL(W)
VSING - SNGL(V)
WRITE (14,*) USING,VSING,WSING,N1,N2,N12,M1.M2,MT,C(1,14)
C C
IF (ISTRSS(IS).EO.1) GO TO 80 IF (ISTRS3(IS).EO.2) GO TO 85 IF (MONOO.EO.1.AND.lSTRSS(IS).EO.0) GO TO 85 C
BMODE CONTAINS GLOBAL QUANTITIES BMODE(ITOT 1) - U BMODE(ITOT,2) - V BMODE(ITOT,3) - W IF (INDIC.NE.3) GO TO 110 BMODE(ITOT,4) - N1 BMODE(ITOT,5) - N2 BMODE(ITOT,6) - N12 BMODE(ITOT,7) - M1 BMODE(ITOT,8) - M2 j
BMODE(ITOT.9) - MT 54
Table B.3 (continued) i GO TO 90 C
STRESS CALCULATES INNER AND OUTER FIBER STRESSES OF C
MONOCOQUE AND SEMI-SANDWICH CORRUGATED SHELLS.
80 IF (ISTRSS(IS).EO.1) CALL STRESS (E1,E2,E12,K1,K2,K12,T(l),Z(I),
ilS,1, TIN, TOUT.CURV1,CURV2)
BMODE(ITOT.1) - U BMODE(ITOT,2) - V
(
BMODE(ITOT,3) - W BMODE(ITOT,4) - SIG11 BMODE(ITOT,5) - SIG10 BMODE(ITOT,6) - TAUI BMODE(ITOT,7) - SIG21 t
BMODE(ITOT,8) - SIG2O BMODE(ITOT,9) - TAUO l
C i
GO TO 90 85 CONTINUE BMODE(ITOT.1) - U f
BMODE(ITOT,2) - V l
BMODE(ITOT,3) = W BMODE(ITOT,4) - E1 BMODE(ITOT,5) - E2 BMODE(ITOT,6) - E12 BMODE(ITOT,7) - K1 BMODE(ITOT,8) - K2 BMODE(ITOT,9) - K12 i
C 90 CONTINUE IF (l.NE.lRING(IS,1K)) GO TO 100 C
CALCULATE HOOP FORCE, IN-PLANE MOMENT, OUT-OF-PLANE MOMENT C
AND TOROUE IN DISCRETE RING.
IF (ILOOP.EO.1) 1 CALL RINGF(IK,KK,RS,RDS.CURV1,CURV2,FN,US,VS.WS,CHIS,FF, i
1 PALL (LINDX3),NTOT,NHARM,lVEC,RHF,TNR TMR.TMRX, IPOS (1), PALL,RVAR)
IF (ILOOP.EO.2) l-
.1 CALL RINGF(IK,KK,RS RDS.CURV1,CURV2,FN,US,VS,WS.CHIS,FF, 1
1 PALL (LINDX3),NTOT,NHARM,lVEC.RHFIX,TNRFIX,TMRFIX,TMXFIX, 1 IPOS (4), PALL,RFIX)
IK - IK + 1 l
KK - KK + 1 100 CONTINUE C
PERFORM SUPEROSITION FOR THIS PARTICULAR AXIAL STATION C
INEXT - IDIST(IPOINT)
NNDIST - NDIST IF (NDIST.EO.0) NNDIST - 1 NNCIRC - NCIRC IF (NCIRC.EO.0) NNCIRC - 1 IF (ILOOP.EO.1) 1 CALL SPOSE(ICALL,1,lTOT,lALL FN.BMODE,PSTS.CHIS, i
f
Table B.3 (continued) 1 S M O D E.TM O DE,15,NTHETA,lPOINT,1NEXT, LOUT,N DI ST,NCI RC,lS,l ANALY, 1 IVEC,N1,N2 lPOS(7), PALL,NNDIST,NNCIRC)
IF (ILOOP.EO.2) 1 CALL SPOSE(ICALL,1,lTOT,lALL.FN,BMODE,PFXD.CHIS, 1SMODE.TMODE.15,NTHETA,lPOINT,1NEXT,10UT.NDIST NCIRC.lS,lANALY, 1 IVEC,N1,N2,lPOS(10), PALL,NNDIST,NNCIRC)
C IF (ITOT.EO.INEXT) IPOINT - IPOINT + 1 110 CONTINUE 15 TOT - 15 TOT + 15 + 2 C
STORE PRESTRESS OUANTITIES FOR BUCKLING ANALYSIS (INDIC-4)
RETURN END A
r C-DECK MODE SUBROUTINE MODE (M3.NVEC,lALL,SMODE TMODE.BMODE,RHF,RHFIX, 1 RVAR RFIX,lLOOP,BBB)
C C
CALLED FROM MAIN.
C MODE SHAPE OR DISPLACEMENTS AND STRESSES CALCULATED FOR C B DitG AND VIBRATION OR UNEAR STRESS ANALYSIS.. INPUT C EIGENVECTOR IS F, C
OUTPUT IS U,V,W, AND STRESS RESULTANTS. THESE ARE NORMALIZED IN C
BUCKUNG AND VIBRATION PROBLEMS (ElGENVECTORS) AND DIMENSIONAL C
IN UNEAR NONSYMMETRIC STRESS PROBLEMS.
l C
DISPLACEMENTS AND STRESS RESULTANTS OBTAINED FOR NVEC
~
C EIGENVALUES C
DOUBLE PRECISION F DOUBLE PRECISION XX DIMENSION XX(3000),FF(6250),XSINGL(3000)
DIMENSION F(3000),BMODE(IALL,9),BBB(10000),DS(1000),1W(1000)
DIMENSION SMODE(9000),TMODE(9000),X(1000),lDIST(1000) i CO M M O N/AM P FAC/O M EG DR,YLATRL,YAXIAL,BDAM P.Af A PLTD.G AM P(200)
COMMON /RESTRT/lREST COMMON /AMPFC2/IMPHl(200)
COMMON /RGTEMP/TNR(98),TMR(N TMRX(98) l COMMON /ALLRNGl NTOT,lPRE COMMOt#NLINS/NSTART,NFIN CO M M Ot#S EG S/NS EG,M2.15(95),12,12G COMMOt#0 SPACE / IDS COMMON /STRHD/NSHEAD COMMON /OUTP/NPRT COMMOfflNSTAB/INDIC COMMOtWNOHARM/NHARM COM mot &SMO AD/IM OAD(100), LPM ODE (95),lPSMD(95)
COMMON /NCONDS/NCOND COMMON / WAVES /NO N NMIN,NMAX,INCR 56
- l 1
4 Table B.3 (continued) --
l COMMON /IDEIG/IDM ODE (500),NVECTL,ElG ENW(500)
COMMON /TWORHS/ITWO,1DMOD2(200) i COMMON /LOADHM/NLPOS(95), LAB (95),lDAB(95),lLPOS(12,95), PALL (6000)
COMMON /ISTUFB/KK,15 TOT,1C,NUNK,KLAP,1 CALL COMMOWLOCATMILOCP,lLOCB,lROWCP(98),lROWCB(98),llWP,IlWB COMMOWGMASS/GM(500)-
COMMON /INTRVU INTVAL(95),10UT,NDIST,NCIRC NTHETA COMMON /SHEUISHL(95),1WAL(95),lTHK(95),1 ARC (95),lLOAD(95)
COMMOWGLOBAUIDZ,1ZRING(98),lSAVE(20),1 NOTE
- i DIMENSION RHF(*),RHFIX(*),RVAR(*),RFIX(*)
l COMMON /RESPO2/ANALY,WSPEC(40)
COMMOWNVVlB/NVlB j
CHARACTER *20 WORD 1(3) l CHARACTER *36 WORD 2(3)
DATA WORD 1(1), WORD 1(2), WORD 1(3)/
1
- HARMONIC EXCITATION
- 1 ' RANDOM EXCITATION I
1
- SHOCK 7
C DATA WORD 2(1), WORD 2(2), WOW %
}
1 'N*g/(OMEGA"2
- BETA)
}
1 ' SORT [ OMEGA *SPECD/(2* BETA)]O/CKEGA"2',
1 1 *2*N*g/ OMEGA"2 7
l C
1 C THIS WAS ADDED BY FOUAD AND RAMA j
C 1
WRITE (14,*)NVEC
-l C
j 10 FORMAT (1P9E13.4) 11 - 12 + 2*NSEG j
M32 - M3*2 -
C RECOVER GLOBAL EQUATION NO.S CORRESPONDING TO W.
_i CALL GASP (IW,ll,3,IlWB)
C RECOVER ELEMENTAL ARC LENGTHS DS.
a CALL GASP (DS,12,3, IDS)
IF (INOTE.EO.1) CALL GASP (BBB,lSAVE(1),3,lSHL(1)).
IF (INDIC.NE.3) GO TO 82 C
FIND ARRAY X CORRESPONDING TO MERIDIONAL DISTRIBUTION. STRESSES :
C OR STRESS RESULTANTS ARE PRINTED OUT FOR THESE MERIDIONAL C SIKID6 -
-20 CALL IFIND(IDlST,X) l 40 CONTINUE j
K - NTHETA*NCIRC'9 j
IF (K.LE.9000) GO TO 50
- NTHETA - 2*NTHETA/3 l
GO TO 40 -
l 50 CONTINUE IF (NDIST.EO.0) GO TO 65
- DO 60 I'- 1,9000 5
60 SMODE(I) - 0..
1
~
65 CONTINUE IF (K.EO.0) GO TO 75 q
57 l
a
B h
Table B.3 (continued)
DO 701-1,K 70 TMODE(i) - 0.
'R 75 CONTINUE C
BEGIN LOOP OVER NO. OF EIGENVALUES FOR EACH HARMONIC N IN C
STABluTY OR VIERATION ANALYSIS.
C BEGIN LOOP OVER NUMBER OF HARMONICS IN NONSYMMETRIC STRESS C-ANALYSIS 80 CONTINUE IF (IREST.NE.2) GO TO 82 IANALY - ANALY WRITE (6,81) WORD 1(IANALY), WORD 2(IANALY) 81 FORMAT (r DYNAMIC RESPONSE ANALYSIS. 7/
1' N - LOAD FACTOR (NUMBER OF GRAVITIES)7 1*
g - ACCELERATION OF GRAVITY 7 1* OMEGA - NATURAL FREQUENCY l'1 RADIANS /SEC7 1'
BETA - DAMPING COEFFICIENTY 1* PARTICIPATION FACTOR:7 1*
P/2 -integra!(mass
- mode *(base motion)/[2*(generalized mass))
17 if RESPONSE TO *,A20/
1
- FORMULA FOR MULTIPUER,m(i)
- ,A36)
C WRITE (6,76) 76 FORMAT (r FREQUENCY PARTICIPATION LOAD FACTOR.. DAMPING 1-SPECTRAL MULTIPUER - AMPUTUDE7 1
(HERTZ)
FACTOR, P/2
'(N)
COEFFICIEN 1T DENSITY.
(m)
' FACTOR-m*P/27)
C 82 CONTINUE DO 250 Il-1,NVEC AMPLTD - O.
IF (IREST.NE.2.OR.INDIC.NE.3) GO TO 85 FREQU - EIGENW(NVECTL+11)
CALL RESFAC(FREQU,FMULT.FNGVAL,DAMPNG,SPECD)
AMPLTD - GAMP(NVECTL+ll)*FMULT WRITE (6.83) FREQU, GAMP(NVECTL+ll), FNGVAL, DAMPNG, SPECD, FMULT, 1
AMPLTD 83 FORMAT (1PE12.4.1PE16.4.1PE16.4.1PE14.4,1PE13.4,1PE13.4,1PE14.4) 85 CONTINUE IF (INDIC.EO.3) GO TO 90 FN - FLOAT (N)
GO TO 100 90 ' FN - FLOAT (NSTART + (ll-1)*lNCR)
C RECOVER THE SOLUTION VECTOR F 100 ' CONTINUE 1F (ILOOP.EO.1) CALL GASP (F.M32,3,lDMODE(NVECTL+11))
IF (ILOOP.EO.2) CALL GASP (F,M32.3,lDMOD2(NVECTL+ll))
105 FORMAT (/r NVEC.II,M3,M32,lDMODE(II),F(IM3),IM3-1,M3)
- 517/
1 (1P10E12.3))
C IF (INDiC.EO.2) CALL GASP (XX,M32,3,IMPHl(NVECTL+11))
58-L.:
r L.
Table B.3 (continued)
C IF (INDIC.NE.3) GO TO 110 110 WMAX - 0.0 KK -1 15 TOT - 0 IC - 4 NUNK - 7 KLAP - 4 ICALL - 1 IF (INDIC.EO.3) GO TO 210 C
C NORMAllZE EIGENVECTOR (BUCKLING AND VIBRATION PROBLEMS ONLY, C
INDIC.NE.3.)
J-1 1-0 120 1-l+1 IF (J.GT.NCOND) GO TO 140 IF (1.EO.lROWCB(J)) GO TO 130 GO TO 140 130 1 - I + IC j~
IF (i.GT.M3) GO TO 150 J-J+1 140 FI - F(l)
WMAX - AMAX1(ABS (FI),WMAX)
IF (1.GE.M3) GO TO 150 GO TO 120 i
150 CONTINUE IF (WMAX.EO.0.) WMAX - 1.
DO.1601-1,M3 160 F(l) - F(lyWMAX DO 1701-1,M3 FI - F(l)
IF (ABS (Ff).EO.1.0) GO TO 180 170 CONTINUE
- GO TO 200 180 IF (Fl.GT.O.) GO TO 200 DO 1901-1,M3 190 F(I) - F(l) 200 CONTINUE
- FMULPY - 1.0 IF (N.EO.0) FMULPY - 2.0 IF (INDIC.NE.2) GO TO 210 GM(NVECTL+11) - FMULPY*3.1415927* ABS (GM(NVECTL+11)/WMAX"2)
C i
DO 205 l - 1.M3 XSINGL(1) - SNGL(XX(I))*FMULPY*3.1415927/WMAX 205 CONTINUE 210 IPOINT - 1 ITOT - 1 ITOTL - O l
IF (INOTE.GT.1) CALL GASP (BBB,lSAVE(1),3,lSHL(1))
j l
I 59
Table B.3 (continued) lADD - 2 DO 2401-1,NSEG C
C MODAL DISPLACEMENTS AND STRESSES FOR EACH SEGMENT FOUND IN C
LOCAL C
151 - 15(1) 220 IPOS 43*lTOTL + 1 IEND - IPOS + 43*l51 - 1 IF (IEND.LT.10000) GO TO 230 IF (INDIC.NE.4) CALL GASP (BBB,lSAVE(IADD-1),1,lSHL(IADD-1))
CALL GASP (BBB,lSAVE(IADD),3,lSHL(IADD))
ITOTL - O lADD -lADD + 1 GO TO 220 230 CONTINUE CALL LOCAL (IW,1,DS(ITOT),151,FN.F BBB(8*151+1POS),BBB(22'15t+1POS) 1 BBB(33*l5l+1POS),BMODE,NSEG,BBB(IPOS),BBB(36*l5f+1POS),1NDIC, 2BBB(39*151+1POS),lALL,II.SMODE.TMODE,lPOINT,lDIST,lANALY, 31 LOOP,lLPOS(1,1), PALL,FF RHF.RHFIX,RVAR,RFIX XSINGL)
ITOT - ITOT + 151 ITOTL - ITOTL + 151 240 CONTINUE
\\
C C
h.......................................................................
C IF (INDIC.EO.2) GAMP(NVECTL+ll)-ABS (AMPLTD)/(GM(NVECTL+11))
...................................... n.................................
i C
C PLEASE NOTE: THE ABOVE FORTRAN LINE WAS MODIFIED (1/92) BY THE
]
C RESEARCH TEAM OF STRUCTURAL ENGINEERING C
DEPARTMENT (Civil ENGG.)
C AT IOWA STATE UNIVERSITY, SO THAT THE PROGRAM CALCULATES C
THE CORRECT VALUES OF
- MODAL PARTICIPATION FACTORS
- FOR l
C VIBRATION PROBLEMS PERFORMED WITH INDIC - 2.
C C
THE CORRESPONDING FORTRAN LINE IN THE ORIGINAL CODE WAS C
IF (INDIC.EO.2) GAMP(NVECTL+ll)= ABS (AMPLTD)/(2.*GM(NVECTL+11))
C C
C*"*"*"*"*""*"""""""""""**"""*""*""""""""""
C C
IF (INDIC.EO.2) WRITE (14,*)
1 EIG ENW(NVECTL+11),G M(NVECTL+ 11), GAM P(NVECTL+ll)
C-60
Table B.3 (continued)
I IF (INDIC.NE.4.AND.INOTE.GT.1) 1 CALL GASP (BBB,lSAVE(IADD-1),1,lSHL(IADD-1))
l IF (INDIC.EO.3.AND.ll.LT.NVEC) GO TO 250 C
IF (INDIC.NE.4.AND.INOTE.EO.1) 1 CALL GASP (BBB,lSAVE(IADD-1),1,lSHL(IADD-1))
'i NN - IALL*3 -
C STORE MODE SHAPES f
IF (INDIC.NE.3) CALL GASP (BMODE,NN,1,lMOAD(ll+NVlB))
IF (INDIC.NE.3) GO TO 250 jl NN - IOUT*9'NDIST IF (IREST.NE.2) GO TO 247
-IF (IANALY.EO.1) GO TO 247 j
IF (NN.EO.0) GO TO 244
'l INBEG - LOUT *NDIST*3 + 1
.j C
WRITE (6,*)* 1ANALY,10UT,1NBEG NDIST,NN *,
C 1
. IANALY,10UT,1NBEG,NDIST,NN
.[
DO 2431 - INBEG NN j
C WRITE (6,*)* l.SMODE(1) *,1,SMODE(I)
]
C. IF (MOD (1,10UT).EO.0) WRITE (6,*)* NEW COLUMN...'
t 243 SMODE(1) - SORT (SMODE(I))
244 NK - NTHETA*NCIRC 9
{
IF (NK.EO.0) GO TO 247 INBEG - NTHETA*NCIRC*3 + 1.
{
. DO 2451 - INBEG,NK-245 TMODE(1) - SORT (TMODE(1))
247 CONTINUE C
STORE MERIDIONAL STRESS OR STRESP RESULTANT DISTRIBUTIONS.
C STORE CIRCUMFERENTIAL DISTRIBUTIONS i
i!
C STORE RING LOADS NN1 - IOUT'NDIST*9 NN2 - NTHETA*NCIRC'9 NN3 - NTOT*NHARM*4 C
.l 3
IF (ILOOP.EO.1) THEN IMOAD(100) - O t
IF (NN1.NE.0) CALL GASP (SMODE.NN1,1,lMOAD(100))
ll
'I lMOAD(99) = 0 IF (NN2.NE.0) CALL GASP (TMODE,NN2,1,lMOAD( 99))
.l IMOAD(96) = 0 IF (NN3.NE.0) CALL GASP (FF, NN3,1,lMOAD( 96))
ELSE IMOAD(98) = 0 IF (NN1.NE.0) CALL GASP (SMODE.NN1,1,lMOAD( 98))
IMOAD(97) - 0
- IF (NN2.NE.0) CALL GASP (TMODE,NN2,1,lMOAD( 97))
i IMOAD(95) !
. ENDIF NN3,1,lMOAD( 95))
[
IF (NN3.NE.0) CALL GASP (FF, 1
C
.l i
250 CONTINUE I
61 q
=-
j
' Table B.3 (continued)
RETURN END C-DECK OUT2 SUBROUTINE OUT2(NVEC.NTOT,BMODE.SMODE TMODE,EGV,lLOOP)
C CALLED FROM READIT, WHICH IS CALLED FROM MAIN ~
C ESSENTIAL OUTPUT FROM CURRENT CASE IS PRINTED.
COMMON /INTRVU INTVAL(95),10UT,NDIST,NCIRC.NTHETA COMMON /ITERS/ ITER COMMON /SMO AD/IMOAD(100),1PMODE(95),lPSM D(95)
COMMON /SHEL/ISHL(95),1WAL(95),lTHK(95),lARC(95),lLOAD(95)
COMMON /FORCESN(98),HF(96),FM(96)
CGMMON/INSTAB/INDIC j
COMMON / SEGS /NSEG M2,15(95),12,12G
'i COMMON / WAVES /NO N,NMIN,NMAX,1NCR COMMON /LSTEPS/ISTEP COMMOWSFLAG/MONOO COMMOWEIGENV/P,OMG2, RHO COMMON /TEMTUR/ TEMP DTEMP COMMON / STEPS / STEP, STEP 1(98), STEP 2(98), STEP 3(98)
COMMON / DETER /DET,NEX COMMON /GMASS/GM(500)
COMMON /IDElG/lDM O DE(500),NVECT1,EIG ENW(500)
COMMON /ROTATN/ OMEGA,DOMEGA COMMOWAMPFAC/OMEGDR,YLATRL,YAXtAL BDAMP,AMPLTD,GAMP(200)
COMMON /NVVIB/NVIB -
COMMOWlOCHOZ/lOYES(50),10SEG(95)
DIMENSION BMODE(1000,9),SMODE(9000),TMODE(9000),EGV(50),X(1000)
DIMENSION VERT (98),HORIZ.(98),FMOM(98)
K - INDIC + 3 GO TO (10,20,50,90,200,210,220),K 10 CONTINUE C
INDIC - - LOAD INCREASED IN STEPS UNTIL DETERMINANT CHANGES C
SIGN. THEN INDIC CHANGED TO - 1.
C-CALL OUTLOD(INDIC,lTER.DET,NEX,P TEMP,V,HF FM NTOT. OMEGA)
GO TO 230 20 CONTINUE-C INDIC - STABluTY ANALYSIS WITH NONLINEAR PREBUCKING C
AND SEARCH FOR MINIMUM BUCKUNG LOAD WITH CIRCUMFERENTIAL C
WAVENUMBER,N.
C CALL OUTLOD(INDIC,lTER.DET,NEX,P, TEMP.V,HF FM NTOT. OMEGA)
WRITE (6.30) 30 = FORMAT (////15X,82H PREBUCKUNG DISPLACEMENTS AND STRESS 1 RESULTANTS.
1 CORRESPONDING TO CRITICAL LOAD
)
CALL OUTPEE(IPMODE(1),lARC,12,15,NSEG.BMODE,0)
IF (IOYES(4).EO.1) GO TO 230 62
t i
si Table B.3 (continued)
?
WRITE (6.40)N 40 FORMAT (////19H BUCKUNG MODE FOR 17,23H CIRCUMFERENTIAL WAVES 1).
CALL OUTBUC(IMOAD(1),lARC 12,15,NSEG BMODE,1NDIC) 4 GO TO 230 50 CONTINUE I
C AXISYMMETRIC NONLINEAR STRESS ANALYSIS GO TO 230 4
l 90 CONTINUE C
INDIC - 1,2, OR 4--BUCKLING AND VIBRATION ANALYSES IN WHICH l
C MANY EIGENVALUES AND EIGENVECTORS FOR EACH N MAY BE OBTAINED.
d C-IF (INDIC.EO.1.OR.lNDIC.EO.4) WRITE (6,100) 100 FORMAT (//33H BUCKUNG LOADS AND MODES FOLLOW )
i IF (INDIC.EO2) WRITE (6,110) 110 FORMAT (//24H VIBRATION MODES FOLLOW
)
WRITE (6,120)N l
i 120 FORMAT (/35H CIRCUMFERENTIAL WAVE NUMBER, N -
17/)
WRITE (6,140)(ElGENW(NVECTL+1),1-1,NVEC) l 140 FORMAT (* EIGENVALUES -Y(1P10E12.3))
IF (INDIC.EO2) WRITE (6,145)(GM(NVECTL+1),1=1,NVEC)
{
145 FORMAT (/' GENERAUZED MASS -7(1P10E12.3))
-}
C""""""*""""""""*"*""""""*""*"""*""""*""
C C
THE FOLLOWING UNES WERE ADDED (1/92) BY THE RESEARCH TEAM OF.
C STRUCTURAL ENGINEERING DEPARTMENT (civil ENGG.) AT IOWA STATE
[
C UNIVERSITY.
C' C
~l IF (INDIC.E02) WRITE (6,9991)
-l 9991 FORMAT (/54H PLEASE NOTE THE FOLLOWING VALUES OF PARTICIPAT!ON 1)
IF (INDIC.EO2) WRITE (6,9992) i 9992 FORMAT (54H FACTORS ARE BASED ON THE MODIFIED VERSION OF THE _
1) l IF (INDIC.EO2) WRITE (6,9993) 9993 FORMAT (54H BOSOR4 SOFTWARE, WHICH IS INSTALLED ON AMES LAB -
1)
_j IF (INDIC.EO2) WRITE (6,9994) 9994 FORMAT (54H CADVAX.
)
IF (INDIC.EO2) WRITE (6.9995)
I 9995 FORMAT (/54H THE MODIFICATIONS WERE MADE BY THE RESEARCH TEAM -
1)-
IF (INDIC.EO2) WRITE (6,9996)
-l 9996 FORMAT (54H OF STRUCTURAL ENGINEERING DEPARTMENT (civil ENGG.)
-1AT
_ ) -.
i IF (INDIC.EO2) WRITE (6,9997) t 9997 FORMAT (54H IOWA STATE UNIVERSIT(. THE MODIFICATIONS WERE MADE
_11N
{
}'
l IF (INDIC.EO2) WRITE (6,9998) -
9998 FORMAT (54H SUBROUTINES " LOCAL" &
- MODE" OF MODE UBRARY.
)
l C-C c........................................................................
j e3
Table B.3 (continued)
- C C'
~
IF (INDIC.EO.2) WRITE (6,147) (GAMP(NVECTL+1),1-1,NVEC)
.147 FORMAT (/' PARTICIPATION FACTORS '/(1P10E12.3)).
DO 1901-1,NVEC NVECTP - NVECTL + l EGV(1) - EIGENW(NVECTP)
IF (INDIC.NE.1) GO TO 170 i
PRESS - P + (EGV(I) - 1.)* STEP
- j TEMPT-TEMP + (EGV(1) - 1.)*DTEMP
-[
OMEGAX - OMEGA + (EGV(1) - 1.)*DOMEGA
=l IF (NTOT.EO.0) GO TO 170 DO 150 J-1,NTOT l
VERT (J) - V(J) + (EGV(I) - 1.)* STEP 1(J)
HORIZ(J)- HF(J)+ (EGV(I) - 1.)* STEP 2(J)
{
' 150 FMOM(J) - FM(J)+ (EGV(1) - 1.)* STEP 3(J)
WRITE (6,160)I 160 FORMAT (//45H CRITICAL LOAD COMBINATION FOR EIGENVALUE NO. 13, 119H
[
FOLLOWS
//)
j
- CALL OUTLOD(INDIC,lTER.DET,NEX, PRESS TEMPT. VERT,HORIZ,FMOM,NTOT, 1 OMEGAX)
L!
170 CONTINUE IF (IOYES(4).EO.1) GO TO 190 WRITE (6,180)I 180 FORMAT (//30H MODE SHAPE FOR EIGENVALUE NO. 13,9H FOLLOWS -)
r
~ CALL OUTBUC(IMOAD(1+NVIB),lARC,12,15,NSEG BMODE,1NDIC) 190 CONTINUE -
GO TO 230 200 CONTINUE C
INDIC - VIBRATION MODES r
GO TO 90
[
210 CONTINUE
-l C
INDIC - LINEAR NONSYMMETRIC STRESS ANALYSIS '
NNDIST - NDIST
.l, IF (NDIST.EO.0) NNDIST - 1 jl NNCIRC - NCIRC IF (NCIRC.EO.;., NNCIRC i CALL OUTNON(IOUT,NDIST,NCIRC,NTHETA,lMOAD,1NTVAL,MONOO SMODE, 1TMODE,X,lLOOP,NNDIST,NNCIRC) t l
GO TO 230
' 220 CONTINUE C
INDIC - BUCKLING WITH NONSYMMETRIC PRESTRESS OR WITH -
l C
PRESTRESS ANALYSIS BYPASSED I
GO TO 90
- 230 CONTINUE RETURN-END I
l
.j i
64 m6'
'M-a
.p*
-b,
, - SHIELD BUILDING ANNULUS l
OONTAINMENT (STEEL) i 4-1
.t h
i 100'l.R 1
1 D lt
- 49 REACTOR REACTOR BUILDING BUILDING t-
$.}
ELEV.
1._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ [ *' _.
!$l TRANSmON REGION Figure 2.1 Cross Sectional Elevation of the System 80+* Containment p
f f
65 u--
l:
ELEY.
_ _ _ _ _ _ _ _ _ _ _ _ _ _. - - - -. 2 4 3. 6 5
i l
i
)
I 167.5 j
V.....
yy-...~
~
- - - - - - - 14 3.1 122.8
- _. ~ ~ ~ ~ ~ ~ 1 13.2 ARC LENGTH 90.1 86.3 SPRAY HEADER SYSTEM ELEV. 24 3.6 SPRAY HEADER SYSTEM CONSTITUTES ONE SEGMENT EOUIPMENT HATCH AND PERSONNEL LOCK 1 ELEV.143.1 TO 167.5 PERSONNEL LOCK 2 ELEV.113.2 TO 122.8 ELEY. 86.3 TO 90.1 TRANSITION REGION Figure 2.2 BOSOR Axisymmetric Model Of the System 80+* Containment 66
1 25000 I
(
- - + sigma i 4mer n i
ligma i outer Ebei
- - - = - - - + -
15000
-4 Gigma 2 inner FLke 10000 I' -
~
ligma 2 Outer Fbbi, d'.
m l
5000 4
- c +-
- - + - - -
o
[?.'M 4:i:-;i=====>2 O
A 9
-4 8 - i-] +---
-5000 M
i :1 $1A m
- ni' E-10000 3
i --- 4 --
+
' a{%
~&
i
- 'l ag g.$5990 :
.+.
I, e
bm
-+
a-
-20000 ' --- h i
-25000
/:I F
--- i Af J
l
-30000
~+-
m-
-i-
- I i
i 35000 g.4__;._
_4
.t
/
'l
-40000 7-
- e-
+- -
- -+
l
-A5000 g-----
-r-
- + -
r v
-50000 86.3 94.7 111.3 136.3 Elevation (ft.)
Figure 2.3 Extreme Fiber Stresses Due to a Temperature Differential of 220 F (indicating only the first two segments) 67
)
25000 20000
- 80= ' * *= 4 -<
.-+--
Strre1OJeFt.!
". - - ~+<
15000 som a w. re. {
'"# ~
10000 sv= 2 os rt !
5000 g- --
L'y + 7
--- -= * =.== = = = =
- O
,e IF
-5000
,A g
aQ/
/
E-10000
-e 25 o
I 0-15000 o
b(n t-
-20000 i
-25000
- - - - - - - - - +
L ----
-30000
--t-s-
-35000
-40000
+
-45000
-50000 86.3 94.7 111.3 136.3 Bevaton (ft.)
Figure 2.4 Extreme Fiber Stresses Due to a Temperature Differential of 40"F (indicating only the first two segments) 68
d-1.000 4
A-.
- .v--.
... +.....
....g
+
T a
j':
.. i}
l
~
100
-v" y.
t
,o o
tT 30 A...
.4.
a2 i.
o C
e i
O
- p a
r b
~-
3 10 o
o l
~d b
.i..:
o 3
~~-
- .I-
- a-e--
- + -
C) s i
i
~
1 i
?
+
i i
1 O.3
~'-
l iY DIRECT 10fJ Z DIRECTION l X DIRECTION 3
0.1 0.1 0.3 1
3 10 30 100 Frequency (eps) -
i i
Figure 2.5 Responso Spectra for Operating Basis Earthquake for Soil Condition B4 i
l i
I 69
.i
- - + - - -
4-1,000
. i 300 4--.------
4---
t-
- b '
Q.
i 100
- - + ---- +.! --
4---
a-~
m
- /
.t:
-oo (T
30
- f' - - * ---
<=-
c O
33 i*
co
-k 10 o
oo.
<C r-
- B b
o o
4 - - --
a 3
/-
- - - - + -
to I
- l t
l 1
- + -
+
0.3 iY DIRECTIO, N Z DIRECTION i X DIRECTION 5....
i 0.1 1
0.1 0.3 1
3 10 30 100 Frequency (eps) i 4
Figure 2.6 Response Spectra for Safe Shutdown Earthquake for Soil Condition B4 l
t s
70 l-i
1 1
i f
I i.
l-1,000
+ - - - - - - - '
+---
t e
+"
1 t
+.:.
- .F ' - v' 300 r --
- i 100
-+-
A - - +-
~ ' " -
n 1
o en tr i
-4
= - - - - '-
i 30
- = - - '
1
/
c t
o a
v a
a W
3 10
- + + -: -
- - + -
[
o o
s b
[
i o
o
. : s ct 3
+
- -t- -
--r---~--~+----
- + -
to a
e 1
T
- f I
0.3
--i-
-+
4 j'
i lY DIRECT 10td Z DRECTDid j X DIRECT)ot4 l
....-4 i
O.1 i
O.1 0.3 1
3 10 30 100 j
Frequerry (cps)
- l i
i
~ [
Figure 2.7 Response Spectra for Safe Shutdown Earthquake for Soil Condition l
C1 e
' 5 r
i f
I r
I i
71
)
I 1,000
.s
. 4
- 1..
-+ +
300
~
~.
5....
- s. s
.~.
l 100
+. <
6 e.
no I
e i
a p
30
- +f t-o C
l o
=
t 4
y 2
10
=
o o<
a bo E
3 i
v-
+- --& ---
+ - -
~
e i
i i
1
+
k r
e
.. N...
.. k --
-..g......:.
}
..n.....
..3..,
l Y DIRECTION ZDIRECTION X' DIRECTION O.1 0.1 0.3 1
3 10 30 100 Frequency (eps)
Figure 2.8 Response Spectra for Safe Shutdown Earthquake for Fixed Base Condition
+
9 k
.r
+
72 i
.j i
s 1
1 l
16.000
! O deg.
i i
t 45 deg.
1 i
' - - - - +
14,000 1
i i
i i
!90 deg.
i 12,000
~
i i:
it i
i i:
j kjo,oog _l(
... +.
4 4
.o o
M s-3 s.-
m
- + -
- 8,000 M
CC a.
o o
b
-1 co 6.000 9.
+-
\\ '.,
i t,
'\\ *.
n-p..
4.000
-- - + N.; -
- + - - - - - -
s 1
~~
i
' ::... - + - - - -
+ -
- 2.000
- + - -
~ : m.
o 86.3 1 21.2 162.2 202.4-234.7 257.0 Elevation (ft.)
i i
Figure 2.9 Comparison of SRSS Meridional Stress Resultants at 0 deg..'45:
.i i
deg.' and 90 deg. Meridians.
?
t N
l I
r
3 16,000
=O deg 45 deg 14,000 e-
+
go def 12,000
- k - +
+-
t 4.
e C
+
m 10,000
-h. - -
+ - -
n
+
o u
5 e
a m
O Q QQQ
.-.h.~,
..4..
C t
e 5
m Od i
(0
'., r.
6,000
-if.
+-
i s.
~.
l s..'..
4,000
- - - - - +
t
. i...
.t
+.,
p A
.._;- n--
R.
2,000 4-n-
'.~.
-Q r
O i
86.3 121.2 162.2 202.4 234.7 257.0 i
Bevation (ft.)
i i
Figure 2.10 - Comparison of SRSS Circumferential Stress Resultants at 0 deg.,
45 deg., and 90 deg. Meridians -
b 1
k I'
74
E i
If 80
\\
i 60 TS o<
O 8 g
.2-.._..
Oz hooto 1
20 1
l N-284 value (predicted) '= 9.20 psi
y--------.
x O
O 5
10 15 20 WAVE LENGTH PARAMETER K Figure 3.1 Complete Sphere with Extemal Pressure - Variation of Imperfection Wavelength l
75 i
80 I
- - - - - + -
60
+ - - - -
TS l
O l
Om
.+.
z U
O 3m 20 l-
- -i -
_ _ _-284 value (predicted) =_9.20 psi N
O O
O.5 1
1.5 2
AMPLITUDE Wo Figure 3.2 Complete Sphere with Extemal Pressure - Variation of Imperfection Amplitude 76
COMPARISON : BUCKLED SHAPE Vs ORIGINAL GEOMETRY (COMPLETE SPHERE) a BAPERFEcT SHELL i
s l
BUCKLED SHAPE w
o o
b i
z
[3EGMENT2 SEGMENT 5 1
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%C FDRu 335 U3. NUCLEAR REGULATORY COMMISSION
- 1. REPORT NUMBER eat!mc2.
7J"'ut,,, l1". @ "-
w.m BIBLIOGRAPHIC DATA SHEET NUREG/CR-5957 is am~ct n on er,, mma IS-5083 2.TGd AND SUSTOLE System 80+D Containment -- Structural Design Review
,"'"**"S",',
May 1993
- 5. AUTHOR (Si
- 6. T YPE OF REPORT Technical
- 7. PE Rt00 COVE RED toncounae Damar L. Greimann, F. Fanous, R. Challa, D. Bluhm
- 8. P FORMING ANIZATION - N A8/ E AND ADDR E SS !rf %RC, awear Deesea. Onere er Nesvoa. u.s m.nev anguderary commasen. sea wap aaerras. de coar.c.or.psu,em Ames Laboratory Iowa State University Ames, 1A 50011
- 9. SPONSORING ORGANIZATION. NAME AND ADDRE.SS (tr N#C erpe ~5eme as eaose";Jr oarwer. passe msCD-se offee or peren. ul m:mer Aepuerry comm uoa.
c ena mastsaa eserveLJ Division of Engineering j
Office of Nuclear Reactor Regulation U.S. Nuclear Regulatory Commission Washington, DC 20555
- 10. SUPPLEMENTARY NOTES
- 11. ABSTR ACT (2ao. orar er esi A review of the structural design of the Combustion Engineering (CE) System 80+* steel containment was completed. The stress analysis and the evaluation of the structure against buckling were performed by using BOSOR4 and BOSORS finite difference software, respectively.
I The CE System 80+* containment was modelled as an axisymmetric shell consisting of different segments and mesh points with the additional mass of the penetrations and appurtenance being smeared around the circumference. The transition region was modelled using elastic spnngs with a foundation modulus of 180 lbs/in'. The stresses due to the individua! loads (dead loads, intemal and extemal pressures and temperatures) were computed using the stress analysis option in the BOSOR4 program. The stresses from individualloads were combined according to ASME Code into stress inter.sities. Service Level B loadings produced a 20 percent over-stress in a small zone just above the transition region. All other stress intensities were within allowable limits. For the System 80+, the perfect she!! with an elastic material was initially analyzed. The calculated factor of safety values were 2.3 (Level B) and 1.59 (Levels C and D). Finally, sensitivity studies were conducted to investigate the effects of mesh size and transition zone -
stiffness on the controlling buckling load, u xE y WORD $ictScRwTOR$ <uir r
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u.,m.ma n.1.u.e er structural design Unlimited steel containment x stcaw ww'w a n,,,, e,
Unclassified n, an.m Unclassified Ib. NUMBE R OF PAGE5
- 16. PRICE WRC PORM 336 (24W
UNITED STATES srtciat rounm etAss :Att NUCLEAR REGULATORY COMMISSION POSTAGE AND FEES PAD WASHINGTON, D.C.- 20555 0001 "8""C PERMIT NO. G-07 OFFICIAL BUSINESS PENALTY FOR PRIVATE USE, $300 i
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