ML19308A328
| ML19308A328 | |
| Person / Time | |
|---|---|
| Site: | Atlantic Nuclear Power Plant  |
| Issue date: | 03/31/1979 |
| From: | Masri S, Seide P, Weingarten V INTERNATIONAL STRUCTURAL ENGINEERS, INC. |
| To: | |
| Shared Package | |
| ML19308A325 | List: |
| References | |
| CON-NRC-03-77-131, CON-NRC-3-77-131 NUDOCS 7905030051 | |
| Download: ML19308A328 (106) | |
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MiBR$$$I$MRR 790503005/
s BUCKLING CRITERIA AND APPLICATION OF CRITERIA TO DESIGN OF STEEL CONTAINMENT SHELL INTERNATIONAL STRUCTURAL INGINEERS Glendale, California 91206 Date Submitted - March 1979 Prepared for Structural Engineering Branch Division of Systems Safety U.S. Nuclear Reactor Conunission Under Contract NRC-03-77-131
I e
PREFACE This report, prepared by Internctional Structural Engineers.under . -
Contract NRC-03-77-131, su==ari:cr. the current state of the art -
used for buckling analysis and for the design of nuclear. power -- . ..
plant steel contain=ent shells. Recomendations are =ade for .__ :
adopting a new buckling design criterion that is conservative . .
and uses the =ost up-to-date analytical methods available in -
the literature.
Technical =onitor for the Structural Engineering Branch of the Division of Syste=s Safety of the Nuclear Regulatory Comission was A. Hafiz. Project Manager for International ' ructural Engi-neers was Bengt Mossberg.,< The report was written by P. Seide, V.I. Weingarten, and S.F. Masri. Additional contributions were
=ade by C.D. Babcock, who perfor=ed the experi= ental testing of Appendix A, and A.P. Gelman, who did the nu=erical stress and buckling analysis.
v
ACKNOWLEDGMENTS
]
During this study a number of meetings were held with the following engineers in private industry, who contributed valuable information ,
frem their experiences related to the design and construction of ,
contain=ent vessels: _
Dr. C. Babcock, California Institute of Technology .
Dr. R. Citerley, Anamet Laboratories Dr. N. Edwards, Nutech Dr. P. Cou, General Electric Dr. J. Love, General Electric Mr. C.D. Miller, Chicago Bridge & Iron Dr. D. Sheikh, Ebasco -
Dr. L.J. Sobon, General Electric Dr. J. Tsai, Offshore Power Dr. A. Walsenko, C.F. Braun & Co. ~
Infor=ation requested for the specific area of shell buckling was generously given by the following authorities in the field: -
G. Ashwell ' University College, Cardiff, Wales j E. Axelred Munich, West Germany
_y M. Baruch i
Technion, Haifa, Israel M. Esslinger D.V.L., Braunschweig, West Germany J
R.M. Evan-Iwanowski Syracuse University, Syracuse, N.Y.
N.J. Hoff Rensselaer Polytechnic Institute, Troy, N.Y. ~
W.A. Nash University of Massachusetts, A=herst, Mass.
E.E. Sechler California Institute of Technology, Pasadena, Calif.
i G.J. Si=itses Georgia Institute of Technology, Atlanta, Ga. i J. Singer Technion, Haifa, Israel '
V. Tvergaard Technical University of Denmark, Lyngby, Denmark l
The kind assistance of these several contributors is gratefully '
acknowledged.
8 iv
TABLE OF CONTENTS Section
__h 1 INTRODUCTION . . . . . . . . . . . . . . . 1-1 2 CONTAINMENT LOADING CONDITIONS . . . . . . 2-1 2.1 Background . . . . . . . . . . . . . 2-1 2.2 Mark III'Cantainment for a BWR . . . 2-1 2.3 Description of Steel Containment . . 2-4 2.4 Loading Conditions . . . . . . . . . 2-4 2.5 Loads and Loading Combinations . . . 2-6 2.6 Loads on a Typical BWR Steel Containment Vessel . . . . . . . . . 2-10 3 STRESS ANALYSIS . . . . . . . . . . . . . 3-1 3.1 Computer Codes . . . . . . . . . . . 3-1 3.2 Methods of Stress Calculation . . . 3-1 3.3 Sample Stress Analysis Calculations 3-6 4 BUCAING ANALYSIS OF NUCLEAR CONTAINMENT VESSELS . .. . . . . . . . . . . . . . . 4-1 4.1 Determination of Buckling Loads . . 4-1 4.2 Nonlinear Analysis . . . . . . . . . 4-1 4.3 Experimental Results . . . . . . . . 4-2 4.4 Linear Bifurcation Analysis . . . . 4-2 4.5 Literature Survey . . . . . . . . . 4-3 4.6 A Tentative Method of Stability Analysis . . . . . . . . . . . . . . 4-36 5 CONCLUSIONS AND RECOMMENDATIONS. . . . . . 5-1 6 REFERENCES,. . . . . . . . . . . . . . . 6-1 Appendix A COMPRESSIVE BUCKLING OF PRESSURIZED THIN CYLINDRICAL SHELLS WITH A PENETRATION . . A-1 v
i e p ILLUSTRATIONS I
Figure ~
--. Page 2-1 h
Typical PWR with Steel Containment . . . . 2-2 2-2 A Recent Form of BWR Containment . . . . . 2-3 2-3 Typical Ice ConIdenser Unit . . . . . . . . 2-5 -
3-1 Finite Element Representation of a Contain=ent Vessel . . . . . . . . . . . . 3-4 _
3-2 Containment Vessel Usec'. for Sa=ple -
Analysis . . . . . . . . . . . . . . . . . 3-7 m
3-3 Cross Section of Stiffening Rings . . . . 3-9 _)
3-4 Deflection Profile . . . . . . . . . . . . 3-12 3-5 Some Calculated Mode Shapes for Sample -
Containment Vessel . . . . . . . . . . . . 3-14 _
4-1 Buckle Mode Shape . . . . . . . . . . . . 4-4 -
4-2 Buckling under Longitudinally Varying Axial Stress . . . . . . . . . . . . . . . 4-6 4-3 Suckling under Circumferentially Varying Axial Stress (Bending) . . . . . . . . . . ' [i 4-7 _.
4-4 Buckling.under Circumferentially Varying Axial Stress (Thermal Stress) . . . . . . 4-8 j 4-5 Lower-BoundReductionFactorsforIso-tropic Circular Cylinders Subjected to Uniform Compression or Bending . . . . . . 4-9 -
4-6 Buckling under Longitudinally and Circum-ferentially Varying Pressure . . . . . . . 4-11 ,l 4-7 Effect of Load Variation on Buckling ?
Pressure (L/a = n, v = 0.3) . . . . . . . 4-12 l
4-8 Circumferential Wind Pressure Distribution on Cylindrical Tank . . . . . . . . . . . 4-13 4-9 Buckling under Longitudinally Varying Hoop Stress . . . . . . . . . . . . . . . . . . 4-14 vi
ILLUSTRATIONS (CONTINUED)
Figure Page 4-10 Critical Circumferential Stress Distribution (R = 10 in., L'= 3.14 in.,
t = 0.0331 in., v = 0.3) . . . . .. . . . 4-15 4-11 Cylinder under Axial Load, Bending Mom 27t, and Lateral Contentrated Loads . .. . . . 4-17 4-12 Critical-Load Combinations for Cylinder 4 4-18 4-13 Normalized Critical-Load Curves for Several Models . . . . . . . . . ... . . 4-19 4-14 Effect of Spatial Distributicns Si through S6 on Critical-Load Curves for Cylinder 4 . . . . . . . . . . . ... . . 4-20 4-15 The Effect a rcular and Square Cutouts on the Buckling of a Circular Cylinder Loaded by Central Axial Compression . . . 4-21 4-16 Effect of Circular Hole on Buckling of a Cylinde under Combined Tccsion and Axial Compression . . . . . . . . . . ... . . 4-22 4-17 Comparison of Results of Linear Bifurca-tion Theory and Experimental Results . . . 4-23 4-18 Effect of Reinforcements around 45-Deg Cutout (t = 0.014, R = 6.06) . . . . . . . 4-25 4-19 Overall Dimensions of Stiffened Cylinder . 4-25 4-20 Effect of $dge Reinforceme'nt on Buckling Load . . . . . . . . . . . . . . . . . . . 4-26 4-21 Collapse Load Interaction Curve for a Cylinder under a Ramp-Step Pressure . . . 4-28 4-22 Collapse Pressure versus Pulse Duration and Axial Dead Weight for a Cylinder . . . 4-29 4-23 Sinusoidal Collapse Pressure versus Frequency for a Cylinder . . . . . .. . . 4-30 4-24 Sinusoidal Collapse Pressure versus Frequency for a Mylar Cylinder . . . . . . 4-31 vii
ILLUSTRATIONS (CONCLUDED)
Figure Page 4-2: Stability Boundaries for Cylinders ~
Subjected to Static and Sinusoidally Vary- _
ing External Pressure . . . . . . . . . . 4-32 n
4-26 Model for Dynamic Buckling . . . . . . . . 4-33 '
4-27 Infinite Step L'oading . . . . . . . . . . 4-34 I
4-28 Finite Step Loading . . . . . . . . . . . 4-35 I 4-29 Buckling of Spherical Shell under External ~
Pressure and Concentrated Loads at the _
Poles . . . . . . . . . . . . . . . . . . 4-37 4-30 External Buckling Pressure of a Cylindrical Shell with Hemispherical Ends 4-38 4-31 Comparison of Various Reduction Factors . 4-41 FABLES l Table 2-1 Steel Loading Combinations for Containment J
Vessel Design . . . . . . . . . . . . . . 2-7 t 3-1 Computer Programs Used by Industry for Static and Dynamic Stress Analysis . . . . 3-2 3-2 Averaged Properties of Plate Elements l in ContainEent Vessel Model . . . . . . . 3-10 t
3-3 Containment Vessel Natural Frequencies . . 3-13 i
3-4 Maximum Absolute Displacements, Meridian Parallel to Direction of Horizontal Ground ,
Mction . . . . . . . . . . . . . . . . . . 3-21 3-5 Maxi =um Absolute Displacements, Meridian Perpendicular to Direction of Horizontal '
Ground Motion . . . . . . . . . . . . . . 3-22 3-6 Maximus Absolute Stress at Selected Locations on the Head . . . . . . . . . . 3-23 3-7 Maximus Absolute Stresses at Selected Locations on the Cylinder . . . . . . . . 3-24 viii
SECTION 1 INTRODUCTION A steel containment vessel is usually a welded steel structure composed of a cylindrical shell attached to a doce structure and a containment base place. Because steel containment vessels provide an important pres- -
sure barrier for nuclear reactors, their design should resist the mos- -
adverse combination of loadings to which they might conceivably be subj ected. ,
The primary function of the containment structure is to localize the effects of a Loss-of-Coolant Accident (LOCA), which is an assumed accident resulting from a sudden break in the reactor cooling system.
The onset of the LOCA condition L=:ediately subjects the containment shell to asymmetric internal dynamic pressures with very high magnitudes.
Static, ther=al, seismic, and other loadings are postulated to act simul-taneously with the LOCA, but the LOCA is usually the most critical loading condition for both shell stress intensity and buckling criteria.
The importance vf these structures is reflected in the U.S. Nuclear Regulatory Commission's Standard Review Plan (SRP), Section 3.8.2 (1} ,, _ _
which establishes procedures for the verification of steel containment vessel design. The present NRC structural acceptance criteria require that contain=ent-vessel design comply.with the limits specified la NRC's Regulatory Guide 1.577 [2} and with part of the ASME Boiler and Pressure vessel Code.$ [3] The acteptable limits given in the SRP are compared with the proposed allowatAe limits for t'ae following major parameters:
- a. Primary stresses (general membrane, bending, etc.)
- b. Secondary stresses
- c. Peak stresses
- d. Buckling criteria Numbers in brackets designate references listed in Section 6.
" Design Limits and Loading Combinations for Metal Primary Reaction Contain=ent System Components."
Section III, Division 1, Subsection NE, " Class MC Components."
1-1
In establishing and evaluating steel-containment acceptance criteria, the NRC reviewer can follow the general lines of the minimum require-ments given in the SRP section " Standard Format and Content of Safety '
Analysis Reports for Nuclear Power Plants." [1} However, if the steel !
containment has unique or new features that are not covered in the
" Standard Format," the reviewer must establish specific requirements _
tailored to that particular steel containment and based on applicable ,
codes, standards, and specifications such as Regulatory Guide 1.57 [2] -.
and the ASME Code [3}. Design limits, loads, and loading combinations specified in the SRP must be covered. The current standard methods for
~]
determining the buckling loads of steel containment vessels that are i subjected to unsymmetrical dyn'amic pressure loads have not been veri-fied by testing or aczurate analysis. Therefore, a reudy is necessary to establish the range of validity of the present ASME Code requirements. j This report su=marizes the results of an investigation into the types _
of loads encountered by a containment structure and the currently used j methods of analyzing stress and buckling. During the study, a number -
of industrial companies were contacted and discussions were held with engineers involved in the design and construction of containment 'l vessels.* In addition, information on the buckling of shells subjected d to nonuniform loading was requested from leading authorities in the field. -
In Section 2 of this report, the critical loading cases used in the design of the containment vessel are discussed. Analysis techniques used to determine the stresses in the contain=ent vessel once the load-ing conditions are known are described in Section 3. The capabilities of a nu=ber of general-purpose finite element and finite difference com-puter programs are summarized. Recommendations are made for the use of these programs to determine the stress intensities in the shell contain-ment vessel. In Section 4, analysis techniques for determining buckling loads of containment vessels are reviewed. Available theoretical and experi= ental data on the static and dynamic stability of cylindrical and spherical shells under nonuniform and localized loads, including the effects of penetrations, are su=marized. Tentative recommendations for buckling analysis are offered. Finally, recommendations for both design and safety factors contained in the report are summarized in Section 5. This section also contains recommendations for future experimental and analyti-cal work required to define the buckling criteria of steel containment vessels mere accurately.
l See " Acknowledgments," p. iv.
1-2
SECTION 2 CONTAINMENT LOADING CONDITIONS ,
2.1 BACKGROUND
The main types of nuclear reactors that use steel containment vessels --
are boiling-water reactors (BWR) and some types cf pressurized-water reactors (PWR).
A typical containment structur'e for a PWR plant is shown in Figure 2-1.
The containment structures of most present-day PWR plants are made of reinforced concrete with a steel liner.
In BWR ;1 ants two levels of containment are available: (1) the primary containment, which encloses the reactor; and (2) the secondary con-tain=ent, which, as in the case of PWR plants, has more or less the same contour as the reactor building.* One of the versions of BWR contain-ments now in use is shown in Figure 2-2.
Regulatory agencies, research organizations, and private companies continue to invest a great deal of effort in accurately defining the dynamic environments to which nuclear power plant facilities are sub-jected. In order to show the sources of dynamic loads that enter into a buckling consideration, this section briefly reviews the general loading conditions that apply to containment vessels, particularly steel vessels.
The loading conditions are discussed in the context of the recently designed and constructed General Electric BWR/6 with the Mark III containment.
2.2 MARK III CONTAINMENT FOR A BWR __ . __
The main components of tre Mark III reactor building structure are the following.
Shield Building: This is.a reinforced-concrete cylindrical shell with a shallow spherical dome roof enclosing the steel containment. The pri-mary function of this building is to limit the escape of nuclear radia-tion to the environ =ent and to protect the containment from external missiles and adverse at=ospheric conditions. It is a seismic Category I structure.
In high-temperature gas-cooled reactors (HTGR) there are also two levels of containment: (1) the primary containment consisting of a prestressed concrete vessel that holds the reactor core, pumps, heat exchangers, etc. , and (2) a secondary containment, which is a heavily built reactor building similar to those of the PWR and BWR.
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Containment: This is a steel leakage barrier whose main purpose is to pre, vent the release of significant fission products to the shield i building in the ever of an accident.
It is a free-standing, vertical, cylindrical, steel pressure vessel with a flat bottes steel-liner place _
and an ellipsoidal or hemispherical head and is classified as a seismic Category I structure.
9 Drvvell: This is a reinforced concrete cylindrical shell that surrounds i the reactor pressure vessel. It is a seismic Category I structure.
Suppression Pool and Weir Wall: This circular pool of demineralized water =aintains a water seal between the drywell interior and the ~
remainder of the containment. Its primary function is to provide a hert sink for venting the nuclear system by safety relief valves. It y' also helps in reducing the nonuniformities of dynamic load distribution during safety / relief-valve operations.
Upoer Containment Pool: This is a rectangular stainless-steel-lined -
pool that assists the dryvell roof slab in withstanding internal pressure leading during a LOCA.
~~l
2.3 DESCRIPTION
OF STEEL CONTAI' MENT -
The main types of steel contain=ents are as follows:
- a. S. eel BWR contain=ents utilizing the pressure-suppression ap9 concept (such as the Mark I, II, and III) b.
Steel PWR contain=ents utilizing the pressure-suppression l concept with ice-condenser elements Gee Fig. 2-3) -J
- c. Steel PWR dry contain=ents 2.4 LOADING CONDITIONS The main loading conditions to which a contair' ment vessel is subjected d are these:
- a. Internal and external pressures -
- b. I= pact loads, including rapidly fluctuating pressure
- c. Dead weight of the. vessel and its normal contents under all -
conditions, including additional pressure due to static and dynamic head of liquids d.
Wind loads, snov loads (if not protected by a sh! id building),
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- e. Reaction of supports .
- f. Temperature effects
- g. Reactions to water jet and steam impingement
- h. Earthquake loads
- 1. Pool swell loads generated by LOCA or safety relief valve (SRV) 7, 2.5 LOADS AND LOADING COMBINATIONS Loads and load combinations that should be considered in the design of steel containments are specified in the ASME Code [3] and the Regulatory Guide 1.57 [2] . Loads applicable to steel containments usually include __
the fellowing:
- a. Loads encountered during preoperational testing
.q
- b. Loads encountered during normal plant startup, operation, i and shutdown (including thermal loads due to operating te=perature); hydraulic loads (in pressure-suppression con-tain=ents usitg water); dead loads; and live loads
- c. Loads that occur during severe environmental conditions such as the operating basis earthquake (OBE) or wind loads (if not protected by a shield building) -
- d. Loads that occur during extreme environmental conditions '
such as the safe shutdown earthquake (SSE) or tornado loads (if not protected by a shield building)
- e. Loads due to abnor=al plant conditions, such as loss-of-coolant accidents (LOCA) and other accidents involving high-energy pipe ruptures
- f. Loads sustained after abnor=al plant conditions, such as flooding of the contain=ent after a LOCA i
Typical steel-loading combinations for containment vessel design are .2 l su==arized in Table 2-1. I i
.J 0
2-6
, TABLE 2-1. STEEL LOADING COMBINATIONS FOR CONTAI! MENT VESSEL DESIGN [2] (1) D+L+F +T ' t t (2) D+L +T +R (3) D+L +T +R + Feq (4) (a) Large pipe break accident (LBA) D+L +T +R +P + Feq o a a. a (b) Intermediate pipe break accident (IBA) D+Lt+T i+R a t a+ p ta + Feq o (c) Small pipe break accident (SBA) . . . _ _ _ D+Li+T ai+R i+P a i a+ Feq o (d) Long-term design event ___
+P + Feg, D + L" + (* + R (5) D + L* + T e" + R e* + P e + Feq o
(6) (a) LBA D+L t+T,i+R[+P + Fegs (b) IBA D+Li+T t+R a i+P a i a+ Fegs (c) SBA D+L +Ta I+Ra i &Pai + Fegs (d) Long-term design event . _ D + L* + T,' + R + P, + Fegs (7) D + L" + T +R +P + Fegs (8) (a) LBA D+Li+Ta i+R a +P a + (Y r +Y j + Y m) + Fegs NOTE: Footnotes indicated by *, t, and # are listed at the end of Table 2-1. 2-7
TABLE 2-1. (CONTINUED) (b) IBA '- D+L &Ta i+R ai+P Ia + (Y r +Y j + Y ) + Fegs m (c) SBA - A + + D+L'+T'+R a a '&P.'a + (Y e ' Y j + Y m) + Fegs - (9) D+L+Ft + Feq, _ where - D = Dea 1 loads and their related moments and forces, includ-ing any per=anent equipment loads and prestressing loads, if any. I Fg = Loads generated by post-LOCA flooding of the contginmene. _ Feq = Effects of operating basis earthquake. - Feqs = Effects of safe shutdown earthquake. L = Live loads and their related coments and forces, including any movable equipment loads and other loads . that vary with intensity and occurrence, such as tem-parary or varying portion of hydrostatic pressures, ~ and pressure differences due to variation in heating and cooling and outside atmospheric changes. Live loads include both the direct and feedback effects of safety relief valve (SRV) actuation, as noted in the applicable loading combinations. ,
=
P, Pressure equivalent static load within or across a com-part=ent or building, generated by the postulated break, j and including an appropriate dynamic factor to account for the dynamic nature of the load. P a includes the feedback effects due to pressurization of compart=ents or areas during the postulated break. P, = Design external pressure. R =
- Pipe reactions under ther=al conditions generated by the -
postulated 'ereak and including R. o Ra includes the pipe reaction due to feedback effects of P' a R = Pipe reactions due to thermal conditions during the event causing P. e NOTE: Footnotes indicated by *, , and # are listed at the end of Table 2-1. 2-8
TABLE 2-1. (CONCI.HDED) R =
. Pipe reactions during normal operating or shutdown conditions, based on the most critical transient or steady-state condition.
T
=
Thermal loads under thermal conditions generated by- - the postulated break and including T. o T
=
Ther=al ef fects under thermal conditions during the event causing Pe. T = Thermal effects and loads during normal operating or shutdown conditions, based on the most critical transient or steady-state condition. Y = Jet impingement equivalent static load generated on a d structure by the postulated break, and including an appropriate dynamic factor to account for the dynamic nature of the load. Y, = Missile impact equivalent static load generated on a structure by or during the postulated break, such as pipe whipping, and including an appropriate dynamic factor to account for the dynamic nature of the load. Y
=
Equivalent static load on the structure generated by reaction of the broken high-energy pipe during the postulated break, and including an appropriate dynamic factor to account for the dynamic nature of the load. Notes: Includes the effects of the full spectrum of SRV events. " Includes the effects of the SRV events, pressure transients, and thermal transients. Includes the design pressures and temperatures for the struc-ture. Does not include an SRV event since the RPV is depressurized. 2-9
2.6 LOADS ON A TYPICAL BUR STEEL CONTAINME!rr VESSEL
- a. Design Pressure and Te=peratures ,
_q (1) Design internal pressure 15 psig (103.5tPa) at 185' F =ax (85 OC) l (2) Design external pressure 0.8 psig (5.5 kPa) at 185 0F max (85 OC) , (3) Lowest service me:a1 temperature 30 0F (1.1 CC) (4) Internal hydrostatic head 24 f t 4 in. (7.4 m)
~
(design life of,40 years) -
- b. Operating Pressure and Temperature
]
(1) Steady-state operating temperature
'. a ) Normal ~I 90 0F (32 CC) _;
(b) High 120 0F (49 OC) q (c) Low 60 0F (16 OC) ' (7) Operating internal pressure in the air volume is 0 to 2 psig (13.8 kPa) with a hydrostatic head of 24 f t 4 in. in the internal suppression pool. (3) Operating external pressure is 0 to 0.8 psig at 230 CF (110 CC) with a hydrostatic head of 24 ft 4 in. ~ in the internal suppression pool.
- c. Dead Loads (1) Weight of steel of the containment vessel I (2) Crane weight ,
(3) E=pty weight of attached piping ,l (4) Weight of miscellaneous components (for example, ladders, platfor=s, etc.) attached to the shell
- d. Live Loads (1) Crane lifting loads (130-ton capacity)
(2) Crane dynamic loads (20% of crane reight and load applied ! at top of the crane rail) ! (3) Design floor loads for walkways, ladders, and platforms (The live load used for the design of the floor section of the pe"sonnel locks is 250 lb/ft2 [12 kPa].) 2-10
- e. Mechanical Piping Loads Reactions produced by external and internal piping .
- f. Thermal Loads Those produced by the presence of axial temperature gradients within the steel containment
- g. Construction Loads In addition to the usual construction loads, the following temporary loads are considered:
(1) Snow loads before the completion of the shield building (2) Wind loads during the erection of the vessel before the completion of the shield building (3) Load on the ellipsoidal head of the initial u-in. (0.15 m) layer of concrete poured for the shield-building dome
- h. Missile Loads No external missile loads are considered, since the steel containment vessel is protected by the shield building; but some internal missile loads are possible.
- i. Loss-of-Coolant Accident Loads (LOCA)
(1) Loads determined by analysis of the transient ptessure and temperature effects that occur during LOCA (2) Jet impingement loads produced by the failure of a high-energy line J. Seismic Loads The intensities'of ground acceleration corresponding to the OBE and SSE are a function of the nuclear facility site location. In addition to inertia loads, ground acceleration can generate loads due to sloshing effects in the suppression pool. The types of loads discussed above indicate that the stress disbribution within the containment vessel is asyesetric. The asymmetry may be due to disturbances in the shell geometry in the form of penetrations and cutouts or to asyn=etric loading such as earthquake or wind loads, transient pressure loads that occur during LOCA, or localized loads occurring during a pipe-rupture accident due to jet or pipe impingement. 2-11
The rational design of the containment vessel requires a knowledge of the detailed stress state in the structure as the first step in the deter =1 nation of safe buckling loads. The types of stress analyses ,._, required and the computer codes available are discussed in the following i section.
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i I _1 2-12
SECTION 3 STRESS ANALYSIS ' 3.1 CONTUTER CODES __ Static and dynamic stress analyses of containment vessels are generally performed with co=puter programs. A number of these programs are cur-rently used by industry for containment vessel design. Their. capabilities are described in Table 3-1. . 3.2 METHODS OF STRESS CALCULATION _ _ _ _ _ .. __ The stress analysis of containment structures may be performed by dividing the containment vessel into finite elements , as in Figure 3-1. A general analysis code can take stiffeners, rings, variable skin thicknesses, variable material properties, cutouts, and penetrations into account. Static stress analyses require the solution of a large number of simulta-neous equations for each load state. Dynamic analyses involving time-dependent loading histories and structural inertia effects can be performed by step-by-step integration of the coupled equations of motion for each s degree of freedom, by modal analysis using mode superposition, or by modal analysis using the response spectrum method. The effects of thermal exci-tation, geometric stiffness, and support movement can also be considered in the analysis. , Which code is used depends on the type of problem, the geometry of the con-tainment vessel, and the degree of approximation that the user is willing to accept. In most instances a linear analysis program is sufficient, since the ef fects of nonlinearities alone are small prior to buckling. A one-dimensional computer code is limited to shells of revolution that contain no cutouts or penetrations. In the case of'a nonlinear one-dimensional code, the load is axisymmetric only. Linear one-dimensional codes, however, can consider circumferentially varying loads. The loads are expanded into Fourier series in the circumferential direction. Each Fourier component is ,then associated with a meridional variation of load, the effects of which are found with the one-dimensional code. The stresses and displacements due to each load Fourier series component are then added to yield total stresses and displacements. It may be more convenient, however, to use one of the available two-dimensional shell computer codes, initially to avoid having to find the Fourier series expansion of the loading and finally to avoid having to add up Fourier components of stress and displacement. The circumferential-direction approximations of the two-di=ensional analyses are offset by the usual restrictions of one-dimensional analyses to a limited number of Fourier components. The cquation-solving techniques used make it possible to solve the one large set of simultaneous equations associated with two- ' dimensional codes, although at the cost of more computer : e than that ' 3-1
.---im . -
TASLE 3-1. COMPUTER PROGRAMS USED BY INDUSTRY FOR STATIC AND DYNAMIC STRESS ANALYSIS t Code (Source) Description ANSYS Linear and nonlinear static, and dynamic analysis "
, of general structures. Anisotropic =aterial. .I L.b. Swanson Soil / structure interaction. Nonlinear collapse
.snalysis Corp. analysis. Nonlinear equations solved by use of -'
Dap'.. VM incre= ental method without equilibrium check. o/0 Pineview Dr. Elemencs: rod, straight beam, conical shell,
~,
Elizabeth, PA 15037 axisy==etric shell, flat membrane, flat plate. , shallow shell, axisy==etric solid, thick plate, solid 3-D. Substructure capability. ASHED 2 Evaluates time-dependent displacements and - stresses of complex axisy==etric structures University of subjected to any arbitrary static or dynamic ~] California, loading or base acceleration using Fourier com-Berkeley ponents for loaders. Three-di=ensional axisym-NISEE/Cempu te r cetric continuus represented either as Applications axisy= metric thin shell or as solid of revolu- _j Davis Hall tion or as co=bination of both. A.xysy= metric Berkeley, CA 94720 shell discretized as series of frustrums of cones, and solid of revolution as triangular or quadri- ! lateral toroids connected at their nodal point I circles. Can consider five cases of loading: dead Icad, arbitrary static load, arbitrary dynamic load, horizontal and vertical component of earthquake acceleration record applied at base of finite ele =ent model. l BOSOR4 Analysis of stress, stability, and vibration codes of seg=ented, ring-stiffened branched Lockheed Aircraft shells of revolution and prismatic shells and Co rp . panels. Performs large-deflection axisy==etric Dept. 52-33, stress analysis, small-deflection nonaxisyc=etric I Bldg. 205 stress analysis, codal vibration analysis with ! 3251 Hanover St. axisy==etric nonlinear prestress included, and Palo Alto, CA 94303 buckling analysis with axisy==etric or nonsy==et-ric eccentric load paths, internal supports, arbitrary branching conditions, and a " library" of wall constructions, such as layered ortho- t tropic, layered fiberwound, corrugated semi- j sandwich, etc. Based on finite-difference energy method (similar to finite elenent method but ; generally more rapidly convergent). Inverse power iterations with special shifts used for eigenvalue analyses. Newton-Raphson method used for solu-tion of nonlinear proble=s. 3-2
TABLE 3-1. (CONCLUDED) Code (Source) MARC Linear and nonlinear static and dynamic analysis of general structures. Anisotropic material. MARC Analysis Bifurcation with linear or nonlinear prestress. Research Corp. Nonlinear collapse analysis. Inverse power Dr. Pedro liarcel iteration for eigenvalues. Nonlinear equations 260 Sheridan Ave. solved by incremental method with equilibrium Suite 314 check.. Elements: rod, straight beam, curved Palo Alto, CA 94306 beam, conical shell, axisymmetric shell, flat membrane, curved membrane, flat plate, deep shell, axisymmetric solid, thick plate, thick shell, solid 3-D, crack tip. NASTRAN Linear static and dynamic analysis of general structures. Anisotropic material. Fluid / MacNeal-Schwendler structure interaction. Bifurcation solution Co rp. for linear prestress. Inverse power iteration Dr. Richard MacNeal for eigenvalues. Elements: rod, straight beam, 7442 N. Figueroa St. conical shell, axisymmetric shell, flat mem-Los Angeles, CA 90041 brane, flat plate, axisymmetric solid, solid 3-D. Substructure capability. SAP 6 Linear static ano dynamic analysis of general structure. Anisotropic material. Bifurcation SAP Users Group solution for linear prestress. Determinant Dept. of Civil search and subspace iteration for eigenvalues. Engineering Elements: rod, straight beam, flat plate, U.S.C. axisymmetric solid, thick plate, thick shell, University Park boundary, solid 3-D, super elemeat substructure Los Angeles, CA 90007 capability. STAGS Br'nched a or segmented general shells. Cutouts, framework shell combination. Layered anisotropic Lockheed Aircraft shell wall. Simple input for special shell Co rp . 5 ,l's. Discrete or smeared stiffeners. Plas-Dept. 52-33, t. . Nonlinear collapse, bifurcation with Bldg. 205 linest p. =t sss. Finite difference energy 3251 Hanover St. method for shell structure. Modified Newton for Palo Alto, CA 94303 solution of nonlinear equations. Inverse power iteration for eigenvalues. 3-3
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associated with the nu=erous smaller sets of equations obtained with the - one-dimensional codes. If the shell surface is not axisy==etric, a two- . dL=ansional co=puter code =ust be used of course. w The =ethod of analysis of the penetrations and effects of cutouts in a I shell of revolution depends on the size and reinforce =ent of the hole. d If the hole is s=all, it is possible to analyze the pri=ary effects of loading with one-di=ensional analyses and to correct for the presence 'l of the hole by a two-dimensional analysis'of a relatively small surround-ing region. The analyses of [4 to 6) indicate that around an unreinforced hole in an isotropic shell, a region having a radius of the order of five ~ hole-radii is affected. If the hole is large enough, the stress state _l in the entire shell will be affected. It would appear to be more advan-tageous in this case to abandon the one-dimensional analysis and to q proceed directly to a two-dimensional analysis. I J It is possible to neglect the presence of the hole if the reinforce =ent is designed to alleviate the acce=panying stress concentration. In this case, one-dimensional computer codes would be sufficient. For d cylinders under external pressure, the ASME Pressure Vessel Code (3c] calls for the cutout volume to be replaced by edge reinforcement. This _ value presumably is sufficient, yet some of the investigations of the sub-ject (7 to 9] cast doubt on the adequacy of this rule of thu=b. On the other hand, studies of buckling of cylinders with reinforced holes, dis-cussed in Section 4, indicate that using the rule of thu=b is sufficient to prevent the degradation of the critical stress of a cylinder without a hole. The question of adequate reinforce =ent of holes in cylindrical - shells thus appears to require further investigation. 3.3 SAMPLE STRESS ANALYSIS CALCULATIONS l Discussions with industry representatives have indicated that one-dimensional shell-of-revolution computer programs are popular for the ' dynamic stress analysis of nuclear containment structures. There appears ' to be so=e reluctance to use two-dimensional codes, possibly due to a belief that it is very costly or possibly due to industry's relative inex- > perience in using two-dimensional codes for this purpose. To determine } the cost of a two-d1=ensional shell analysis, we ran a sample set of static " and dyna =ic stress analysis calculations for the contain=ent vessel of Figure 3-2. Results fro = these sa=ple calculations illustrate the type of results and accuracy of the solution that can be obtained fro = a two- a dimensional finite ele =ent analysis. The progra= used in the sample con-tain=ent vessel analysis by International Structural Engineers was SAP 6. However, uany of the other general-purpose progra=s could have been used with equal success.
'l, 3.3.1 GENERAL DESCRIPTION OF THE SAMPLE PROBLEM The containment vessel of the example proble= is assumed to be a cylin-drical shell 120 ft (36.58 =) in diameter and 150 ft (45.72 =) high, 3-6
e 4 0 0
- 32 o 300 I 16 200 g
._n .- :
200 I n - 200 t ~ i 200 o b ~ a 200 l- 1 200 I ._ _ 200 l p ._ _ u 200 , 4 F 200 l m _ _ _ . 720 ALL DIMENSIONS IN INCHES.
-~
FIGURE 3-2. CONTAI2iENT VESSEL USED FOR SAMPLE ANALYSIS 3-7
closed by an ellipsoidal head 30 ft (9.14 m) high. The ellipsoid radius / - hei,ght ratio is thus 2:1. The base of the cylinder is assu=ed to be rigidly clamped. The vessel is shown in Figure 3-2. The cylindrical portion has three thickness courses, each of which is 50 ft (15.24 m)
- in depth. The first course, beginning at ground level, has a thickness ]
of 1.75 in. (44.45 ==), tha second course thickness is 1.5 in. (38.10 cm), I while the third is 1.25 in. (31.75 cm). The thickness of the head is uniform at 1.5 in. (38.10 =m). Six circumferential stiffening rings are located on the cylinder. Their locations and dimensions are given in Figures 3-2 and 3-3. For simplicity, the rings are assumed to be iden-tical and located in three sets of two. Each of the two rings of a set " is located at the one-third point of a thickness course. The geometry and .. loading of the exa=ple problem' vere arbitrarily selected for deter =ining computer cos:.s, and the tabulated deflections and stresses are not appli- 7 cable to any actual design or loadings. 3.3.2 MATHEMATICAL MODEL Usually, the containment vessel geometry is nearly axisy= metric. How-ever, it cannot be modeled by axisy= metric finite elecents because it l generally contains penetrations, attach =ents, asy==etric stiffening, etc. Because of the possible co=plexities of contain=ent-vessel geometry and of the loading functions, it was decided to use SAP 6, a general purpose finite-ele =ent computer program, to perform the various analyses , required. The elements used for cost of the vessel are thin-shell quadrilaterals. However, it was necessary to use triangular shell elements near the apex of the ellipsoidal head. In order to take advantage of symmetry about the z-axis, only one-half the vessel is modeled. Figures 3-la and 3-lb show the location of the nodes. It night be noted that the model and element nu= bering scheces used are chosen entirely for the convenience of providing input to the ] ' program and for reviewing output. This is possible because SAP 6 con-tains an auto =atic bandwidth minimi:er that optimizes the model number-ing internally. Optinization is, of course, necessary to keep the cost j of the co=puter run dcwn to a minimum. The effect of the circunferential rings was taken into account by s= earing the stiffening ring to obtain an equivalent orthotropic shell for each course. The rings could have been included as discrete elements, newever. Each course was then divided into three rows of plate ele =ents, as indicated in Figure 3-la. The. averaged properties of the plates are tabulated in Table 3-2. The final =athematical codel used in both the static and dynamic analy- _l ses consisted of 170 place elements and 1038 degrees of freedom. 3-d
ja 1" -- -- 1 16" -- t-32" - - - - FIGURE 3-3. CROSS SECTION OF STIFFENING RINGS 3-9
1 TABLE 3-2. AVERAGED PROPERTIES OF PLATE ELEMENTS IN CONTAINMENT I VESSEL MODEL ' C C C g, , 22, C 33' Course in. psi psi psi psi q 6 6 6 6 1 1.75 4.564 x 10 1.092 x 10 3.64 x 10 1.288 x 10 2 1.5 4.032 x 10 6 9.52 x 10 5 3.13 x 10 6 1.092 x 10 6
]'
3 1.25 3.528 x 10 6 7.8 x 10 5 2.604 x 10 6 9.24 x 10 5 For all cases A = 96 in. I = 11,272 in. - b = 22.25 in. h = 14.76 in . O i w 3-10 '
- 3. 3,. 3 STATIC ANALYSIS The static analysis was performed for the vessel under an applied .
external pressure of 1 psi (7000 Pa). The maximum hoop stress developed in the cylindrical portion of the shell occurred in the top course, where the thickness is 1.25 in. (31.75 mm). The hoop stress was SIS psi (3.6 MPa) compressive (the membrane stress approximation of PR/t for this case is 576 psi [4 MPa]). - The meridional stress in the cylindrical portion was fairly uniform, varying from 203 psi (1.4 MPa), to 288 psi (2 MPa). The ellipsoidal head developed a =aximum tensile stress in the circum-ferential direction near the edge. The value of this stress was 95 psi (0.67 MPa). The tensile stress decreases towards the center, becoming compressive at a distance of about 545 in. (13.8 m) from the center. From this point inward, the circumferential stress becomes increasingly compressive to a maximum value of 505 psi (3.5 MPa) at the center. The meridional stress varies from 250 psi (1.8 MPa) com-pressive at the outer edge to a maximum compressive value of 506 psi (3.5 MPa) at the center. The possibility that the head would buckle under the above stress state should be noted. The deflection profile of a meridian cut through the vessel is shown in Figure 3-4. The maximum deflection obtained in the cy1.indrical portion was 0.011 in. (0.28 nm). The maxi =um deflection in the ellipsoidal head occurred at the center and was equal to 0.052 in. (1.32 mm). 3.3.4 SEISMIC ANALYSIS The vessel was subjected to a ground motion time-history dynamic analysis in order to simulate the effect of an earthquake. The tech-nique used was the normal-modes method. The numerical integration in normal coordinates was carried out in the SAP 6 program using the Wilson theta method with theta = 1.4. As a first step in the sefsnic analysis, it was necessary to calculate the system natural frequencies and mode shapes. The eigenvalue extrac-tion technique used in SAP 6 was subspace iteration [10). The first 15 natural frequencies (Table 3-3) and the associated mode shapes were calculated. The first three mode shapes are shown in Figure 3-5. The simulated earthquake consisted of both a horizontal and a vertical ground acceleration applied simultaneously to the basa of the contain-ment vessel. The horizontal direction was parallel to the plane of sy= metry. A normalized version of the recorded El Centro earthquake was used for this analysis. The maximum amplitude was scaled to 0.18 g, and the integration ti=e interval was 10 msec. Duration of the motion was limited to 12 sec. 3-11
7
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3-12
TABLE 3-3. CONTALNMENT VESSEL FATURAL FREQUENCIES Mode Frecuenev. Hz 1 5.1 2 5.5 3 5.6 4 6.5 - 5 7.0 6 8.2 7 8.9 8 10.0 9 10.0 10 10.4 11 10.5 12 11.3 13 11.3 14 11.8 15 12.0 3-13
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]
~_
# / /I \ N %
&/l \\n 1 .
/~\ \ \7 \ '
/ \ \ U \ .,
~
\ l \ l
\] I' l\ / b
\J L 7 l'\ / ,
X l l \ / i ! (c) Second ecde, elevation FIGURE 3-5. (CONTINUED) 3-16
WE"*'ain,. ....n M -. .. N
,V
\ \
\
l _ _ _ _ _ _ _ _ _ . . _ . q \ \
.NN *
.4 s.
\
'A -
3 (d) Se.cond mode, plan FIGURE 3-5. (CONTINUED) 3-17
9 e e oe**w D Li;""l ;"in,. i.:,n
-]
ff , i s
\
w
/ ,
i i \ \%
\ N\
//I \
./ \? .i IN Yl\ '
I \l) ' ' {\ \ u I i , i i \l
\i\/ ~
\i\l
- ~,
8
\U\ NJ ,
K\ l1
/\ \
/ /1x
.l l
(e) Third mode, elevation
.IGURE 3-5. (C0hIINUED) 3-13
Se5 Y * $E E r.s.sut
,w
,/ N
'/ ,
I
. i l
\
P
/
\ ( p
~ - -
- 3 (f) Third mode, plan FICURE 3-5. (CONCLUDED) 3 - l'.'
Because of the complexities of the geometry of the vessel, it is diffi- ' , cult to characterize the displacement response. Table 3-4 contains
*the -'v4 u' absolute displace =ent of nodes along a ceridian that is in the plane of sy==etry of the vessel. This plate *s
. parallel to the -
direction of horizontal =otion. Table 3-5 shows the maximum displace- ,!
=ents obtained by nodes on a =eridian that is perpendicular to the plane of sy= metry. The x-direction respor.se, which is perpendicular to q the direction of horizontal motion, is due to axisy==etric model l response.
It should be noted that these m'v4 "~a do not generally occur at the same instant in ti=e. In addi. tion, they occur only ,nce and are usually not representative of the general level of displacement occurring during the earthquake. ~ Stressas of selected elecents are tabulated in Tables 3-6 and 3-7. The-ele =ents were chosen along two =eridians in planes parallel and perpen- 1 dicular to the direction of horizontal motion. 3.3.5 COST
.)
The actual cost of static and dynamic analyses of the type performed for this sample problem is, of course, dependent on the efficiency of _l the s ructural analysis co=puter program and the computer that are used. ._, For the case in point, the computer was an IBM 370/158. The containment vessel was somewhat coarsely modeled--consisting of 170 place ele =ents and 1038 degrees of freedom. The cost was minimize' ey taking advantage of sy==etry and by bandwidth minimi:ation. The cost to International Structural Engineers (ISE) of the static analy- '; I sis was $30. Although this only included one loading, many other lo.ds could have beea run s'.=ultaneously for only a slight increase in cost, ., using the sa=e Gause.an reduction of the stiffness = atrix. The cost of the dynamic analysf a to ISE was more expensive, $150, since it entailed an eigenvalue solntion in addition to the time-history integration. The cost of these an. lyses is quite modest when compared to the overall cost , of engineering '.or a vessel of this magnitude. It is evident that even _1 when a more rafined codel'is used, as would be for an actual design, the cost would ',e greater but would not be prohibitively high. ( i l 3-20
TABLE 3-4. MAXIMUM ASSOLUTE DISPLACEMENTS, MERIDIAN PARALLEL TO . . DIRECTION OF HORIZONTAL Gl.0CND FOTION Displacements, in. Horizontal Vertical Location Node- () (y) 2 1.28 0.097 4 4.06 0.097 Cylinder (+z) 6 6.91 0.221 8 5.50 0.346 10 1.87 0.227 102 1.75 0.069 104 2.82 0.108 7 *# 106 2.63 0.215 108 2.11 0.231 110 1.94 0.251 122 1.92 0.164 144 1.95 0.247 166 1.95 0.080 Head 183 1.96 0.027 176 1.95 0.011 154 1.94 0.012 132 1.92 0.016 3-21
TABLE 3-5. MAZIMUM ABSOLUTE DISPLACEMENTS, dRIDIAN U PERPENDICULAR TO DIRECTION OF HORIZ0!TIAL GROUND MOTION .
~~
' Displace =ents, in.
Horizontal Horizontal Vertical Location Node (x) (z) (y)
'].i 52 0.975 0.924 0.082 54 3.53 1.03 0.016 Cylinder -
56 1.33 1.38 0.109 __ 58 2.06 1.73 0.065 - 60 0.144 1.82 0.080 127 0.002 1.89 0.017 -
~
149 - 0.004 1.93 0.034 Head . 171 0.013 1.95 0.161 183 -- 1.96 0.027 I I 3-22
TABLE 3-6. MAXDiUM AESOLUTE STRESS AT SELECTED LOCATIONS ON THE HEAD
~
Circu:sferential Meridional Moment, Moment, Stress, in.-k Stress, in.-k Element ksi in. kai in. 91 2.6 0.03 1.3 0.15 111 0.82 0.06 0.34 0.15 131 0.45 0.55 0.93 0.26 166 1.2 1.00 3.00 0.58 95 2.22 0.16 1.16 0.19 115 0.32 0.03 0.24 0.06 13: 1.92 0.67 0.87 0.27 168 4.0 1.34 4.4 1.51 e D 3- 23
TABLE 3-7. MAXIMUM ABSOLUTE STRESSES AT SELECTED l LOCATIONS ON T'43 CYLINDER
~1
.. }
Circumferential Meridional Mo=ent, Moment,
-~~
Stress, in.-k Scress, in.-k Element ksi in. ksi in. - l 1 16.0 0 36 7.4 0.63 21 2.4 0.70 7.2 0.46 31 2.2 0.48 7.0 0.31 51 0.95 0.25 3.18 0.19 _i 61 2.6 0.20 9.7 0.20 - 81 1.3 0.16 6.9 0.14 5 5.9 0.31 6.4 0.41 25 2.3 0.96 6.0 0.40 35 1.9 0.96 5.4 0.54 I 55 0.93 1.92 2.7 0.83 65 1.65 0.98 6.3 0.33 85 0.26 0.27 7.0 0.09 l 3 - 24
, ., SECTION 4 BUCKLING ANALYSIS OF NUCLEAR CONTAINMENT VESSELS
- 4.1 DETERMINATION OF BUCKLING LOADS Critical loads on structures may be obtained by a variety:of methods ~~
and computer programs. The ASME Pressure Vessel Code [3b} stipulates, - ~ for example, that values of critical buckling stress of nuclear con-tainment vessels may be determined with the use of one of the following-methods:
- a. Rigorous nonlinear analyses that consider the effects of gross and local buckling, geometric imperfections, nonlinear 1-ties, large deformations, and inertial forces (dynamic loads only)
- b. Classical (linear) analyses reduced by margins that reflect the difference between theoretical and actual load espabilities
- c. Tests of physical models under conditions of restraint and loading like those the configuration is expected to experience Although each of these three methods is correct in principle, the cur-rent state of the art of shell tuckling analysis is such that the three methods can produce widely different values of critical load.
4.2 NONLINEAR ANALYSIS .___ _ _ _ _ _ . Within the present state of the art of shell analysis it may be possible to determine quite accurate critical loads by means of a nonlinear analysis, provided that an almost precise description of the shell geometry (includ-ing imperfections) and edge conditions can be furnished. However, since the correlation of theory and experiment in the few cases where the it.itial shape was accurately deterpined has not been consistently satisfactory, shape imperfections do not completely explain the discrepancies between theory and experiment. Support conditions may introduce significant errors as well; and these are much = ore difficult, if not impossible, to ascertain completely and accurately. To neglect the effects of geometric imperfec-tions entirely is equivalent to a linear bifurcation analysis without the imposition of a reduction factor. The expense of a nonlinear investigation in terms of computer time for a geometrically " perfect" shell is thus not generally justifiable. In view of the wide range of buckling loads for shells of a given nominal geometry, what would be required for design purposes when imperfections 4-1
are unknown is an expected distribution of the variation in shape imperfec- ' tions and edge conditions from which an expected critical load could be
,obta'ined with a nonlinear analysis. This approach requires the measure-ment of the imperfections of a large number of shells, the calculation of m the associated critical loads, and, finally, the determination of statistical l design values. The use of critical loads appropriate to the quality of construction demonstrated by a particular contractor would be oossible. ,,'
Such measurements are unavailable at the present time, however. Although international attempts to obtain measurements of geometric imperfections and esti=ates of' support conditions have been initiated, they are in a ' very preliminary stage. The danger inherent in relying on the results of nonlinear theory is, then, the possibility of using critical loads based on an unconservative distribution of imperfections.
-}
a 4.3 EXPERIMENTAL RESULTS - The experimental approach can yield much the same information as the nonlinear analytical approach if the same attention is paid to details of the shell's shape and edge conditions. Tests of physical models will yield useful data for the design of safe structures if the tests are carefully - devised and if a sufficient number of tests is conducted. However, shell structures such as the cylinder and sphere combinations considered for nuclear reactor contain=ent structures can be extremely sensitive to differ-ences in shape and edge conditions.
]
a For example, it is well known that for carefully made speci= ens, some axial compression cylinder-model tests yield results close to 90% of the classical buckling value; on the other hand, speci= ens with greater imperfections yield results less than 20% of the classical value. Thus, a great many tests of ostensibly identical representative structur9.s may be required for an adequate determination of the expected scatter band of test results. Too few tests and too much J care in construction may yield unrepresentative and unconservative criti-cal load values.
]
4.4 LINEAR BIFURCATION ANALYSIS An alternative approach to the deter =1 nation of design data is to rely on the results of a sufficiently large number of tests to define a
" knockdown" factor or coefficient in a design equation based on a sin- I plified linear theory. The knockdewn factor combines the effects of '
nonlinearity and i= perfections in geometry and edge conditions. If enough tests are perfor=ed, a statistical analysis of the data can be
=ade from which design factors having an appropriate confidence level can be chosen, although the precise correlation between imperfections and buckling load will not be known.
.i 4-2 .
The theory used is a linear bifurcation analysis, which is useful as a much less expensive analytical method for deternining the dependence of critical loads on various parameters. However, the method usually
- cannot indicate the effects of geometric shape imperfections. To per-form a linear bifurcation analysis, the linear two-dimensional stress analysis results discussed previously can be used as the input to a computer program that investigates bifurcation buckling. The results of -
such an analysis are valid under the assumption that the initial shape of the shell has the idealized assumed shape (cylindrical, spherical, etc.). - and input stress distribution just prior to buckling. Computer codes that' will perform linear bifurcation analyses with a two-dimensional prebuckling stress state are STACS, NASTRAN, SAP 6 (Version 2), and MARC. Generally, the approach used would be to neglect dynamic effects, in which case the stress distribution at a particular time would be multi-plied by a factor A, say, and used as input i a bifurcation buckling code. The results of the analysis would be a critical value of A. If A is greater than unity, the stress distribution is safe. If A is equal to or less than unity, the distribution is unsafe. As an example, the assumed nuclear containment vessel of Section 3 was investigated for - buckling under an instantaneous seismic analysis stress field that appeared to yield maximum compressive stresses. These stresses were saved on disc during the dynamic stress analysis performed in Section 3.2, using the , SAP 6 program, and were then used in a two-dimensional linear bifurcation analysis. The inverse iteration method is used in SAP 6 to obtain the critical buckling load. The buckling load and buckling mode shape are shown in Figure 4-1. The cost to ISE of performing one bifurcation analy-sis on the containment model (shown in Fig. 3-2) on an IBM 370/158 com-puter using the SAP 6 program was on the order of $100. The drawback to the use of the linear bifurcation analysis method at the present time is the lack of experimental data for the determination of knockdown factors. While uniform stress or load states in cylindri-cal and spherical shells have been extensively investigated both theo-retically and experi=entally, little is known about the effects of nonuniformity of loading or stress state. Thus, the possibility exists for using an unconservative knockdown factor in conjunction with the results of a linear analysis. 4.5 LITERATURE SURVEY _ _ _ _ _ _ __ It is obviots then that all three methods suggested by the ASME Pressure Vessel Code [3b] have linitations or drawbacks, and that alternative methods of analysis must be sought. In seeking to arrive at a tentative conservative method suitable for shell structures subjected to asy= met-ric, dynamic loading, a literature survey was =ada of the state of the art of shell stability. The literature on the stability of shells is vast. (See, for example, [11]). Numerous surveys of methods for the calculation of buckling loads for certain standard static loading conditions on idealized shell 4-3
+m e y N-v \/\/ s 8 g\ N/ \
//\ s\ j
/
// /
'\ / -
2
~
m
~-
\ \l ~-% N: x N NI {N je 7\ N.N N N-
\!ANs ya s
,Q \
[k 6 'N N Y x) 2 K N5 i m x xy ! J p N' N \ c\ ~ x
' =
s W
,/ s
'N
\w % 'x!
N .
~
~ m,; y s 4-4
structures are available [12 to 15c}. Nuclear containment vessels may be subject to a variety of asymmetric loadings, however. Among these are concentrated loads due to the static and dynamic effects of attachments, . jet impingement loads and thermal stresses due to possible pipe ruptures, dynamic loads due to seismic horizontal ground motion or to wave action on offshore platforms, compartmentalized internal pressure loadings, and unsymmetrical external or internal pressures due to blast loading or to hydrostatic forces from the accidental tipping or sinking of offshore reactor containment vessels. In addition, discontinuities --- such as openings (whether reinforced or not) may result in an unsymmet-rical stress distribution, even with a symmetrically applied load. The - stress intensities due to those loadings may be large enough to result in buckling and catastrophic failure of the containment vessel. The-following summarizes what was found about dynamic stability, buckling under nonuniform stress states due to distributed or localized loadings, and perforations. 4.5.l CYLINDRICAL SHELLS 4.5.1.1 Axial Stress Statec Most of the theoretical evidence suggests that the maximum axial stress governs the buckling of cylindrical shells when the stress state is predominantly axial. An investigation [18] of buckling under axial stress varying linearly from zero to a maximum value-(Fig. 4-2) yields a critical maximum stress equal to the critical uniform axial stress when the shell is sufficiently long (L /Rh 2 > 500). For lesser lengths or greater thicknesses the critical. maximum stress is greater than the critical uniform stress; for example, about 25% greater when L2 /Rh = 100. A similar importance of the maximum axial stress is fcund for cylindrical shells under pure bending (Fig. 4-3) and bending and axial compression [19] or, indeed, under any axial stress that is uniform axially but varies circumferentially [20-26] (see, for example, Fig. 4-4). If the stress state does not vary drastically over a dis-cance about equal to a buckle half-wave length, the critical maximum axial stress and the critical axial stress for uniform loading are almost identical. - The experimental data available cover primarily the case of uniform bending and axial compression. Ther=al buckling tests on shells heated along longitudinal strips provide further data for axial stress states that vary circumferentially. Experimental results for axially varying stress states are given in [27] for a cantilevered cylindrical shell under an end transverse force. All of the test results indicate that critical maximum stresses for nonuniform axial stress states tend to be larger than critical uniform axial stresses. In Figure 4-5, for example, lower bound curves of test results for bending and uniform axial com-pression are compared. The more localized the axial stress, the better is the agreement between theory and experiment. It has been conjectured, but not conclusively proven, that the larger critical nonuniform axial stresses are due to the probable different locations of the maximum stress and significant imperfections. The logical conclusion would be 4-5
-I K,= ?:r-D? ~~ ^
1 000 8 l I.i 6 l 6 i i '
~
'l l 100 E Uniform sheer q
,S I L? Ka l' l
' ' p Unifera .
.e , ; comp res sion-E
! f
.t,. t I
-. g l ~ 1 10 100 t oco
,, ~l z..g m ..>
FIGURE 4-2. BUCKLING UNDER LONGITUDINALLY VARYING AXIAL STRESS [18] ._ a
.I 4-6
1.10 , , g ig g
- 1 000 (Soo j:00 E. 3 ,
7 y 100 A/ y 4,
.u s ' ~
\\ A l $y\- -
' \
)l " . -
y'
=
\ / \,
i g i/ y
/ 'y 75 M
E',':,1 M/ UV4 0 8 01 0.02 0.03 0.04 0.05 0.06 0.01 0.04 0.d 0.10 LisA FIGURE 4-3. BUCKLING UNDER CIRCUMFERENTIALLY VARYING AXIAL STRESS (BENDING) [19] 4-7
i i
.I l '
iI .
- SYMMETRIC BUCKLING Heaw wgg W / 3 o
# l Z ,
d r i l p .-
. , k, I
x _. s 3 10 26/(re)I/3 FIGURE 4-4 BUCKLING UNDER CIRCUMFERENTIALLY VARYING ~ AXIAL STRESS _j (THERMAL STRESS) [24]
~1 s
9 h A 4-8
1.0 ~ _ 09 -~
'-~~
& ~ -
~~~
2.z. ~ _ _. _ g j PURE BENDING _ '}' _ _ _ , _
'h
- h (guNIFORM CbMPRESSION
---- M 0.6 w E -
--. - 5 0.5 g w _ _ .
C \ w
---M 0.4 %
$ \
- - - a: 03 \ _ _ _ _ . . . _ _ . . _ _ _
N
--~ 0.2 -
~~ ~~'--~ '
~~- '- - ~~ 2 4 6 8 2 10 g 10 2 4 6 8 10 3 15 R/t FIGURE 4-5. LOWER-BOUND REDUCTION FACTORS FOR ISOTROPIC CIRCULAR CYLINDERS SUBJECTED TO UNIFORM COMPRESSION OR BENDING {l5a] 4-9
that the use of the critical uniform stress as a measure of the critical - cax1=um axial stress is conservative. With the lack of = ore extensive experimental data, no other generalization is possible. . 4.5.1.2 circumferential Stress States _ . _ _ _ _ For cylindrical shells with stress states.that are predominantly circum - ferential, as when under external pressure, the buckling phenomenon is --
)
_ associated vita longitudinal wave lengths and with circumferential wave 1------- - lengths that depend on the length of the shell. For thin, moderately -_ long shells, the circumferential buckle half-wave length is small. The- o results reported in [28] and [29] indicate relative insensitivity of critical m'vi-" stress to the circumferential distribution of stress. - In [28], the theoretical maxi =um critical pressure for circumferential band loading with a varying pressure distribution is shown to be approxi-
=ately the same as that for s1=ilar band loading by uniform pressure - - -
(Figs. 4-6, 4-7). In [29], the =axi=um experimental pressure due to a quite variable wind loading distribution (Fig. 4-8) is experimentally r-about 40% greater than the critical uniform pressure but, interestingly enough, close to the theoretical uniform critical pressure. This lack of dependence of the maxi =um value of the circumferential stress on the ~ circu=ferential distribution can be expected if the circumferential stress does not vary significantly across a length equal to the buckle 'I half-wave length for axisy==etric stress distributions. .] If the pressure distribution is uniform circumferentially but varies longitudinally, it becomes difficult to determine any general rules of thumb for an esti= ate of the critical buckling pressure. In [18] and [30], theoretical results are obtained for a normal pressura distribu-tion that is uniform circumferentially but varies linearly in the - longitudinal direction, such as would be encountered on a cylinder - i=mersed on end in a fluid medica such as water or oil (Fig. 4-9) .* The critical equivalent uniform pressure is found to be not the maximum , pressure but the length averace of the positive pressure distribution,_-_ _. _ ! that is,.the total positive pressure force per unit of circu=ference . acting on'a longitudinal-generator, divided by the total length of the j shell. If the pressure is entirely positive, this corresponds to the ! average pressure. The =axt=um circumferential stress in the shell can then be considerably greater than the critical uniform stress. The theoretical results reported in (31] to [331 also indicate that the - uniform circumferential buckling stress is a lower bound to the critical
=aximum stress for a nonuniform, predominantly circumferential, stress state.
The amount of conservatism depends on the location of the maximum stress and upon the shape of the circumferential stress distribution (Fig. 4-10). The more localized the stress distribution, the higher the critical stress. The critical stress also increases with the distance of the maxi =um stress from the center of the cylinder. l I 4-10
0 ' 0.5 , O, g I l + 01 -( p,,3,,0
/* '
~ ,
s [ , 2
~ - -
\ -
/ '
r + x -
. / sV rw N , .*
>s
\ v
/
% o 0.i
. LATERAL PRESSURE = p P(4) = PO+Pg COS 4 P = Pj/(P0+P) 3 ita ax o l *ja/n i I RELATIVE IN(,REASE IN BIJCKylNGj PREpSUILE AT p = p.5, L/a = y, b/L = 1,v = 0 3 FIGilRE 4-6. BUCKLING UNDER LONGITUDINALLY AND CIRCUMFERENTIALLY VAlYIbG P ES RE [33) i !
l
\'
! ) i
. 4 hi 5000 i . .
>>+'_
.- 0
~
O
~
---05
- ! %_ a/h
( 500 DN -
~
U - N w s
- 100 '
c N N . . R N e w %*~~- 200 . . - j - 50 s s
~
s N s
%s % %
l 400 600 _] 10 _ %'~-- 1000 - 5- ' ' ' O.1 0.5 '.0 1
.b L
FIGURE 4-7. EFFECT OF LOAD VARIATION ON BUCKLING PRESSURE (L/a = n, v = 0.3) [33] _ 4-12
8 0 S S NEGATIVE PRESSURE 60-80 r POSITIvg pggggggg / FIGURE 4-8. CIRCUMFERENTIAL WIND PRESSURE DISTRIBUTION ON CYLINDRICAL TANK (29] 4-13
5
,1 l i
,1 L -- - .
~ . .
i II,..,,,. _,
- t. . _ _ _
e 1000 , _1 _ . . e.......... e s. s ' l t it
' t i iis :j S,ebel[ . . .f,'.g e
, -(i er) x ,.
. e , , ,. . ,. .
,ai.s1 y e 100 _
. !* i '
i f .. u
- p. ..
.o
- f= , s -
e .!* . , , ., ,i.. i , ,i _ _ _ ,a 6 . - g i ! 'l l ! ...,-,..Ja . ; A _. , , , P P. 1-e 1 10 100 1 000 10 000 100 000 L Z'obb Rt gs -- - - . .* (bie>t 1 FIGURE 4-9. BUCKI.ING UNDER LONGITUDINALLY VARYING HOOP STRESS [18] 4-14
39, ca . 48,800 Pte CohaMI. 37..e Pts Ttu u,.oo rs caa . - - rero. ...co .
,. es. um -
FIGURE 4-10. CRITICAL CIRCUMFERENTIAL STRESS DISTR"m'JIONS [31] (R = 1,0 IN., L = 3.14 IN., t = 0.0331 Lt., v = 0.3) 4-15
There are apparently no grperimental results for cylinders with varying 7
- circumferential stress other than the test results of [29]. The only con- ;
.clus' ion that seems reasonable and conservative is that the critical uniform ,
circu=ferential stress can be used as a measure of the critical maximum ' , circumferential stress. This approximation is more conservative than using the critical uniform axial stress as a measure of critical maximum ~l axial stress. 4.5.1.3 Localized Normal Loads - The effects of localized loadings have been treated by a number of writers.- In [34] to [36].the effect of radial concentrated loads (Fig. 4-11) on the stability of cylindrical shells in arial compression or bending is studied ' experimentally. The results indicate that the critical axial stress is - - independent of the radial load up to some limiting value that depends on the imperfectness of the shell (Figs. 4-12, 4-13). Thereafter, the critical - axial stress is reduced. The small number of test results and configura-tions tested make it difficult to arrive 3r general rules of thumb. The - - results indicate that location and nu=ber of concentrated loads are of little significance (Fig. 4-14) provided that the loads are sufficiently far apart and sufficiently far from the cylinder ends. It should be noted that the phenomenon of axial load reduction due to con-centrated lateral loads cannot be precicted with the use of a linear ] analysis. The buckling process is one of snapthrough from a nonlinear prebuckled state, and this requires a conlinear analysis from the initia- - tion of loading. At present, unlike the prcble=s discussed above, there are no theoretical results available. 4.5.1.4 Holes in Cvlindrical Shells 3 An unreinforced hole in a cylindrical shell has much the same effect - on a shell in axial compression as has a concentrated radial lead [37-46]. It has been shown that an i=portant para eter is the ratio of some representative hale dimension r and the attenuation length of edge effects. The represenettive length is the hole radius for circular holes and one-eighth of the perimeter for rectangular and square holes. For small hole-parameters (r//EE 3 0.5) the critical compressive stress is unchanged (Fig. 4-15). As the hole-parameter varies in the range 0.5 3 r/ DEI ) 1.6, there is a rapid decrease in - the critical axial load. Finally, for large values of the hole parame-ter (r/r'Rh > 1.6) the critical compressive stress is again relatively insensitive to hole size. Si=11ar results have been obtained for shells under combined torsion and axial compression (Fig. 4-16). Attempts to obtain analytical results by means of linear bifurcation ! theory have yielded values that are somewhat largc in the small-hole _[ range and quite conservative in the large-hole range, owing to the importance of nonlinear effects (Fig. 4-17). 7 4-16
l' J { { hh @b [_^ (:71, ::: ty, -r !.
,[y:- i(
, : ;It-Y
'T-l-$ . %.~
-Trg dt"
, . .xF di y.: 'i. T/-t' :/i:
" N. // ~-%
sQl , U , ,' , , b& / '4 ' si si s, 4 s, s. ALL LAT[RAL LCCS HAVE WACNITUC[ Q FIGURE 4-11. CYLINDER UNDER AXIAL LOAD, BENDING MOMENT, AND LATERAL CONCENTRATED LOADS [36] 4-17
t J m 7 7 e . , i i 1200 - 'Ist :ArI _ CALAP5E
- 12/ F/67 SNAP Q (2/7/67 8L
~ ~ ' '
I !4 sa J~ J
/i
.s 800 -
/>.. >
g 1'_
- 5. v -
5 600 - -
- s 3 %
M 4cC- % _ g % y . 2x - -
- i , , , , . ,
a - 0 002 C04 C06 008 ClO 012 0 14 0.6 0 13 RACIAL t. CAD 0, !b
'l FIGURE 4 12. CRITICAL-LOAD COMBINATIONS FOR CYLINDER 4 (36]
l t I
}
4-18
to . , , , i i io i , , , e x -
.. T- ..
\ . -
1
,,6 .-
- ,=
A
\
A - v . -
.a e
i s j 8 as - - - 3 os 3a4 - N - a4 - - 0.3 \-[N % - q) - 3 2 '3
.2 - ,... _ - - -
a2 - s' - - an vs 6 10
= '"
., l _
.i - _
tascocx 2.0 3a1 LS 0 0 04 Mg Q42 16 0 20 Q24 03 0 0 03 al2 06 0 20 Q 24 328 CR/tr3 OR/0:3 (a) Unpeessurized, spatial dtstribution Si (b) Pressurized, spatial distribution Si 9 FIGURE 4-13. NORMALIZED CRlrICAL-LOAD CURVES FOR SEVERAL MODELS [36] 4-19
7, l
.J I0 l I, i l i !
as l as w = .-
\
ar _
\\
.g
., _ . -l N\ _
QS -
\ ,, g, g -
g3 E a5 - Ih - 3
,\\ k a 0.4 - -
% .p=opd 0.3 - _
$NnAL 0:57A"suncs
---si --
Q2 -
$2 - -
- . _ $3 _. .
54 gi _ _
. ss _ ..
- .. - s s _ _
0 ' ' ' ' I I O 204 0 08 0.12 0.:6 G20 024 023 ' ORl8r0 i 1 FIGURE 4-14. EFFECT OF SPATIAL DISTRIBUTIONS Sy THROUGH S ON 6 CRITICAL-LOAD CURVES FOR CYLINDER 4 [36]
.1 4-20
.8 -
t -
.6 -
s CIRCULAR CWCWS - P
-4 P
. s --
et ' g,
-k
.2 SQUARE CWOWS ___ g i ' ' , , i 0 1 2 4 5 6 FIGURE 4-15. THE EFFECT OF CIRCULAR AND SQUARE CUTOUTS ON THE BUCKLING OF A CIRCULAR CYLINDER LOADED BY -
CENTRAL AXIAL COMPRESSION f39] 4-21
.9 C %M -
.3 -
g - Q D ~ ' - " g APPROXIMATE PURE TC%5 ION
.7 VALUES WITH AXI A'. DEAD -
T D.. 4 . g LOA 0 EXTRAPOLAT'O TO ZERO ( .6 - O EXPERIMENTALLY MEASURED
, ' '* l RESULTS WITH AN AXlAL !
DEAD LOAD OF 7.2113 ,' t
.5 -
.l
}
*r , , , , f , , ,
Q l 2 3 4 5 6 1 3 7 (a) Pure torsion i
.I r. J
~
.6 -
]
I CL f"O
~
d (L13 _ LO5s
.2
$_ g, 2.5
- 1 05 0 -
f f f
~
0 .1 .2 .3 4 .I .6 .7 9 YCL (b) Torsion and axial compression I TIGURE 4-16. EFFECT OF CIRCi1LAR HOLE ON BUCKLING OF A CYLINDER U'iDER COMBINED TORSION AND AXIAL CCMPRESSION [39] 4-22
.9 i 3<,
t
~-
i
' REPRESENTATIVE SELL WITH RA - 400 NASTRAN O LOCAL BUCKLING
^
*I ~
RESULTS i a CENERAL COLL! PSE 6 \ '
, ag
{ .5 3
' ~
g as fBC. wlES OF EXPERIMENTAL 4 // RESU;f 5 FOR 16 SHELLS
- b ~
f ,.
~ 2 -
.1
_, I t t t t i I O 1 2 3 4 5 6 7 7 (a) Circular cutouts g REPRESENTATIVE SHELL
, WITH SQUARE CUTOUTS
_ 5, O LOCAL BUCKLINC -
, a CENERAL COLLAPSE
'6u i _. .
hNASTRAN p _ ',RESULTS P ' CL "
~
800NDARIES OF EXPERIMENTAL RESULTS FOR 12 SHELLS - o __.
.2 -
M _ f f f f I i _ 0 1 2 3 4 5 6 7 r (b) Square and rectangular cutouts FIC'_ ..I 4-17. COMPARISON OF RESULTS OF LINEAR BIFURCATION THEORY AND EXPERIMENTAL RESULTS [39] 4-23
The analyses of [47 to 49] indicate that it is possible to counteract - the, deleterious effect of perforations by suitable edge reinforcement.
. The*results available are sketchy, so that they should be looked upon only as qualitative indications of the effects of stiffening. Precisely <
what reinforce =ent dimensions are required is difficult to ascertain, since results have been obtained only for specific shells. In.[47], conocoque cylinders having a rectangular cutout with and without edge stiffening were analyzed by the nonlinear STAGS prograc...It was found- y that it was more advantageous to reinforca only the edges in the direc-. _ . . _ . - tion of the load (stringer reinforcement) than to reinforce all four . . :. edges (picture frame reinforcement) . Using only a fraction of the u - - - r recoved cylinder area as stiffening is sufficient to bring the buckling ; load of the cylinder back to the range of values obtained experimentally for uncut cylinders (Fig. 4-18). It is interesting to note that the , more locali:ed the stress distribution, the better the agreement between : theory and experiment. This result is, of course, to be expected in the light of the discussion of buckling under axial stress states. ., I Somewhat better indications are given by the results of [49], which ' investigates the effect of cutout edge stiffening on a cylinder with rings and stringers. Only a single cylinder with two sizes of square l cutouts is studied (Fig. 4-19). The results show the effect of a J constant area picture-frame type of edge stiffening on the buckling - load as the ratio of the volume of edge stiffening to the cutout volume is increased (Fi;;. ,-20) . The buckling load of the uncut cylin-der is seen to be obtainable with edge reinforcement volumes ranging from 0.5 to 1.0 times the cutout volume. There are no experimental - , , results for such shells. Since the agreecent between theory and experi-ment is known to be quite good for uncut, suitably stiffened shells, l however, the results given should be accurate. "_ Current reinforcement-area requirements of the ASME Pressure Vessel Code [3c] call for a mini =um reinforcement volume equal to twice the cutout volu=e for shells designed for internal pressure, and half of that for shells designed for external pressure. These amounts of edge stiffen-ing would be adequate to prevent degradation by buckling due to axial . stress for the particular cutouts of [47] and [49] . Indeed, the shell designed for internal pressure e uld have sufficient edge reinforce =ent 1olume for larger openings.. If the shell were designed for external -l pressure, however, the edge reinforcecent volu=e might be inadequate for large openings. The adequacy of the design for internal edge .f reinforce =ent that permits one-half the usual area to be used for longitudinal edges may be questienable for some opering sizes as well. The effect of penetrations, that is, a hole filled with an object such . as a pipe, was studied experimentally as a part of the current investi-gation. The results obtained by C. Babcock of the California Institute j of Technology are reported in Appendix A. The few experimental tests , conducted indicate that for s=all holes, a rigid penetration is the 4-24
n ae . - - - . . . . . . sm an o,u e' RANGE OF EXPERIMENTAL "8 RESULTS FOR UNCUT CYLINDERS " 'f ,..J y 0,4 -
#~"""~" ""~
e,, rusa + - .. - T -> + " 03s c2 - -
~~
o COMPUTED VALUES OF P er 08 -
+ EXPERIMENTAL RESULTS F01 STF INGER REINFORCEMENT
- EXPERIMENTAL RESULT FOR PICTURE FRAME REINFORCEMENT
~ ~ ~
0 0.005 0 010 0.015 0.020 0.02S REINFORCEMENT THICKNESS, s, FIGURE 4-18. EFFECT OF REINFORCEMENTS AROUND 45-DEG CUTOUT [47] (t = 0.014, R = 6.06) h .1" R - 57. 3" N e
- w
- 2a " L %"
1 2a
}-
-.h FIGURE 4-19. OVERALL DIMENSIONS OF STIFFENED CYLINDER [49]
4-25
l l
- 5 e
- _ h 2 s d
e -
. _ 0 n e
/ '
2 f f 6
/
]
i
/ i ,,
O I t s 4 9 a 21 4 - 5 TA r [ 2 2 1 R e D . g A E n 0 f M i 1
/ - 0 U r G =
L t =. 1 O s N
/ V I.
I d n K 5 C
/ '
0 a g n U l i
=
/ i N R O 5 0 5 0 5 0 0 "
1 1 0 5 0 5 0 < 5 N 0 9 9 8 8 7 )
. . 7 6 b E 1 1 1 1 0 0 0 0 ( M 0 0 O E
o u . C t " ~ 4" g3 o2 U!= ! R O F N I E R E
- G =
t D 5 E 2 F - O
' l T -
. - 0 l C e E f - 2 h s
F F E
/ .
O d 5 I e f 2 a "2 1 4 2 1
. T A
R f n e 0 2 f - . E i 4 M t
' 0 U s E R
/ 1 L O r U V e G
/ g I F
. 5 n ,
/ '
0 i r
' S t
0 5 0
'0 I
0 5 0 5 0 2 1 1 0 0 9 9 5 8 ( a
. . 8 1 1 1 1 1 0 0 0 0 o.g a O o 'd Q S oz >
=m iM
, egyivalent of an unreinforced hole. For larger holes, there is a significant i=provement in the buckling stress with a rigid penetration where the cylinder is not pressurized. As the internal pressure is - increased, however, the cylinder with the unreinforced hole exhibits as good , or even better, buckling strength characteristics. 4.5.1.5 Dynamic Buckling For loads that are functions of time, design procedures may require little modification. Dynamic application of external pressure appears to present no difficulties, since both theory and experiment indicate-increases in maximum dynamic pressure above the static buckling pressure for many load histories of practical i=portance. Step-function pressure distributions and impulses of various shapes are examined in [50, 51], for example. Step loading has little effect (Fig. 4-21), whereas impulse loading yields peak pressures considerably larger than critical static pressures (Fig. 4-22). Sinusoidal pressure / time distributions with zero average pressure are investigated experimentally in [50] and yield peak collapse pressures that range from 90% to 350% of the static buckling pressure for an axially loaded shell (Figs. 4-23, 4-24). Similar trends are obtained when the sinusoidally varying pressure is combined with a static pressure. A theoretical investigation (52] of the stability of a cylinder subjected to a sinusoidally varying pressure combined with the static critical pressure indicates that the somewhat greater peak pressure can be sustained even in this case (Fig. 4-25). The results are suspect, since a linear theory is used rather than a non-linear theory. Nevertheless, the results are in qualitative agreement with the experimental results for a pure sinusoidal variation. Axial compressive load dynamic effects are of much more importance. The results reported in (53] and [54] for a simplified structural model of a cylindrical shell (Fig. 4-26) indicate that the critical dynamic axial load applied as a step-function may be significantly less than the static buckling pressure. The more imperfect the cylindrical shell, the smaller the ratio of dynamic and static loads (Fig. 4-27). However, the dynamic load may be increased above the critical static load without effect if it is applied for a sufficiently short time (Fig. 4-28) . It should be noted in addition that the dynamic buckling load is. always 70% or more of the static buckling load. The reduction can be disregarded safely if a dynamic stress analysis is used to determine - _ _ . the =aximum axial stress, which is then applied as a static uniform stress in the structure. 4.5;2 SPHERICAL SHELLS A few results are available for spherical shells other than for spherical caps. An extensive review of the buckling of spherical caps is given in (55). Results for nonuniform loading have been obtained 4-27
- 90 yg __ i ___ _ __ 60 - s k _. ! _ so OQ 1
) -
PIAK g -- 8 f O d
---_h40 -
t 3
\ .,
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O I~ 30 - - - - ( _m A mI A . _ _ _ . STATIC 4 h e C.CC75 INCH
- .'JCAN CYNAMIC @
A/4e 534 PCAX CYNAMIC @ , 0 QCt 002 0.03 004 CCS 0.06 0.C 7 { OlFFERENTI AL pag 3$ygg.pgg j TICURE 4-21. COLLAPSE LOAD INTERACTION CURVE FOR A CYLINDER UNDER A RAMP-STEP PRESSURE [50] 4-28
4 50 .
;r a
PEAX PRESSURE f h --sucxuno STATIC 350
}Y th I~
300 M
~
f250 m u E 200
& 1 L a \
ISO $ \ ,
\
\ \
100 l 1 AXIAL LOAD, P Pos72.6 LB. 24.49 L:EL @ R/:534 3 50 3** O "*' ^ 41.52 La @ h so.OO751N. 1 0 0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 PULSE DURATION (t). SEC FIGURE 4-22. COLLAPSE PRESSURE VERSUS PULSE DURATION AND AXIAL DEAD WEIGHT FOR A CYLINDER [50] 4-29
IC6 o I
~
o o 9' 90 - I o ' " w',.,,-,
\ o o
o 80 o I o 8 o 8 o 70 ; _ _ _ _ _ 3 o o h o o
- 60 .o U l o 11 0
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- to -
b e O.CCS INCH I I R/h a 800 L/R e 2 l0 P, a36.3 (83 I P e 20 (83 * ~ 0.011 a082 act3 0.014 C015 0.C16 CtFFERCNTIAL PRL53UME-933 FIGURE 4-23. SINUSOIDAL COLLAPSE PRESSURE VERSUS FREQUENCY FOR A CYLINDER [50) 4-30
SCO- - h e O.COS INCH R/h a 800 L/R e2 ' I Po
- 33.2 LBS Pe 20 LSS
- 600- , O N
a. dl 0 a 8
$ 8 0 0 x
taJ
,(
e w E s E o I O
?nn _ _ _ .
O I O o 0 - 0 0.01 0.02 0.03 DIFFERENTIAL PRESSURE-PSI FIG'JRE 4-24. SINUSOIDAL COLLAPSE PRESSURE VERSUS FREQUENCY FOR A MYLAR CYLINDER [50] 4-31
1.0 ._..
~
r/L - 0.8 05--- '/h '
/ ._ _
~
1.0 1200s [ -
~
~
UNSTABLE' I / 0.2 - .
} Q sf ,W \ s'
~, / -
0 , A8LE , , , , , , . , _ o 0.05 0.10 0.15 0.20 0.25 0 30 w/w , (a) Lateral pressure 1.0 r/L r/h 0.8 ' 05--- 1200 - 100O [ / 0.6 , f 0 j 0.4 ' _ 0.2 20O l
^
s Y/ [}
// - . - _ .
~
/ /ll l 0
. 'i 1
STABLE 1 0 0.05 0.10 0.15 0.20 0.25 0 30 w/w CJ) Hydrostatic pressure FIGURE 4-25. STABILITY BOUNDARIES.FOR CYLINDERS SUBJECTED TO STATIC AND SINUS 0IDALLY VARYING EXTERNAL PRESSURE [52] 4-32
I t . Af(t)
~ , , , , ,
E*E -. w 2=h F 7 s K(( +a(2) (QUADRATlC SPRING) Ac =y ,,,,,,l,,,,, F= K (+ b(3) (CUBIC SPRING) FIGURE 4-26. MODEL FOR DYNAMIC BUCKLING [54) 9 4-33
i QUACRATIC /
,/ .
f(t) '\
/
j _, A o j CUBIC T' f
/
' ~
0 =
.t .5 ' ' ' '
O xs : Te AC= BUCKLING LOAD OF PERFECT STRUCTURE As= STATIC BUCKLING LOAD OF IMPERFECT STRUCTURE AD= DYNAMIC BUCKLING LOAD FIGURE 4-27. INFINITE STEP LOADING [54] 4-34
~ , - ,
I
' 'l C_ . ,. . .
_ A L- %,f 6'ZSBv-3,_byi[ff} {fQs'. 1% y}.7 a y
~ ' Nh,'r?'d.4$i . - g , _7__ _ R L
, : h ,9 Q/[)'f
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a 1
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i i i 10 - i T I " T
- I5 8-1 -
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J J, l As 4 f 1 r I 4 ._ ( F(t) i - l 2
\ t
* \ T L" 0 2 ._
' ' t.0
' ' .8 for i < T A
.6 f(a) = 1 O .2 A for t < 0,1 > I O _1.
f(r) = 0 Ac FINITE STEP I.0ADING [54] FIGURE 4-28. '
for co=plete spherical sheli: under concentrated loads without [56, 57] and with (58} external pressure. These results are also applicable to deep, truncated shells or caps. For a spherical shell under an inwardly directed point load, buckling of the structure into varying numbers of waves ocnurs, but the shell does not collapse and continue to carry increasing load. An outwardly directed concentrated load results in buckling in eight circumferential waves. For thin shells,- the nu=ber is apparently independent of radius / thickness ratio. An inwardly directed, increasing concentrated load produces changes in critical uniform external pressure similar to results for. concentrated-lateral load and axial compression for cylindrical sh alls (Fig. 4-29). Initially, the critical external pressure is relatively insensitive to concentrated load values. As the load increases, the critical uniform pressure decreases significantly and then levels off and becomes insensi-tive once more as the load increases still further. Experi= ental and theoretical results are in reasonable agreement. Some dynamic stability results are available l591 for pressure step loading on a co=pl;te spherical shell. An approx 1= ate analytic solution yields a dynamic buckling pressure 19% less than the static buckling pressure. Similar results have been obtained for spherical caps. An interesting investigation indicative of the interaction between cylinders with henispherical closures (Fig. 4-30) is reported in [60]. The_ external buckling pressure of the combination is found to be greater than the buckling pressure for a cylinder with stiffening rings, except for very short cylinders. The hemispherical shells were assumed to have a thickness ranging from one-half the cylinder thickness to a value equal to the cylinder thickness. No other end support was provided. This result would suggest that buckling is more apt to be critical for the cylinder than for a hemispherical end closure. 4.6. A TENTATIVE METHOD OF STABILITY ANALYSIS . _ _ _ _ _ _ _ _ _ . _ _ _ _ Although there is little available on specific reduction factors for variable stress states, the foregoing literature survey indicates that to use knockdown factors associated with uniform stress states is con-servative. The literature survey also. suggests that ncglecting the effects of dynamic loading on buckling--that is, the assumption that the internal stress distribution at a particular instant of ti=e is quasi-static--is definitely conservative for circumferential stress states. The use of dynamic stress states as quasi-static stress states appears to be conservative for axial load states if the dynamic axial stress is greater than about 140% (1/0.7) of the axial stress obtained with a static load application. 4-36
-a - ., ,
4 I 3 .- y- 1.00 -- l-n ,, p g .-
,. . .. . n u 88 e ** 85- '
0.75 P
, y' '*****
g _,- I$< m - '4'..'... *
',, ,_,I i-_ N-L,e;.a e' (
g _ _-
- O.M -
P,. g q', 4A C' j s L , e
~
- rr , i.
,.,,,, ra i
es 7 w 0 - N 0.25 h y ,' N i, \ , y s si s g p 0
, I,' . E!
2
\ 3 0
1 4 0 0.2 0.4 0.63.0.8 1.0 1.2 J' O 0.47 0.94 1.22 1.55 1.80 1.96 Central point load versus additional central de8ection P' for a series of constant values of the external pressure. Pressure versus central deficcaion: experimental results. Numbered points give values of the dimple radius r,' rP p r ' P' = Et t p' = EV 12(1 - u') J' = J a r,' = [12(1 - u')] O= Vrt FIGURE 4-29. BUCKLING OF SPilERICAL SiiELL UNDER EXTERNAL PRESSURE AND CONCENTRATED LOADS AT TIIE POLES [58] l - l l
- i 'l l ! I '!
i . l. i j l j i I
; i '
l . I. i f . ,
~
I -
. i.. . .
e 5 Ib p/cm') l l es IIP /**'I 4=lult
- N,,,,
1 8* 5 g
, 4 na s (np/ca')
8 " I " 88 *
; ,,. -.,, 10 - -th -- -
10 s ,,, , Q, s s n=8 uule ' s 5 e .s e , x 5 en *,
-TTi
% e 1 s.-s3 .
6 Q '-Js
~
c- - -
, 's, e ,s i s ,,,
s a s
', \N
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s - g {2:. a.5 - e.s N,
'N ,-
r g o.cs o.1 c.s 5 ,I.cs o.: e.s a x 5 e.c5 o.: s.s s 5 UR L/R g,j g _ l CYLINDRICAL SilELL WITil liEMISPilERICAL ENDS CYLlHDRICAL SilELL WITil STIFFENING RlHGS AT Tile ENDS l
~
2 FIGURE 4-30. EXTERNAL BilCKLitiG PRESSURE OF A CYLIllDRICAL SilEl.i. WITil llEMISPIIERICAL ENDS l60) _ m j h 8 I i g g e ! l ; i . i l
~
l f 8
- l I
With these considerations in mind, the following is recommerded as a conservative method of buckling analysis.
- a. An accurate two-dimensional linear stress analysis of the shell including the effects of stiffening, cutouts, penetrations, and dynamic loading is made. The analysis should be shown to be accurate by comparing the stress state for finite element or finite difference models having increasingly larger numbers of subdivisions.- -
. b. For a particular f.nstant of time, the compressive axial and circumferential stresses anel the positive or negative shear' stresses at each nodal point are increased by dividing each by a suitable knockdown
, factor for uniform axial, circumferential, and shear stress. Tensile _
axial and circumferential stresses should not be increased since, in general, they serve to increase the buckling load. Thus, the knockdown factor for these stresses is vaity. A knockdown factor greater than unity may be used with tensile streeses for an even more conservative result.
- c. The shell model use.d in the most accurate stress analysis is analyzed for linear bifurcat:.on buckling under the increased stress distribution. In this analysis the modified stress distribution is multi-plied by a constant factor, say A, which is then determined as the lowest eigenvalue of the buckling pioblem. A value of A greatar than unity then i
indicates that the structure is safe. A value of A equal to or less than unity indicates that the strtcture is unsafe.
- d. A sufficient numbe of " time snapshots" of the stress state must be considered to ensure the Loclusion of the worst possible condition.
In the discussion of the preceding method, nothing concrete has been stated about what knockdown factors should be used. Experimental data that can be used to establish reduction factors for axial, circumferential, and shear stress states for cylinders are available in publications asso-ciated with the aerospace industry. In [lSa], for example, the following values are recommended for the reduction of calculated stresses of unreinforced isotropic cylinders: ( ) Axial Compressica: 1 - 0.901 1-expi77fR/tI
\
1 _ /l Exteraal Pressure: 0.75 Torsion: 0.67 The external pressure reduction factor was actually recommended for hydro-static pressure, which includes an axial compressive stress equal to half the circumferential stress. Since the effect of the axial stress is negligible except for short cylinders, the value of 0.75 may be used for compressive circumferential stress alone. Values of knockdown factors for buckling of fabricated cylinders under axial compression and external pressure are given in [16] and [17}. 4-39
- I Although considerable data exist for stiffened cylinders and have been
' co= pared with theoretical results [61, 62], no specific reco==endations
~ :or reduction factors have been =ade. It is known that the reduction factor is larger for stiffened shells than for unstiffened shells, but a lower-bound curve as a function of significant parameters has not been established.
y Design rules for ring and stiffener sizes to prevent general instability are given, however, in [17] . A conservative recoc=endation for stiffened shells is the use of reduction factors for the unstiffened
---~shell- that has a pertinent uniform buckling load equal to that of the stiffened shell.
Since reduction factors for circumferential and shear stress are independent of thickness, all that needs to be established is the equivalent shell for uniform axial stress. In that calculation the effects of cutouts would be omitted. In the buckling analysis of stiffened shells, the nu=ber of longitudinal stiffeners or circumferential rings may be large, in which case they =ay be " smeared" and the shell treated as a continuous structure. Local buckling sht 21d be considered as an alternative possibility if the buckle half-wavelength stiffener spacing.in the appropriate direction is close to the ring or For local buckling between closely spaced longitudinal stiffeners, the reduction factor may be taken as 1.0. Studies su==arized in [62] suggest that it is important, if at all feasible, for rings to be treated as discrete stiffening ele =ents.
=
= Thd is 3.
factor of safety specified in the ASME Pressure Vessel Code [3b] If the reduction factors given above are reduced by this value and co= pared with values suggested by Part (b) of the / SME Code provi-sions for buck.O ng [3b]and [62] , i.e., a reduction fautor of 1/10 for axial co=pressit- and 1/3 for external pressure (Fig. 4-31), the two values for axial compression are about equal for shells with a radius-to-thickness ratio of 500 or so, whereas the ASME Code value for external hydrostatic pressure is actually larger than 1/3 of the NASA reco== ended reduction factor. A safety factor of 2 would cake the reduced NASA circum-ferential stress about equal to the recommended ASME value. It is felt that a safety factor of 2 is sufficient to achieve a conservative design for all states of stress;if applied te reduction factors obtained as the
'j minimum of experimentally obtained data. It is satisfactory, however, to abide by the experience emb.odied in the ASME Pressure Vessel Code [63]
and to use reduction factors of 1/10 and 1/3 for axial and circumferential stress, respectively, without an additional safety factor. Since the " experi= ental reduction factor for shear stress is about equal to that of circumferential stress, the same factor of 1/3 might be used for shear. For deep spherical shells under external pressure, the recocmended reduc-5 tion factor given by the ASME Pressure Vessel Code [62] is 0.1, whereas the recoc= ended NASA value [15c] is 0.14. Here again, the ASME value is O '
]
sufficient, while a safety factor of 2 should be used with reduction factors experimentally obtained data. determined as minimum values obtained from experimentally l 1 J i 1
! 4-40
) _ _ _ _ _ . . ...
.m . .
I ~ 10 l l l 1 I I AXIAL COMPRESSION
---- EXTERNAL PRESSURE 0.8 -
REDUCTION FACTOR RECOMMENDED 0.6 - IN NASA SP-8017 o
"TH 0.4 - -
N ~~ / _ 1/3 NASA C2 - VALUE ' .. AS CODE Q I I I ' I I - 0 100 200 300 400 500 600 700 R/t FIGURE 4-31. COMPARISON OF VARIOUS REDUCTION FACTORS 4-41
SECTION 5 CONCLUSIONS In the light of the preceding sections, a number of conclusions about
- the current state-of-the-art of the analysis of nucicar containment. - - -
vessels for buckling may be reached. . . .
- a. The criteria in , art (a) of the current ASME Pressure Vessel Code [3a] are inconsistent and may permit the selecticn of the least con-servative method of analysis. While present analytical methods are sufficient to yield accurate buckling loads of " perfect" nuclear contain-ment structures, there are not enough experimentally determined data on either expected imperfections of shape and support conditions or expected knockdown factors for nonuniform stress states to per=it these methods to be used indiscriminately,
- b. The criteria of part (b) of the current ASME Pressure Vessel Code [3b] are conservative for the loading conditions for which they are defined. As presently defined, however, they are not applicable to com-bined loadings of stiffened structures.
Since some conservative method of buckling analysis of containment vessets is necessary, the f* Llowing tentative procedure is recommended: A general-purpose computer program such as NASTRAN, MARC, SAP 6, or ANSYS is used to perform an accurate linear elastic dynamic analysis for the nuclear containment structure in question. Such two-dimensional analysis codes can take into account ite=s such as stiffening, penetrations, cutouts, non-uniform loads, and combined loads. The stress state at a given instant of time is then modified by dividing the axial, circumferential, and shearing _ _ _ _ _ stresses respectively by a knockdown f actor appropriate to uniform axial, _ circumferential, and shearing stress acting separately. The modified stress state is now applied as a quasi-static prebuckling s tress state for which a linear bifurcation analysis is performed, using the same model that was used to perform the stress analysis. A sufficient number of such bifurcation analyses for different time stations is performed to ensure the determination of the worst stress state. ' The writers feel that the method reco= mended for analysis of buckling stress should be established as the method of first choice. Although current industry practice is to dismiss two-dimensional dynamic stress and buckling analysis as too impractical or too expensive, the preliminary results of the present report indicate that these analyses are practical and can be performed at reasonable cost. 5-1
The costs of running such programs may be drastically reduced by using the latest computing methods, utilizing minicomauters, microprocessors, end computer graphics. The SAP 6, NASTRAN, ANSYS, and MARC programs are or will shortly be available on minicomputers, for example. Preliminary runs, using the SAP 6 program on the VAX minicomputer, indicate that the costs of two-dimensional buckling analysis of nuclear containment vessels may be further reduced. Thus, techniques for the theoretical determination of buckling loads based on two-dimensional bifurcation theory and using avail-
. able computer programs should be studied to achieve maximum efficiency.
'Diey must be accompanied, however, by an extensive and well-thought-out test program to establish reliable knockdown factors that are applicable to shells with a wide variety of nonuniform stress states, including the effects of penetrations, cutouts, and localized loads.
Such a program would serve to establish the analysis of nuclear containment vessels on a more rational basis and would enable the degree of conserva-tism of the design to be assessed more accurately. At the same time, it is recognized that a simpler method such as design curves might be prefer-able for preliminary design. The results of the recom= ended program of investigation would help to develop such a simplified method, possibly along the lines of that suggested in (64}, which, however, has too many uncertainties that render its adequacy questionable. Another fruitful area for further study is that area dealing with the com-bination of loading conditions of different probabilities of occurrence.
. Methods based on a straight addition of worst possible conditions are extremely conservative, of ;ourse. However, rational methods for combining different loading conditions, based on the use of probability statistics of the loading in conjunction with modern methods of risk analysis, should be studied to reduce the conservatism to more reasonable, but still allow-able, levels.
5.2 RECOMMENDATIONS -. . . - - The following recommendations are made for necessary future work en deJign criteria for buckling of: nuclear containment vessels: (1) A method has been proposed for the verification of the buckling capability of nuclear containment vessels. The proposed method should be tested on actual contain=ent designs, using typical design loading conditions to deter-eine the degree of conservatism of current design methods. (2) The investigation ahould include a study of modeling techniques necce*,_y to adequately represent the structure, particularly penetrations and their associated edge stiff-ening. Amounts of stiffening required to prevent buckling load degradation should be studied. The effect of attach-ments such as piping on buckling loads should be investf-gated to determine their importance in the design Process. 5-2
(3) Further studies should be made to refine knockdown and safety factors for combined loading and for nonuniform-loading experiments performeo in conjunction with analysis to help develop appropriate knockdown factors, and to help refine or modify the proposed method for buckling verification. Particular attention should be paid to the deternination of appropriate knockdown factors for stiffened structures and to the determination of the most important design parameters. (4) The possibility of devising simplified methods of buckling analysis should be investigated. This investigation should include anal,.*c studies of values of design parameters that would permit such simplifications as the neglect of penetrations and the use of one-dimensional computer codes. The development of design curves or equations that would include the effects of.penetratiens, stiffening, and asymmetric loading should be explored. (5) The use of moderr. methods of Tisk analysis, together with the probability statistics of various types of loading, should be studied to determine a conservative method of load combination that is more reasonable than the sum of worst-possible conditions. The rationale for such methods should be particularly studied and presented in such a way as to inspire reasonable confidence in their use. 5-3
SECTION 6 REFERENCES
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___ . ' , __ 12. Gerard, G. and Becker, H., Handbook of Structural Stabilitv. Part III. Buckling of Curved Plates and Shells, NACA IN 3783, 1957_._ _
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J. Aero Sci., 27:11, November 1960, pp. 854-858. ._____ ,
- 21. Lundquist, E.E. , " Strength Tests of Thin-Walled Duralusin Cylinders in Combined Transverse Shear and Bending," NACA TN 523, 1935.
- 22. Johns, D.J., "On the Linear Buckling of Circular Cylindrical Shells
_. _._under Asy==etric Axia{ Compressive Stress Distributions," J. Roy. Aerr. Soc., 70:672, December 1966, pp. 1095-1097. ._
- 23. Abir, D. and Nardo, S.V. , "Ther=al Buckling o f Circular Cylindrical
_ _ _ _ _ _ _ Shells under Circumferential Tersperature Gradients," J. Aero. Sci., 26:12, December 1959, pp. 803,808.
- 24. Hoff, N.J.; Chao, C.; and Madsen, W.A., " Buckling of a Thin-Walled Circular Cylindrical Shell Tested along an Axial Strip," J. Acol.
Mech., 31:2, June 1964, pp. 253-258,
- 25. Ross, B.; Mayers, J.; and Jaworski, A., " Buckling of Thin Cylindrical Shells Heated along an Axial Strip," Proc. Soc. Exo. Stress Anal.,
22:2, 1965, pp. 247-256.
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- 27. Peterson, J.P. and Updegraff, R.G., " Test of Ring-Stiffened Circular Cylinders Subjected to a Transverse Shear Load," NACA IN 4403, September 1958.
- 28. Almroth, B.O., " Buckling of a cylindrical Shell Subjected to Non-uniform External Pressure," J. Appl. Mech. , 29:4, December 1962, -
pp. 675-682. 29. Esslinger, M. and Geir B ,"Beulen von Schalen," in Deutschen Forschungs-und Versuchsanstalt fur Luf t-und Raumfahrt E.V. , Institut fur Fluggeugbau, Braunschwieg, 1975.
- 30. Brush, D.O. and Almroch, B.O. , " Thin Shell Buckling Analysis . . . A General Expression for the Second Variation of the Strain Energy,"
Proc. 4th U.S. Nat. Cong. Appl. Mech., Vol. 1, 1962, pp _497-505._____ __
- 31. Hoff, N.J.,
" Buckling of Thin Cylindrical Shells under Hoop Stress Varying in Axial Direction," J. Appl. Mech., 24:2, Septe=ber 1957, pp. 404-412.
- 32. Brush, D.O. and Field, F. A., " Buckling of Cylindrical Shells under a Circumferential Band Load," J. Aero. Sci., 26:12, December 1959, pp. 825-830.
- 33. Almroth, B.O. and Brush, D.O., " Buckling of a Finite-Length Cylindrical Shell under a Circumferential Band of Pressure," J. Aero. Sci., 28:7, July 1961, pp. 573-578.
~
- 34. Ricardo, O.G.; Marreco, D.; and Csoknyai, H., " Lateral Deflection of
- Circular Cylindrical Walls during Buckling," J. Mech. Eng., Sci., 5:3, 1963, pp. 245-257.
- 35. Ricardo, 0.G.S., "An Experimental Investigation of the Radial Displace-ments of a Thin-Walled Cylinder," NASA CR-934, November 1967.
- 36. Okubo, S.; Wilson, P.E.; and Whittier, J.S., " Influence of Concentrated Lateral Loads ots the Elastic Stability of Cylinders in Bending,"
Experimental Mechanics, 10:9, 1970, pp. 384-389.
- 37. Tennyson, R.C.,
"The Effect of Unreinforced Circular Cutouts on the Buckling of Circular Cylindrical Shells under Axial Compression,"
J. Eng. Ind., Vol. 90, 1968, pp. 541-546. _.
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- 38. Starnes, J.H., "The Effect of a Circular Hole on the Buckling of Cylindrical Shells," Doctor dissertation, California Institute of Technology, Pasadena, 1970.
- 39. Starnes, J.H., "The Effects of Cutouts in the Buckling of Thin Shells,"
in Thin-Shell Structures , Theory, Experiment, and Design, edited by Y.C. Fung and E.E. Sechler, Prentice-Hall, Inc. , Englewood Cliffs, N.J. . 1974, pp. 289-304. 6-3
- 40. Brogan, F. A. and Al=roth, B.0. , " Buckling of Cylinder with Cutouts,"
AIAA J., 8:2, February 1970, pp. 236-240.
- 41. Starnes, J.H., "Effect of a Circular Hole on the Buckling of Cylindrical Shells Loaded by Axial Co=pression," AIAA J. ,10:11, Nove=ber 1972, pp, 1466-1472.
- 42. Jenkins, W.C., " Buckling of Cylinders with Cutouts under Co=bined
,,, Loading," M.D.A.C. Paper W.D.1390, McDonnell Douglas Astronautics Co., 1970.
- 43. Toda, S. "The Effect of Elliptic and Rectangular Cutouts on the Buckling of Cylindrical Sh' ells Loaded by Axial Co=pression,"
A.E. Engineering Thesis, California Institute of Technology,1975. 44 Al=roth, B.O.; Brogan, F.A.; and Morlowe. M.B., " Stability Analysis of Cylinders with Circular Cutouts," AIAA J. ,11:11, Nove=ber 1973, pp . 1582-1584.
- 45. Willia =s, 0.G. and Starnes, J.H., "So=a Applications of NASTRAN to the Buckling of Thin Cylindrical Shells with Cutouws," NASA TM-2637, Septe=ber 1972.
- 46. Starnes, J.H., "Effect of a Slot on the Buckling Load of a Cylindrical Shell with a Circular Cutout," AIAA J., 10:2, February 1972, pp. 237-240.
- 47. Al=roth, B.O. and Hol=es, A.M.C. , " Buckling of Shells with Cutouts ,
Experi=ent and Analysis," Intern. J. p lids Structures, Vol. 8, 1972, pp. 1057-1071.
- 48. Palazotto, A.N., " Bifurcation and Collapse Analysis of Stringer and Ring-Stringer Stiffened Cylindrical Shells with Cutouts," Co= outers and Structures , Vol. 7,1977, pp. 47-58.
- 49. Cervantes, J.A. and Palazotto, A.N. , " Cutout Reinforce =ent Stiffened Cylindrical Shells,", unpublished manuscripc.
- 50. Final Report on Buckling of Shells under Dynamic Loads. Reporr.No.
8622-0001-RU-000, Space Technology Laboratories, Inc., October 1961.
- 51. Anderson, D.L. and Lindberg, H.E. , " Dynamic Pulse Buckling of Cylindrical Shells under Transient Lateral Pressure," AIAA J. , 6:4, April 1968, pp. 589-598.
- 52. Yao, J.C., Dyna =ic Stability of Cylindrical Shells under Static and Periodic Axial and Radial Loads, Report No. IDR-169(3560-30)TR-1, DCAS-TDR-62-151, Aerospace Corp. , 27 July 1962.
- 53. Budiansky, B. and Hutchinson, J.W., "Dynacle Buckling of Inperfection-Sensitive Structures," Proc. XIth Int. Cong. Aeol. Mech., Munich, Germany, 1964, pp. 637-651.
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- 54. Budiansky, B. , " Dynamic Buckling of Elastic Structures: Criteria and Estimates," Dyna =1c Stability of Structures, Proc. Inc. Conf. at Northwestern University, October 18020, 1965, Pergamor Press, New York, 1966.
55.__Kaplan, A., " Buckling of Spherical Shells," in Thin-Shell Structures, Theory, Experiment, and Design, edited by Y.C. Fung and E.E. Sechler,
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Prentice-Eall, Inc. , Englewood Clif f s , N.J. , 1974, pp. 24 7-289,~~~ _~.-~
- 56. Ashwell, D., "On the Large Deflection and Buckling Behavior of a
_ ___ _ Spherical Shell with an Invard Point Load," Proc. Svap. on the Theory of Thin Elastic Shells, North-dolland Publishing Co., 1960, pp.._4_3-46. _
- 57. Bushnell, D. , " Bifurcation Phenomena in Spheracal Shells under Concentrated and Ring Loads," AIAA J.,
5:11, November 1967, pp. 2034-2040,
- 58. Sabir, A.B., "Large Deflection and Buckling Behavior of a Spherical Shell with Inward Point Load and Uniform External Pressure,"
J. Mech. Eng., Sci., 6:4, December 1964, pp. 394-404. _ _. . _ _ _ _ _59._ Ho, F.H., " Dynamic Instability of a Deep Spherical Shell," Proc. ASCE, 91:EM-3, June 1965, pp. 233-248.
- 60. Machnig, O. and Handge, P., Zur Stabilitat zylindrischer Berhalter_
mit Ralbkugelboden under Aussendruck, Stahlbau, Bd. 34, Heft 4, __ April 1965, pp. 119-123.
- 61. Singer, J. " Buckling of Integrally Stif fened Cylindrical Shells--
A Review of Expariment and Theory," in Contributionc to the Theory of Aircraft Structures. Rotterdam: Delft University Press, 1972, pp. 325-358.
- 62. Singer, J. and Rosen, A. " Design Criceria for Buckling and Vibration of Imperfect Stiffened Shells," ICA3 Paper No. 74-06, Frco., The Ninth Congress of the International Council of the Aeronautical Sciences, Haifa, Israel, Aug. 25-30, 1974.
_ __ __ 63. American Society of Mechanical Engineers, ASME Boiler and Pressure Vessel Code, ANSI /ASME BPV-III-1-A, Section III, Rules for Construc-tion of Nuclear Power Plant Components, Division 1--Appendixes, 1977 Edition, 1 July 1977, Appendix 7, pp. 224-225. 64 Tsai, J.C. and Kircik, W.J., Buckling Criteria and Application of Criteria to Preliminary Design of the Steel Constraint Shell for the Floating Nuclear Plant, Docket No. STN 50-437, Report No. 7270-RP-16A31, Offshore Power Systems, Florida, December 1975. 6-5
- 65. Sechler, E.E., "Ihe Historical Development of Shell Research and Design "
___ in Thin-Shell Structures , Theory, Exteri=ent, and Desien, edited by Y-C. Fung and E.E. Sechler, Prentice-Hall, Englewood Clif fs , N.J. ,1974, pp. 3-25.
- 66. Weingarten, V.I.; Mcrgan, E.J.; and Seide, P., " Elastic Stability of Thin-Walled Cylindrical and Conical Shells under Axial Compression,"
J. Amer. Inst. Aeronautics and Astronautics _, 3:3, March _196_Supp j 00-505.
. 67 Weingarten, V.I, and Seide, P., " Elastic Stability of Thin-Walled Cylindrical and Conical Shells under Combined External Pressure and Axial Compres ion," J. Amer. Inst. Aeronautt s and Astronautics, 3:5,
_ _ _ _ _ _ May 1965, pp. 913-920.
- 63. Weingarten, V.I.; Morgan, E.J.; and Seide, P., "Eastic Stability of Thin-Walled Cylindrical and Conical Shells under Co=bined Internal
~ ~ ~ ' -
Pressure and Axial Compression," J. Amer. Inst. Aeronautics and _ Astronautics, 3:6, June 1965, pp.1118-1125. _ _ _ _ . _.
- 69. Ishai, 0. ; Weller, T.; and Singer, J. , " Anisotropy of Mylar A Sheets,"
J. Materials, 3:2, 1968, pp. 337-351. . _ _ _ _ _ _ s 6-6
9 APPENDIX A COMPRESSIVE BUCKLING OF PRESSURIZED THIN CYLINDRICAL SHELLS WITH A PCIETRATION by C. Babcock California Institute of Technology A.1 INTRODUCTION _ Reactor contain=ent structures may include a relatively thin shell as part of the containment system. Th.is shell structure can be subjected to a variety of load' y; conditions in the event of abnormal reactor behavior. Some of .hese loading conditions can result in compressive stresses in the shell wall that can lead to buckling. The technical literature contains a vast amount of data on shells and buckling [63], much of which is directly applicable to the problem of reactor containment structures. There do exist, however, noticeable gaps in the problem areas addressed in previous work. The most notice-able of these include buckling of shells under nonuniform loading, geometry other than shells of revolution, and buckling of shells with discontinuities (cutouts, penetration, etc.). It is this last area that will be addressed in this experimental work. Reactor containment shells usually incorporate a cylindrical section. This section will generally be stiffened and also will contain a number of penetrations for pipes carrying steam, coolant, etc. These penetra-tions cause local changes in shell stiffness that will affect not only the local stresser but also the buckling of the shell wall if compressive stresses result from the overall shell loading. The buckling problem with penetrations (or cutouts) has been rather extensively studied for the case of a cylindrical shell with circular, elliptical, or rectangular holes. Most of these works [37, 39, 41, 43, 45, 47] are experimental. studies of the effect of cutouts (with or without local stiffening around the cutout). In the past few years, several numerical studies have also been made of the problem [40, 44, 46, 47,48]. These numerical studies have demonstrated that bifurcation and nonlinear collapse analyses can provide useful prediction tools and that considerable care must be exercised by the analyst in using these tools. The literature on this problem is completely silent on the subject of the effect of penetrations likely to be encountered in a containment shell. For this reason it was decided to conduct a limited experimental study of the problem, using an idealized structure and loading condi-tions. The main purpose of this study was to determine the difference between the effect of a cutout (hole) and a penetration (hard point), which represent the two extreme types of discontinuit'.es. A-1
A.2 EXPERIMENTAL PROGRX4 - - - - - - - . The experimental work was carried out using a shell model structure. The cylinder was made of Mylar (E.I. du Pont de Nemours and Co. Inc., Wilmington, Delaware). This material has been extensively used for model shell structures for buckling studies [39, 41, 43, 45], which treat the problem of cutouts as well as of uniform shells (64-66]. The material exhibits some anisotropic behavior (67], but buckling results are in good agreement with results using other materials for experimental models (37, 41,47}. Based on these results and the si=plicity of model. construction, Mylar was selected as the model material. The loading coisidered in this work is internal pressure and uniform axial compression. These loads represent i=portant load cases and are relatively si=ple to achieve experimentally. Results for nonuniform compression (including bending) can generally be inferred from the uniform axial load case, provided the variation around the circumference is not rapid. Nonuniform internal (external) pressure results can also be approximated from the uniform pressure case, but this procedure is somewhar more 11=1ted due to .the longer buckling-wave length expected with ext.ernal pressure. The scaling parameters for these types of buckling studies have been well established by analysis. The physical parameters of interest and their dimensions are listed below. Parameter Dimension L Shell Length 1 R Shell Radius I h Shell Thickness 1 E Young's Modulus f/1 P Pressure f/1 o Buckling Stress f/1 n Poisson's Ratio a Penetration Radius 1 The appropriate dimensionless parameters are: P I\ R Pressure, P = 7I A-2
( Load, 3- 3(1 - n ) Shell, f, The dimensions for a reactor containment shell fall in the following ranges: L = 1000 + 2000 in. (25.4 + 50.8 m) R = 500 + 1000 in. (12.9 + 25.4 m) h = 1 + 1 in. (0.0284 + 0.0254 m) This gives: f=250+1000, f=1+4 The shell model selected had the following dimensions and properties: L = 7.8 in. (0.2 m) l R = 4.0 in. (O.I m) h = 0.0096 in. (0.0002 m) E = 725,000 lb/in. (4998 MPa) n = 0.3 1 resulting in f=1.95, f=417 which lie near the middle of the range expected in these parameters. The parameter that characterizes the penetrrtion (or cutout) has been shown to be: A = a/ Rh For the reactor containment shell, the ran3e of "a" would be a = 5 + 25 in. (0.1 m + 0.6 m) This gives the following range ~for A: A = 0.08 + 1.12 A-3}}