ML20207Q087
| ML20207Q087 | |
| Person / Time | |
|---|---|
| Site: | Oyster Creek |
| Issue date: | 01/14/1987 |
| From: | Donohew J Office of Nuclear Reactor Regulation |
| To: | Office of Nuclear Reactor Regulation |
| References | |
| NUDOCS 8701230005 | |
| Download: ML20207Q087 (126) | |
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January 14, 1987 Docket No. 50-219 LICENSEES:
GPU Nuclear Corporation Jersey Central Power and Light Company FACILITY:
Dyster Creek Nuclear Generating Station
SUBJECT:
DECEMPER 19, 1986, THIRD MEETING WITH GPU NUCLEAR CORPORATION (GPUN) TO DISCUSS CORROSION OF THE OUTER SURFACE OF THE DRYWELL SHELL On Friday, December 19, 1986, a meeting was held at NRC, Bethesda, Maryland, with GPUN, the licensee, on the corrosion of the outer drywell shell, at an elevation just below the downcomers. This is the third meeting with the licensee on the subject. The summary for the first meeting held on December 1, 1986, was issued December 12, 1986. The summary for the second meeting held on December 10, 1986, was issued January 5, 1987. Figures 1 and 2 in show a vertical cross section of the entire drywell shell and of the region just below the downcomers, including a drainage channel and the sand filled cavity. Dyster Creek was at this time shutdown in the Cycle 11 Refueling (Cycle 11R) outage. is the list of the participants that attended the meeting.
i l contains three figures of the Oyster Creek drywell. Attachment 3 i
contains the handout from the licensee for its presentation. The handout is I
orranged in the order of the licensee's presentation. Attachment 4 contains the draft preliminary stress analysis for the sand cavity or transition zone.
The following is a sumary of the significant items discussed and the actions, c
if any, taken or proposed.
The licensee's agenda for this meeting is the first page of Attachment 3.
A description of the drywell is given in pages 2 to 6 of Attachment 3.
The licensee began its presentation with a discussion on its evaluation of the damage to the drywell shell. This is pages 7 to 11 of Attachment 3.
The data discussed was that presented in the two previous meetings and is summarized on page 7 of Attachment 3.
Each dot represents a measurement made in the shell by ultrasonic testing (UT) measurements. The lowest measurements in bays 5 and 15 are reflections from inclusions in the solid metal and are not indication of corrosion wastage of the shell.
The licensee removed seven cores from the drywell shell. The sample number in l
the left hand column of the table of the core sanple visual examination on l
pages 10 and 11 in Attachment 3 represent the sequence of the core cutting and l
the bay in which the core was removed. Sample 1-19C is the first cutting and was done in bay 19 and in location C with respect to the downcomer. The letters A to 0 are shown with respect to the downcomer in the figure on page 5 of.
The location of the bays by number witn respect to the 10 downcomers is given in the third figure in Attachment 2.
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,.. The second pa.rt of the meeting was a discussion on the mechanism for the damage to the shell. This is pages 12 to 51 of Attachment 3.
This included a presentation of studies made by General Electric (GE) for the licensee and a listing of the other consultants being used by the licensee.
The licensee stated that it has concluded from documentation on the construc-tion of the shell that the red lead paint put on the shell surface extended throughout the surface exposed to the sand cavity. This paint was for corrosion protection.
In the first meeting, it was reported that this paint only extended over the upper half of the shell surface exposed to the sand cavity. Most of the lead (Pb) reported in analyses of the corrosion product is from this paint.
The preliminary results of the licensee's microbiological or bacteriological studies of the effect of microbes in the sand on the corrosion of the steel shell are given in pages 27 to 30 of Attachment 3.
The results given for sulfate reducing bacteria (SRD) are based on only one week for cultures but the results are negative or only weakly positive. This is viewed as indicating that microbiological induced corrosion (MIC) is not a primary corrosion mechanism on the shell.
In its presentation, GE provided a table and two figures on reported corrosion rates for carbon steel under static air saturated conditions. This is page 44 and 45 of Attachynt 3.
The figures on page 45 show the corrosion of~ iron versus pH and Fe, respectively, of solutions of Ferric chloride at 20*C.
The figure on page 46 compares the corrosion for the Oyster Creek (OC) shell over 6 years and 17 years, respectively, in mils per year (mpy) to repnrted corrosion rates.
The reported corrosion rates are from the table on page 44.
The conclusions of the licensee are the following and are given on pages 49 to 51 of Attachment 3.
o Wastage of the drywell plate is the result of an aqueous corrosion process influenced by localized oxygen depletion.
o Although viable bacteria were identified in the sand and corrosion product, no substantive evidence exists as to its involvement in the corrosion process.
o The combination of using a D-Meter for ultrasonic thickness measurements and an "A" scan for qualitative assessment of the plate condition are adequate for engineering evaluations, o
Corrosion is limited to the steel in contact with the sand bed and is present to a significant amount only in bays 11, 13, 17 and 19 and only within drywell elevations 10' 3" to 11' 9".
o The areas of observed corrosion appear to be those areas in which the sand has remained significantly wetted.
o Corrosion rates have been conservatively set at 48 mpy although more typically it would be expected to be approximately 17 mpy.
The third part of the meeting was the licensee's presentation on its structural evaluation of the damaged shell.
This is pages 52 to 55 of
. ~.
. Attachment 3.. This is the structural criteria on page 52, the rodel of the damaged region of the shell on page 53 and the results of stress calculations based on the model on pages 54 and 55 of Attachment 3.
These were discussed in detail in the second meeting.
The licensee stated the drywell was designed for the dead weight (DW), for water flooded to the 75' elevation, for the design basis accident (DBA) conditions and for the operating basis earthquake (OBE) loads. This is the American Society of Mechanical Engineers (ASME) Code level A and B loads which is more conservative than level C.
Level C includes the safe shutdown earth-quake (SSE) loads but has higher allowables than level A and B.
The licensee stated the allowable stress criteria are local membrane stresses (PI) of 28,875 psi and membrane stress plus discontinuity stresses (Pl+Q) of 52,500 psi. The local membrane stress is the limiting stress.
The model of the Oyster Creek drywell shell is a spherical shell with a circumferential notch representing the corrosion wastage over the sand cavity. The stress intensities and meridianal stresses along the shell surface are shown in the figures on page 54 and 55 of Attachment 3.
The ASME Code allowables listed on page 52 are represented as vertical lines on the figures.
The conclusions of the licensee are the following and are given on page 56 of.
o Damage is limited to areas of shell plate in contact with the sand with the minimum mean thickness being 0.87".
o Structural analysis, based upon 0.70" plate thickness in the sand cavity region, is acceptable.
o Corrosion rate is estimated to be 15 mils / year but could be as high as 50 mils / year.
o Sufficient margin exists to justify operation for at least one additional cycle.
This ended the presentation by the licensee. The staff asked if the licensee has considered jet impingement loads on the wasted region of the shell. The licensee stated that most of the surface of the shell exposed to the sand cavity is covered by concrete.
It further stated that the thickness of the remaining steel should be sufficient to take the strain to allow the concrete outside the shell to take the jet impingement loads. The sand cavity is designed to retain the sand in the cavity and the sand should be under j
compressive load from the shell.
f The staff also requested that the licensee discuss the water remaining in the 4
trenches cut into the concrete floor of the drywell. These trenches were discussed in the second meeting. They were two trenches cut to expose the j
drywell shell below the floor but above the sand cavity. This was done to allow UT measurement of the shell and to remove two cores. Water was found to collect in the trenches. The licensee explained that this was reactor grade water which had probably permeated the concrete through cracks in the floor and was seeping from the concrete into the trenches. The licensee stated that t
the shell surface exposed by the trenches did not show corrosion.
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- The licensee submitted its safety evaluation on the drywell shell corrosion to the NRC staff in its letter dated December 18, 1986.
It concluded that the drywell meets the licensed requirements for structural integrity and that operation of Oyster Creek for the next operating cycle is safe.
The next operating cycle will be from the Cycle 11R outage to the Cycle 12R outage.
The Cycle 12R outage is expected to begin in April 1988.
The licensee concluded its letter with its commitment to maintain an intensive effort to continue the following:
o Eliminate the source of any future water incursions into the sand bed, o
Dry the moisture from the sand cushion and/or otherwise render corrosive attack minimal.
o Continue the metallurgical and chemical investigations to determine, if possible, the exact cause of the corrosion.
o Further assess longer term corrective actions that may be appropriate.
o Continue the UT shell thickness test program at future outages of opportunity including forced outages otherwise requiring drywell entry during the next cycle.
The staff stated that it would review the analyses made of the 0.7" notch in the model of the drywell shell, the corrosion rates, and the evaluation in the licensee's letter dated December 18, 1986. After discussion of the drywell analysis of the sand cavity or transition zone by Chicago Bridge & Iron, it was agreed that the licensee would explain in detail this analysis to the staff. This analysis, which should be considered a draft and preliminary at present, is in Attachment 4.
The figures on pages 53 to 55 in Attachn'ent 3 are from this analysis. The remaining meeting was a technical discussion on the details of this analysis.
The staff issued a Safety Evaluation on its review of the analysis of the notch, of the corrosion of the shell and of the licensee's safety evaluation on December 29, 1986, no J
Project Manager OWR Project Directorate #1 Division of BWR Licensing Attachments:
1.
List of Attendees 2.
Drywell Figures 3.
Licensee's Handout for Meeting 4.
Draft Drywell Analysis, Sand Transition Zone
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DISTRIBUTION: See attached page f 7/
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OFFICIAL RECORD COPY
Oyster Creek Nuclear Generating Station cc:
Mr. Ernest L. Blake, Jr.
Resident Inspector Shaw, Pittran, Potts and Trowbridge e/o U.S. NRC 230d N Street, NW Post Office Box 445 Washington, D.C.
20037 Forked River, New Jersey 08731 i
J.B. Liberman, Esquire Commissioner Bishop, Liberman, Cock, et al.
New Jersey Department of Energy 1155 Avenue of the Americas 101 Commerce Street New York, New York 10036 Newark, New Jersey 07102 Mr. David M. Scott, Acting Chief Regional Administrator, Region I Bureau of Nuclear Engineering U.S. Nuclear Regulatory Commission Department of Environmental Protection 631 Park Avenue CN 411 King of Prussia, Pennsylvania 19406 Trenton, New Jersey 08625 BWR Licensing Manager Mr. P. B. Fiedler GPU Nuclear Vice President & Director 100 Interpace Parkway Oyster Creek Nuclear Generating Parsippany, New Jersey 07054 Station P. O. Box 388 Deputy Attorney General Forked River, New Jersey 08731 State of New Jersey Department of Law and Public Safety 36 West State Street - CN 112 Trenton, New Jersey 08625 Mayor Lacey Township 818 West Lace) Road Forked River, New Jersey 08731 Licensing Manager Oyster Creek Nuclear Generating Station Mail Stop: Site Emergency Bldg.
P. O. Box 308 Forked River, New Jersey 08731 i
ATTACHMENT 1 THIRD MEETING WITH GPU NUCLEAR CORPORATION (GPUN)
TO DISCUSS DRYWELL SHELL CORROSION DECEMBER 19, 1986 NAME ORGANIZATION J. Donohew NRC/NRR/ DBL M. Laggart GPUN J. Kowalski GPUN B. Turovlin NRC/NRR/ DBL J. Strosnider NRC/RI R. Blough NRC/RI C. P. Tan NRC/NRR/ DBL 1
G. Lainas NRC/NRR/ DBL R. F. Wilson GPUN J. L. Sullivan GPUN D. L. Hill Niagara Mohawk Power Corp.
J. A. Zwolinski NRC/NRR/ DBL R. Greiner General Electric T. Ahl Chicago Bridge & Iron S. D. Leshnoff GPUN D. K. Croneberger GPUN G. R. Capodanno GPUN B. M. Gordon General Electric B. D. Liaw NRC/NRR/ DBL R. Hermann NRC/NRR/ DBL W. Cristali NJ DEP*
R. Green NJ DEP R. W. Houston NRC/NRR/ DBL
- State of New Jersey, Department of Environmental Protection.
ATTACHMENT 2 Three drywell figures O
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Core Samples Sample Bay /
No.
Location Type Elevation Samples Obtained 1
19C Wastage 11'-3 5/8" Core, sand, bacteriological 2
15A Pit /Inc1 11'-5 1/4" Core, sand; bacteriological 3
17D Wastage 11'-3 3/4" Core, sand 4
19A Wastage 11'-3 3/8" Core, sand, bacteriological 5
11A Wastage 11'-3" Core, sand, bacteriological 6
11A Minor wastage 12'-2 3/4" Core, sand 7
19A Minor wastage 12'-1" Core, sand
CORESWLEVISilALEXAMINATION SamleNo.
ConditionofSteel CorrosionProduct/ Sand 1-19C Generalwastage
. Adherent black scale on steel Non-uniformpenetration
. Scale with metallic lustre on sand
. Scale is dense, magnetic
. Sand was damp, compacted 2-15A Minorsurfacecorrosion
. Red rust on surface Stringerspresentinsteel
. No scale on sand
. Sand was dry, loose
. Space existed between sand and steel O
3-17D Generalcorrosion
. Adherent grey scale on steel Flatsurface
. Some rust spots on steel
. Dense black scale on sand ( 1/4' thick)
. Sand was damp, compacted 4-19A Generalcorrosion
. Adherent black scale.on steel Flatsurface
. Dense black scale on sand ( 1/4" thick)
. Sand was damp, compacted
1 CORE SAMPLE VISUAL DANINATION (CONT'OJ SamleNo.
ConditionofSteel CorrosionProduct/ Sand 5-llA Generalcorrosion
. Adherent black scale on steel flatsurface
. Dense black scale on sand
. Sand was damp, compacted 6-llAH Noevidenceofcorrosion
. Two rust-colored streaks on steel C
. No scale on sand
. Sand was slightly damp, somewhat compacted
. Small gap between sand and steel at top 7-19AH Hinimalcorrosion
. Some black loose deposit on steel L
Flatsurface
. No scale on sand
. Sand was slightly damp, loose
. Smil gap between sand and steel at top
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ENERGY DISPERSION ANALYSES FROM THE SURFACE OF THE PLUG SATLES ELEMENTAL COMPOSITION
- SAMPLE MAJOR MINOR TRACE 15A Fe Pb A1,Si,Co,Mn 19C Fe Cl Al,Si,Mn
- MAJOR 10 wt. %
MINOR 1 wt, %
TRACE 1 wt. %
ELEMENTS BELOW ATOMIC NUMBER OF 11 ARE NOT DETECTABLE.
/
i EERGY DISPERSION ANALYSIS OF THE FLAKE DEPOSIT FROM i
THE 19C PLUG SAMPLE ELEMENTAL COMPOSITION
- MAJOR Fe MINOR TRACE Si,Cl pH DETERMINED BY LITMUS PAPER 4
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MATERIALSLABDATASURRY e
PLUG 15A
-N0SUBSTANTIALWALLLOSSWADOCCURRED.
-THEPROTECTIVEPAINTLAYERISN0LONGERINTACT.
-THEUTINDICATIONWASIDENTIFIEDASAl-0INCLUSIONCLUSTERS
-ANIRONOXIDESCALEWASPRESENTON.TWEWALLSURFACE.
-0XYGENPITTING(
0.005" Deep)ISPRESENTINTHEOUTSIDE WALL 0FTHEDRYWELL.
-THESANDINCONTACTWITHTHEWALLSHOWSTRACEAMOUNTS OFCHLORINEPRESENT.
22
6.
s..
PLUGlSAUTINDICATION ANALYSIS INDICATIONWASFOUNDTOBECLUSTERSOFAl-0 GLOBULAR INCLUSIONSORIENTEDINTHEROLLINGDIRECTIONOFTHE PLATESTEEL.
THEINCLUSIONCLUSTERSWERELOCATED0.490"FROMTHE OUISIDEWALLOfTHELINERANDWEREDISPERSEDINA BAND 0.208" WIDE.
23
. ~..
Drywell Drain Line Water Analysis Sample I Sample II Parameter (ppm)
(ppm)
Na 145 96 K
142 85 Ca 7.5 6.4 Mg 30 11 Al
.33
.02 Ni
<.01
<.02 Fe
<.01
.74 Cr
(.01
<.02 Mn
(.01
.02 Pb
.06
<.02 NH (N) 3.6 3
1 C1 32.5 25 NO:
8.7 6
S0.
153 60 P0.
5 N.D.
F
<1 TOC 51 23.3 Organic Acid
.1 Total Sulfur 153 Conductivity 1100 us/cm 814 pH 8.9 8.7 Alkalinity (HCO3) 130 bh; Samples taken 12/86
F Sand teachate Analysis I
Sand Leachate Sand Leachate Sand Leachate Sand Leachate Firebar-D* Leachate Bay 11 Drain Bay 11 Drain Plug #1 (19C)
Plug #2 (15A)
Analytical I Hr. 60* C 24 Hrs. Room Temp 1 Hr. 90* C 1 Hr. 60* C 1 Hr. 60* C e
Parameters fua/a1 fua/a) fua/a) fua/a1 iua/a1 Na 777 25 25 37 47 K
784 25 20 37 23 Ca 176 30 25 47
< 23 Hg 1936 30 10 10
< 23 Al 0.3 0.5 1.5 39 2.3 7
Ni
< 0.3 0.5 0.5
.33 2.3
}
Fe 0.3 5.0 1.0 82 8.4 Cr 0.3 0.5 0.5
.33 2.3 l
Hn 0.3
- 0. 5' 0.5 3.7 2.3 j
Pb 0.6 1.5 0.5
.33 2.3 i
C1 573 10.5 6.5 45 93 f
NO3 132 2.5 1.5
< 17 6
S04 2850
< 25 32 2a 79 PO4 N.D.
N.D.
N.D.
N.D.
N.D.
F 14 N.D.
N.D.
i i
TDC 1056 39 37 46.6 N.D.
Organic Acids
< 20
< 5
< 5 Tatal Sulfur 2850 B
Conductivity 588 pH 8.46 7.43 7.58 7.02 5.99 Firebar-D is a composite of foam, fibers and concrete
CONCLUSIONS
' MAJORITY 0FCONTAMINANTSARELEACHEDFROMINSULATION
'MAJORLEACHABLECONTAMINANTSARE:
CATIONS ANIONS TOC S0DIUM CHLORIDE P0TASSIUM NITRATE CALCIUM SULFATE MAGNESIUM
' CORR 0SIONPRODUCTSWHENEXPOSEDTOWATERWILLFORMAN ACIDICENVIRONMENT
'HIGHB0RONINDRYWELLBAYDEPOSITSINDICATETHEWATER SOURCE MAY BE THE FUEL P00L O
^-
^-
.^
i Microbiogical Analysis Purpose To determine if microorganisms are involved in the corrosion process occurring in the Oyster Creek Drywell.
Criteria Used Include:
i 1.
Presence of viable cells in or around corroded areas 2.
Presence of conditions at the site that are believed to promote microbiological 1y influenced corrosion (MIC) 3.
Presence of specific types of microbes that have been associated with corrosion
?2'1.
Bacteriological Studies Preliminary Results Sample No.
Tyge Cell Count
- SRB 2-15A Sand (dry) lx10' cells /gm negative Adjacent to Drywell 71% viable 1-19C Corrosion Product 5x10' cells /gm weak pos.
Adjacent to Drywell 50% viable 6-11A Sand (moist) 4x10' cells /gm weak pos.
Away from Drywell 74% viable 4-19A Corrosion Product 6x10' cells /gm negative Adjacent to Sand 40% viable Stained with fluorescein isothiocyanate r
,/
RESULTS SAMPLE FITC SPC SRB
- DESCRIPTION (DATE)
CELLS /q (cfu/g)
(% viable) 7 4
15A 1x10 2x10 negative short rods, (12-6-06)
(71) few filaments 6
4 19C 5x10 3x10 weak +
(12-6-86)
(50) corrosion material visible 7
5 11A 4x10 1x10 g,,,
+
(12-7-86)
(74) similar to 15A 19A 6x106 6
4x10 negttive short cells (12-7-86)
(40)
~
~
attached to i
corrosion strands 1
- tentative l
' ~
.~.:.
~....
Preliminary Microbiological Conclusions o
Viable bacteria are present in the sand and corrosion product o
Some bacteria cells have been observed in contact with corrosion product 4
o Filamentous bacteria have been observed i
o Corrosion products do not suggest microbiological influenced corrosion o
Current evaluation of cultures show either negative or weak presence of sulfate reducing bacteria 3D
k GPU NUCLEAR DRYWELL CORROSION REVIEW BARRY M. GORDON PRINCIPAL ENGINEER-CORROSION PERFORMANCE CHAIRMAN OF NACE COMMITTEE T-2A-NUCLEAR SYSTEMS GENERAL ELECTRIC 804)d5 3 l ~3 A r
\\
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- n. 7.6 -- &2V R_,00 Co
- _D, ao - I, 30 0.R o l
_(~
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__o,00L DA3 l D. Lf I
l Fe t/h_3 2 - 9& 82 7.8. V7 l
8
(,_af 0, 72 I
Gr*
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p.co-o, /4 Ca e.co-o.o3 c.iY-o.48 W
/,J6- /r 93
/r 17-2,Y8 o.oo - o //
fe 12.73 - 1% 40 94 0 7 - 9 7. V S R/o 72 -28 87 4/--St. 75 Pb J
Br o.co - o.3 r o.00- o,ic i.n -i37 C4 t
1 Ti o,0 Y - 0 0 7 o 03 -oro) l Cr
- o. of - 0,0 9
- c. oS - o,to O
PLUG #6 WITH NO CORROSION l
HAS RED LEAD INTACT 0
HIGH S CONTENT h
REFRAI Elfr,TRIC CO.
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77pical 6orce,,fd,krs (W/h)
E/ame N"7 #3 A(I+0) Nay #Y(tn A.) Nuf W4 (lIh-H)
Al S
0,08 0,30
,S 0,23
+
d 3 Vf t-0,/V
- 2. S 7 K
- o. o2 Cs 0.02
/%
/, 63
/i2f oaoi Fe 9V.9'O 18,37 0,44 Pb 77.64 t-D. / f-
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6 0
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CROSS SF.CTION ANALYSIS SIMILAR TO SORFA'C'ES
EEhtRAL ELECTRIC CO.
Nuc!zr Erwrgy Business Group
- NG!ME5RlHG CN.CULATION SHEET NUMBER DATE
/ 2 ~ / 2 ~ E I' SUBJECT By-[ Mar. er SHEET OF Ovder Geek Piq Crusi Snph s n' me,4/
Cu, gan7c F/49 3 I i+ DI Play f lie)
AA o, co - /,03 SE 0.00 - 0,43 3,r/ - ao,3 Y
,5 0.07 - /,72
&I 0,02 - 0.38 0.33 - /,73 K
0,oo-o, o ?
o.oo - o.sc Ca c.co - c.//
o.co - e.44 Mn
/,sf-to,92 o, co - t,So fe 88.32-18.24 6 % A7-13,36
[d or o,0 0 - o. 2 3 or 00 - 0, S2 0,00
,1,18 0
CRUST SPECIMENS CONSIST OF IRON OXIDES AND SAND
__ _ scoa. sy impt _ _.
~
X-RAY DIFFRACTION RESULTS FOR CRUST SAMPLES PLUG #3 (17D)
MAJOR PHASE M 0 34 MINOR PHASE SIO2 M=
PREDOMINANTELY FE WITH OTHER TRACE ELEMENTS PB (RED LEAD PB 0 ), NI, CR (SPINEL TYPE, FCC) 34 PLUG #4 (19A)
MAJOR PHASE S102 TRACE PHASE M034 PLUG #3 AND #4 (MAGNETICALLY SEPARATED)
MAJOR PHASE 90% XFE 0 24 TRACE PHASE
<5% -FE 0 23 TRACE PHASE S0 y2 POSSIBLE (LOW PROBABILITY)
Y FE 023 X = MN OR FE PREDOMINATE PHASES PRESENT IN CRUSTSkMPLESAREFE3 04 AND S Oi2 3'?
MICR0 HARDNESS EVALUATION PLUG #3 (0.905" THICK)(17D)
KN0OP ROCKWELL "B" 168 81 175 84 170 82 169 81 PLUG #4 (0.900" THICK) (19A)
KN00P ROCKWELL "B" 167 81 168 81 165 80 165 80 PLUG #6 (1,150" THICK)(11A)
KNOOP ROCKWELL "B" 169 81 165 80 166 80 165 80 o
TYPICAL HARDNESS VALUES FOR ASTM A212 32
CORROSION REACTIONS OF IRON / STEEL IN OXYGENATED WATER BASICkN0DEREACTION FE + FE *+2E BASIC CATHODE REACTION 0 +2H 0+4E
+ 40H-2 2
RUST FORMATION IN PRESENCE OF OXYGEN:
4 FE+30 +2H O + 2FE 0,H 0 HEMATITE 2
2 23 2
SUPPLY OF OXYGEN RESTRICTED:
4FE+20 +4H 0 + 4FE(OH)2 FERROUS HYDR 0XIDE 2
2 3FE(OH)2 + FE 0 +2H 0+H2 MAGNETITE FORMATION 3g 2
IN THE PRESENCE OF A SALT MCL (NACL,MGCL,ETC) 2 FE ++2CL
+ FECL AT AN0DE 7
HIGHLY SOLUBLE M*0H
+ MOH AT CATHODE FECL +2M0H + FE(0H)2+2MCL REACTS AWAY FROM SURFACE 2
4FE(0H)2+02 + 2FE 0.H 0+2H 0 WITH 0 23 2
2 2
3FE(OH)2 + FE 0 +2H 0+H
- 3g 2
2 WITHOUT 02 NON PROTECTIVE O
DRYWELL ENVIRONMENT CONDUCIVE TO FORMATION OF NON PROTECTIVE OXIDES 30l
CORROSION PRODUCTS OF IRON / STEEL FE0 NH 0 FE 0 'NH 0 FE 0 2
34 2
FE 3FE(OH)2
)
FE 0 +2H 0+H 3g 2
2 02 FE(0H)3 FE(0H)2+1/2 H 0+1/402 2
HYDROUS FERROUS HYDROUS FERROUS HYDROUS FERRIC 0XIDE FERRITE OXIDE FERROUS HYDR 0XIDE MAGNETITE FERRIC HYDROXIDE PH 9.5 PH 7.0 PH 7.0 WHITE (PURE)
BLACK ORANGE / BROWN RED (RUST)
~
GREEN MAGNETIC NON MAGNETIC GREENISH / BLACK
-FE 0 2 3 HEMATITE MAGNETIC YFE 0 73 4
40 m
D F
- "-5"
KEY FACTORS FOR CORROSION AT DRYWELL/ SAND INTERFACE P,0ROSITY (AERATION)
AREAS CLOSE TO DRAIN AREAS CLOSE TO INSULATION GAPS RAND 0M AIR P0CKETS DUE TO BACKFILLING DIFFERENTIAL AERATION CONDUCTIVITY DISSOLVED SALTS CHLORIDE MARINE ATMOSPHERE INITIAL WETTING 0F SAND FIREBAR D M0ISTURE INITIAL WETTING OF SAND LEAKING BELLOWS ACIDITY / ALKALINITY BACTERIA ANAEROBIC AEROBIC OTHER CONTAMINANTS SULFATE CARBON DIOXIDE
[l
POSSIBLE MECHANISM 0F OYSTER CPEEK CONTAINMENT.C0RROSION BACKFILLING OF SAND CREATES AIR P0CKETS OF SAND CUSHION INITIAL WETTING 0F SAND ESTABLISHES ELECTROLYTE MARINE ENVIRONMENT / SAND QUALITY / WETTING SOLUTION ENHANCES ELECTROLYTE BY CL CONTAMINATION BIOLOGICAL ACTIVITY EXISTS BUT THE INFLUENCE OF THE BACTERIA IS UNDEFINED AND APPEARS NOT TO BE A MAJOR CONTRIBUTOR THROUGH ANY CURRENTLY PUBLISHED MECHANISM CORROSION OF STEEL INITI ATES - PB 0 PROVIDES INITIAL 3q PROTECTION TO
$50% OF SAND CUSHION CONTACT AREA PROTECTION DECREASES WITH TIME - BREAKDOWN ACCELERATED BY CO AND/0R SO -
2 q
AREAS WITH 0XYGEN ACCESS ADJACENT TO INSULATION GAP AND DRAIN BECOME LOCAL CATHODES AREAS ADJACENT TO CONCRETE PROVIDED SOME PROTECTION DUE TO LOCAL ALKALINITY CONDENSATION AND LEAKS FROM FUEL P00L BELLOWS CONTRIBUTE AIR SATURATED WATER-PERHAPS LEACHES CL FROM FIREBAR D -
MAINTAIN / ENHANCES ELECTROLYTE SOME AREAS SEE ALTERNATE WETTING AND DRYWING DURING STARTUP/ SHUTDOWN CYCLE CONCENTRATES CL-AT METAL SURFACE SAND MAINTAINS CORROSION PRODUCTS CLOSE TO METAL SURFACE
- REDUCES FURTHER CORROSION CORROSION PROCEEDS DURING
" WETTING" PERIODS (CONDENSATION, LEAKS) OR ON A CONTINUOUS BASIS N
w
PoS61Me Corrosion Met havusm 5f fg O
Oysb Geek DryweJI gg3" g
.c Core %plec.
Red Lead Prnne
- Pa os,
'oc ah ons s
c.no becak do~n -,4 Cot
[
0 _ rec h 2
" / 11
>< Pb 0 F,be g hs 3, Pb 0s + M.0 + H 0 + %.0 3
1 2
2 3
II 1.PbCO3 PblOH)2 We.t ow pocx 4 (o )
e z
Sand 0
deeeAed l
1 hpe< cones,w -
+
p, g p, u ele-10 3
Fe.+,1c l _3 f, C 1, 2
3 Io' Fe.C.lz el.r1oH 4 lfeloH)u+ 1NC 3 0 _ rau h 2
- lowe, In 0_c'* a< ens'.
2 ox,3_i4,_o q, gggg_
Ny he+ stoti
- 4 NoH 4 Fely)t+03_g fep g gg L '0 dep e4e4 a. veas:
S'is%"
i
+
2 3 Fe (09) 2. A l~e g0 +1R O+H.1 9
2 i
Erh At watir..i.#
t t
due to conc. rete O Co z
y corc.ve4e (OnGVt48 D fRs h 45
A A A m t.A CORROSION RATES OF CARBON STEEL UNDER STATIC-AIR SATURATED CONDITIONS WATER TEMP EXPOSURE CORROSION TYPE
- C(*F)
M PERIOD, DAYS RATE, MPY REF DISTILLED +
40(104) 4.5-8 23.8 UHLIG/
NA0H/ HCL WHITMAN 22(72) 4.1-9.5 16.6 UHLIG/
WHITMAN PARTIAL DEMIN 52(125) 62 2.1 BRUSH (0.1/3.6uS/cM) 145 5.6 BRUSH NA 40(104) 12 SPELLER 60(140) 14 SPELLER DISTILLED 25(77) 5.4-6.5 100 1.4 MERCER 40(104) 5.4-7.0 100 2.9 MERCER 60(140) 5.4-8.0 100 6.6 MERCER TAP 40(104) 7.2-7.7 90 53.5 KH0MITCH 50(122) 7.2-7.7 90 54.0 KH0MITCH 60(140) 7.2-7.7 90 61.7 KH0MITCH-CONDENSATE 25(77) 35 8
BREDEN 4
FEEDWATER 45(113) 20 N0E CONDENSATE 45(113) 7.5-11 365-1095 20 WAGNER CONDENSATE 45(113
>6 15 OBRECHT SEAWATER 25(77) s7 3.5 HUDSON SEAWATER 50(122) s7
>50 NACE S0ll MIX 15(59) s7 1456 s]8 UHLIG kl
E F R ~I-W P - l' ~+ +
D e c I4'61 200 x
x k
I 150-
.6 3
e
{ 100,
,4 1
50 2
2 o
O O
m O
2 3
4 5
6 pH serie 1: iron in contact with magnetite serie 2: no contact with magnetite Figure 3-1. Corrosion rate of iron versus pH of solutions of FeCl, at 20'C i
2 200. _ _
w l
}
l E
150 6
d 5
f100
.4 0
1 o
50 2
2 0
0 a
0 1
2 3
4
( FeC1 I M/I 2
serie 1: iron in contact with ragnetite serie 2: no contact with magnetite Figure 3-2. Corrosion rate of iron versus (Fe++) of solutions of FeC1 at 20*C 2
l l
0 IlAGNETITE CAN ACT AS AN OXIDIZER FOR IRON IN CONTACT WITH FECL2 3-3
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CORROSION FORECAST AND RECOMMENDATIONS CORROSION INITIATED DURING CONSTRUCTION AND HAS CONTINUED DURING SUBSEQUENT SHUTDOWNS AND/0R OPERATION i
ESTIMATED CORROSION RATE ~20 MPY MITIGATION POSSIBILITIES DESICCATION CATHODIC PROTECTION BIOCIDE INHIBITORS MONITORING CONTINU0US UT CORROSION METER IN-SERVICE INSPECTIONS ADDITIONAL CORE SAMPLING yr
Consultant's Meeting
Purpose:
o Review available data on core sample analysis, chemical analysis, microbiological analysis and discuss if the data supports the conclusions drawn by GPUN.
If not, develop a defendable set of conclusions.
o Discuss potential methods for inhibiting future corrosion.
Outside
Participants:
o S. Pednekar, Ph.D.
Research Scientist Battelle Columbus L. Nelson Project Manager EPRI A. Harder, Ph.D.
Energy Research Center Lehigh University B. Gordon Principal Engineer General Electric J. Johnson Aquatic Biologist TVA C. Mathur, Ph.D.
Microbiologist York College S. Dexter, Ph.D.
Marine Biologist Univ. of Delaware Chairman, NACE Comm. on MIC
Meeting Results Key Points o
The GPUN core sampling plan was adequate to define the corrosion mechanism and a defendable set of conclusions were drawn.
o Corrosion rate should be relatively stable at this time due to the buildup of a thick oxide scale.
Rates of 10-20 mpy are reasonable; however, they may have been higher initially, o
In the experiences of the Biologists present MIC generally results in a deep pitting type attack rather than the general wastage observed.
o MIC of carbon steel has generally resulted in iron sulfide formation not simply Fe30. as observed at Oyster Creek.
o The annulus represents a relatively closed environment which would not provide a continuous source of nutrients for biological activity.
o Drying of the sand or cathodic protection would be two effective means for mitigating corrosion, o
Corrosion observed can be explained by classical aqueous corrosion mechanisms assuming chloride contamination and oxygen depletion.
o The Firebar-D appears to be the source of chloride contamination, k
p 1, Conclusion Summary 1.-
Hastage of the drywell plate is the result of an aqueous general corrosion process influenced by localized oxygen depletion, the degree t'o which moisture is present, temperature and chloride contamination.
2.
Although viable bacteria were identified in the sand and corrosion product, no substantive evidence exists as to its involvement in the corrosion process, at least in terms of currently publicized mechanisms. However, because of the variable nature of microbial induced corrosion any attempts at mitigating corrosion should consider this mechanism.
3.
0-Meter thickness readings, which initially were thought to be either pitting and later characterized by "A" scan UT as inclusions, were confirmed by metallography to be aluminide inclusions in the carbon steel.
4.
The combination of using a D-Meter for ultrasonic thickness measurements and an "A" scan for qualitative assessment of the plate condition are adequate for engineering evaluations.
5.
Corrosion is limited to the steel in contact with the sand bed and is present to a significant amount (i.e.,.25" -
.35") only in bays 11, 13, 17 and 19 and only within elevations 10' 3" to 11' 9".
As
6.
The areas of observed corrosion appear to be those areas in which the sand has remained significantly wetted.
This wetting most likely occurred during initial construction and then periodically during refueling outages as a result of leakage from the drywell bellows. Documented evidence of such leakage exists since.1980.
7.
Corrosion rates have been conservatively set at 48 mpy although more typically, through review of industry experience and corrosion literature, would be expected to be approximately 17 mpy.
SIRUC111RAL CRITERIA
~
SUSIAINEDLOADSFORWHICHCONIAIhENTISDESIGNED (ASE LEVEL A & B)
'DW0FSIEEL&APPURIENANCES
' FLOODED 1075'ELEV.
'DBA'S 35PSIG,281'F 62PSIG,175'F
'0BE/SSE All0WABLESTRESSCRITERIA ASE SECT VIII, '62 ED, & CODE CASES (1272N-5)
ASE SECT III, DIV 1, SUBSECTION NE P
LOCAL EMBRANE STRESS 1,
1 5x1.1xS
=28875 psi (SECT.VIIIANDSECI.III) 1 P + Q MEMBRAE SIRESS PLUS DISCONTINUIIY STRESSES 13S
=52500(SECI.VIII)OR57750(SECIIONIII)
NOTE: DISCONTINUITYSIRESSESARESECONDARYSTRESSES.
EXCEPIIONS STRESSINTENSITIESUSEDTHROUGH-0UI 1.0RIEXCEEDED(<2X) fi2.
DEC 89 '96 11:39 650 CBI CaxtRQ0K P.92
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=
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Afg/)41 uneav eneav noeav cwe av Aa.lyk -Aedeoed 7tnanea a
enr 1 o*y A
DATE DATE DATE
F DESIGN. BASIS.
n o
-s A - Membran6 Stress Intensity 0 - Surface Stress Intensity (Inside) n
+
. Surface Stress Intensity (Outside) k o
9 O
I r.
J o
1 2
4 d
4 4
g k
N l
p g
^
i 4
i p
D N,
t 1
o R
3 4
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R n$
b
~
'r elev. It '-1 a
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/-
=
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so 7
4 4
I w
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9 S
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/* r Jd " gas lea 944 STRESS INTENSITIES ALONG MERIDIAN OTSTER CREEK EMBEDMENT. CASE 1: P=3S TIMAX)=281 THICK =0.70 MAXIMA
- 32147, 2S2St.
29914.
vwasr
-es er
- =o.,
e u......
0 f.nban Cr-eee ENbeelmee&
J32 1:lA NGH41 nata ystfr4 Yoih
.e=o A ae/yr e - 4 c d << / 7' e = 4 e e 4
/.i
- m
NOT DESIGN BASIS n
.4,- Membrane Stress Intensity g
3 - Surtaos Stress Intensity (Inside)
+
. Surface Stress Intensity (Outside)
N o
h 3
o M
G k
k h
o 5
3 al k
s j
O yn 1
k 2
{
.1 h
Y,3 i
.s
~
y
(
k N
.eier. lt!.3 f
4 e
g 4,
~
O
..n 4
ib W
elev &LMR
-I, I
a 4
4 a
s s
$< ele I* s to Wil
'O ** s ja - J4 " ges le*944 MERIDIANAL STRESS ALONG MERIDIAN OTSTER CREEK EMBEDMENT. CASE 2: P=62 TIMAX)=175 THICK =0.70 W/0 SAND MAXIMA S3897.
16944.
28734.
Cf.nhe Cr-see Ekbeefmea&
.).sg
'QA NGH41 KL m, -
thl)$1 $ %_
ony_ 2_fte__ ____ __ _________
. c....-
CONCLUSION 1.
DAMAGEISLIMITEDTOAREASOFPLATEINCONTACTWITHSAND.
MINIMUMMEANTHICKNESSWITHIN60"LENGTHIS0.87" 2.
STRUCTURALANALYSIS,BASEDUPON0.70"PLATETHICKNESSIN SANDP0CKETREGION,ISACCEPTABLE.
3.
CORROSIONRATEISESTIMATEDTOBE15 MILS /YEARBUTCOULDBE ASHIGHAS50 MILS / YEAR.
4.
SUFFICIENTMARGINEXISTSTOJUSTIFYOPERATIONFORATLEASTONE ADDITIONALCYCLE.
L 4
4 l
.~..s...
U
[
1 6b 4d 9
M eting)
~
DRYWELL ANALYSIS
- ll-SAND TRANSITION ZONE OYSTER CREEK CONTAINMENT VESSEL GPU NUCLEAR CORPORATION PARSIPPANY, NEW JERSEY t
l I
E l
l l
CBI Services, Inc.
December 15, 1986 1
Contract 861172 l
p
== = N e g Qme -r * -
p-e.ev
' r' e "
-'-----ev r--,
o Table of Contents Descriction Pane No.
'Rev.
Introduction 1.0 0
Applicable Codes 1.1 0
Allowable Stresses 1.2 0
Input Loading Description 1.3 0
l Table of Input Loads 1.4 0
.r.-
Mathematical Model of Reduced Thickness Zone 1.5 0
Description of Kalnins Output 1.6 0
Case 1 - Circumferential Stres Plot 1.7 0
Case 1 - Meridianal-Stress Plot 1.8
'O Case 1 - Stress Intensity Plot 1.9 0
Case 2 - Circumferential Stress Plot 1.10 0
Case 2 - Meridianal Stress Plot 1.11 0
Case 2 - Stress Intensity Plot 1.12 0
Analysis of Embedment Zone with Sand Considered to be Ineffective 2.0 0
Case 1 - Circumferential Stress Plot 2.1 0
Case 1 - Meridianal Stess Plot 2.2 0
1 Case 1 - Stress Intensity Plot 2.3 0
Case 2 - Circumferential Stress Plot 2.4 0
Case 2 - Meridianal Stress Plot 2.5 0
Case 2 - Stress Intensity Plot 2.6 0
Conclusions 3.0 0
Appendix A - Kalnins Program Description Al-A14 Appendix B - Original Stress Summaries 1Al-1A4, 1B1-1B4 Appendix C - Stress Printouts C1-C12
, i.
.;;.. : : ~...
Introduction The Oyster Creek Nuclear Power Plant Mark 1 Steel Containment Vessel was designed, fabricated and erected by Chicago Bridge and Iron Company in 1965.
The configuration of the drywell portion is shown on page 1A1 of the attached Appendix B.
The lower spherical portion of the drywell is embedded in concrete at elevation 8'-11 1/4. A sand pocket extends from the point of complete emb.edment upward 3 '.-3 3 /4 to an elevation of 12'-3.
This sand pock *et performs two major functions:
I ~h a)
Provides a transition from the completely embedded portion of the spherical shell to an unconfined portion.
The sand
" springs" help to ease this l
transition.
b)
Provides a suitable means to dissipate the thermal gradient in the meridignal direction.
A recent inspection of the steel shell in the sand pocket region revealed that some degradation of the steel shell had taken j
place at some time during the twenty plus years since completion j
of construction.
Preliminary information indicates that the steel shell may have been reduced from the original 1.154" to as
- I little as.80.90" in thickness.
This report is an assessment of the' stress levels which will exist if the shell is assumed to be reduced to.70 inches around the entire periphery in the sand pocket region. The analysis is f.
performed for the following two cases:
a)
Pressure = 35psig Temperature = 281"F b)
Pressure = 62psis Temperature = 175 F Other normal dead loads and earthquake loads for the operating basis accident are included.
SUBJECT Oy.fder 6 esg [mbedm.,e W"OFFICEM Ns tl41 REFERENCE NO.
MADEBY CHKD BY MADE BY CHKD BY Anel sw-Asdesed 7/uemrsst y
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Aeolicable Codes The Oyster Creek Nuclear Plant Mark 1 Containment. vessel was designed, fabricated and erected in accordance with the 1962 Edition of ASME Code,Section VIII and Code Cases 1270N-5, 1271N and 1272N-5.
The allowable stresses used in this reduced thickness analysis are consistent with the original code of record. Some symbols and clarification have been extracted from the 1986 ASME Section III, Subsection NE Code. The use of these references in no way changes the allowable stress levels intended for the original design. The references used merely reflect the current day interpretations of the stress state and tend to be more consistent with todays analytical tools.
Speci'fic r'efeiences to ASME III, 1986 Edition are:
t-1.
NE - 3221.2 2.
Table NE - 3217-1 3.
NE 3213.10 4.
NE 3221.4 F
l.
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SUBJECT OFFICE REFERENCE NO.
Oys4nr-Creek Emjadmeet O cbu R
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DATE DATE DATE DATE l
WIL/mL N%
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I Allowable Stresses
- Primary Stresses (does not include thermal effects)
Allowable Stresses Ref.
General Membrane 1.1 x 17500 = 19250 psi 1272N-5 5(g)(1)
Local Membrane **
1.5 x 1.1 x 17500 = 28875 pai
'NE-3221.2 Local Membrane 1.5 x 1.1 x 17500 = 28875 psi 1272N-5
- bending,
+
5(g)(2)
Surface Stress ***
3.0 x 17500 = 52500 Table
~$
NE-3217-1 Secondary Stresses (includes thermal effects)
Surface Stresses (P +P +Q) = 3.0x17500 =52500 psi 1272N-5 y
b 5.(f) and NE-3212.4 all actual stresses are either stress intensities per NE 3000 or unidirectional stresses per ASME VIII, and Code Case 1272N-5, whichever is greater a local primary membrane stress is defined as one which does not exceed 1.1.x1.1x17500 =
21175 pai for a distance greater than 1.0,f"IT" Ref. NE 3213.10 if bending moment at the edge is required to maintain the bending stress in the middle to acceptable limits. the edge l;
moment is classified as P.
Otherwise it is clannified as b
Q.
Note:
For urinary stress evaluation - lo' ads include (1)
Internal pressure (2) Dead weight of Steel (3) Dead weight of appurtenances (4) 11% Horizontal Earthquake -
OBE equivalent (5) 5% Vertical Earthquake - OBE Equivalent essentially service Level A in 1983 ASME Code For Secondary Stress Evaluation - loads include all of above plus meridicinal thermal gradient.
SUBJECT OFFICE REFERENCE NO.
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,7 j
Innut Loadina Information The spherical portion of the containment vessel is assumed to be completely embedded at elevation 8'-11 1/4 (point G as shown on i,
sheet 1A1 of Appendix B). This analysis of the sand pocket zone includes a segment of the spherical shell extending up to elevation 23'-6 7/8 (point F as shown on sheet 1A1 in'. Appendix B)
The boundary. conditions at. point F are taken from the tables shown.on sheets 1B1 thru 1B4 as shown in Appendix B.
This consists of the S
values as described below:
(S is the resultant load in t$e meridianal direction in pounds p$r inch)
~
The table on the following page is a compilation of these input loads.
Note:
The earthquake stresses shown on pages 1B1 through 1B4 were originally calculated for a 22% horizontal l
earthquake and a 10% vertical earthquake. This has been assumed to be the equivalent of today's description of a Safe Shutdown Earthquake. Since the 4
1986 Code would permit higher allowable stresses for the SSE included earthquake, the SSE earthquake
~
loads shown have been divided by 2; i.e. 11%-
i horizontal and 5% vertical earthquake to simulate an
[
equivalent Operating Basis Earthquake. These levels are compatible with todays description of the Operating Basis Earthquake. The allowable stresses for the loads in which the OBE is included are lower than those which include the SSE. An assessment of L
I both earthquakes with their respective allowables
[,
indicates that the more critical is the OBE case.
I t
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The CBI Model used in the analysi's of the reduced thickness Oyster Creek embedment zone is described in the figure shown below. Complete fixity is assumed at elevation 8'-11 1/4.
Sand springs are modeled as inward radial forces, the. magnitude of which are dependent upon the magnitude of the shell displacements. The sand spring constant is 274.3 pa'i/ inch of radial displacement.
The attenuation of the thermal gradient in the meridional direction is assumed to be completed within the sand,/281"F emb dment t
zone, that is, tge temperature distribution is 175 at elev. 12'3 and 60 F at elev. 8'-11 1/4.
The embedment, zone is analyz.ed by use of the Kalnins Shgli of Revolution Computer Code. Complete fixigybegins at 36 from vertical axfs.' The model continues to 67.
See Appendix A for the program description.
Boundary loads at elev.
23'-6 7/8 are taken from the original design.*
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SUBJECT OFFICE REFERENCE NO.
Oy,yfa,. 6asw., Embedme,t a ** E** n M4//47 MADE BY CHKD BY MADE BY CHKD 8Y l
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Pati 1
&c.d,.1 y ne. c " + -' + **
8-dd M h' ', M.0.2.503E+04 74508E*03 0.000E*00 1 792E+04 7 508E*03 2.503E+04 4
36.0 1.370E+04 4 111E*03 0.000E+00 9 992E*03 4 111E+03. 1 370E*04 M.0 2 381E*03 7 142E.02 0 000E+00 1 6ME*03 7 142E*02 2 381E*03 g
37.1 1 602E+04 1.910E
- 03 0 000E+00 1.451E*04 1.510E*03 1.602E+04 4M/
37.1 1 342E*04 S.848E+02 0 000E+00 1.283E*04 S.848E*02 1 342E+04 37 1 1 002E+04 -3.398E*02 0 000E+00 1 116E*04 -3 398E+02 1 082E*04 i,
38 1 1 143E*04 -2 207E+03 0 000E*00 1 364E+04 -2.207E*03 1 143E*04 l
38 1 1 309E*04 -1 851E*03 0 000E*00 1 494E*04 -1 891Ee03 1 309E*04 38 1' 1 474E*04 -1 495E*03 0 000E*00 1 42SE*04 -1 495E*03 1 474E*04 t
39 2 1.033E+04 -S.43SE*D3 0.000E*00 1.576E*04 -S.43SE*03 1 033E*01 39.2 1 271E*04 -4 806E*03 0 000E+00 1.752E+04 -4 804E*03 1.271E*04 a
39.2 1.510E+04 -4 177E+03.0 000E*00 1 92 8t
- 04, -4 177E+03 1 510Ee04 40 2 1 201E*04 -0 578E+03 0.000E +00 ' 2+0S9E*04 -8 578E+03 1 201E*04 0{
40 2 1 229E+04 -0.S47E*03 0.000E*00 2.084E+04 -8.S47E*03 1 229E*04 40 2 1 257E*04 -8.516E+03 0 000E*00 2.108E*04 -8 516E*03 1 257E*04 40.2 1 204E*04 -8.491E*03 0 000Et00 2 053E*04 -8.491E+03 1 204E+04 40.2 1 229E*04 -0.471E+03 0 000E*00 2.076E*04 -8 471E*03 1.229E.04 40 2 1 254E*04 -8.451E*03 0 000E+00 2 099E+04 -8.451E*03 1.254E+04 41.3"ll.613E*04l=1 078E*04 0 000E*00 2 690E*04 -1.074E*04 1 613E*04
~
41 3=
1.152E.04 =1 216E*04 0 000E.00 2 398E*04 -1 216E*04 1 182E*04 41 3-7 50SE*03 -1.354E*04 0 000E*00 2 10SE*04 -1 354E+04 7.50SE*03 Jtd >
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DATE DATE DATE DATE wu aset.
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.4.- Membrane Stress Intensity 5 - Surface Stress Intensity (Inside)
+
. Surface Stress Intensity (Outside) l o
I k
It g
4 L
t E(
8 ri 4
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d i.
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/*.56 " 40'C lChybb i
CIRCUMFERENTIAL STRESS ALONG MERIDIAN l
OYSTER CREEK EMBEDMENT, CASE 1: P=35 T IM AX) =281 THICK =0.70 l
MAXIMA 14102.
14951.
18321.
WW5 JECT teADE SV CMnD SY CHAma g Ng, o n. cea<e Abnalmeat J3E aA NGH47 Aulvnr~ Redad Thisunse N!N}se ItjNE
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4.- Membrane Stress Intensity 5 - Surface Stress Intensity (Inside)
+
. Surface Stress Intensity (Outside) o Y ':? '
, \\.
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a a
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0 ze u
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/* - 3d " ges le=944 MERIDIANAL STRESS RLONG MERIDIAN 0YSTER CREEK EMBEDMENT, CASE 1: P=35 T IMAX) =281 THICK =0.70 MAXIMA 23615.
12719.
23 tl75.
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n 4.- Membrane Stress Intensity a - Surface Stress Intensity (Inside)-
n
+
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N
- -N..
a
~
s N
a.i j
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Sc.le /": m est We ss, Ise t l': 3d " moa les,44 STRESS INTENSITIES ALONG MERIDIAN OYSTER CREEK EMBEDMENT, CASE 1: P=35 TIMAX)=281 THICK =0.70 NAXIMA 32147.
25251.
29914.
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,4 - Membrane" Stress Intensity 0 - Surface Stress Intensity (Inside)
+
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o k
9-k 4
'(
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elev. 8 '-11E 4
4 2
3 2
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/*-36"4*wlenp4$
CIRCUMFERENTIAL STRESS RLONG HERIDIAN OTSTER CREEK EMBEDHENT, CASE 2: P=62 T IM AX) =175 THICK =0.70 MAXIMR 12445.
12655.
13124.
uW5 JECT MADE SV CMMO ST CMARSE NO.
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.4.- Membrane Stress Intensity
~
O - Surface Stress Intensity (Inside)
+
. Surface Stress Intensity (Outside)n b(
o s
D
.c
+.
1 D
+
c-N g
O
{
if t(
)
~
.s s
Y, w
5 s
{
e 3
4 o
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1 elev,12 '3 s
g 2
9 8
4 1
g
- A
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tlev. 8 '-!!!d 4
4 2
s s
~ Scale l= to Ksl s+ress,ics;
/* - Jd " oes leapd4 MERIDIANAL STRESS ALONG MERIDIAN OYSTER CREEK EMBEDMENT. CASE 2: P=62 T(MAX)=17S THICK =0.70 MAXIMA 34324.
20108.
33060.
BWDJECT e4 ACE BY CMMO BY CHARGE NO*
C Y. sear-Cnee Ekbeelmu&
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'r_@
j Ngy47 Atlelynt=. Acalm:ed 'T*bteto*Jy th/'bhc PLhitfac
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~ o i .A.- Membrane Stress Intensity 5 - Surface Stress Intensity (Inside) + . Surface Stress Intensity (Outside) o o h ~ 80.-. Q g:, q f y;f ,f, I ,l .v l 4 h e ~ { N. gs y.., g 4 i C elee 12!.3 k = _g ~ l 4 -4 f I =! l 4 -A O [ ,/ e/ev. d '-1/ $ = 4 4 a 3 i Sc.la i"= to gg; s+,,,, u ; /*: Jo"4ec /eapd4 STRESS INTENSITIES ALONG HERIDIAN l OYSTER CREEK EMBEDMENT, CASE 2: P=62 T(MAX)=175 THICK =0.70 ( MAXIMA 24027. 17889. 24935. . u.a.1 0 Me Cr-eee Elesfnoforee& DYg" *$" yj"jjj',"** rivn ~ na-a nieana.a 426 n*GL ~," ,,,,,,a
gy Analysis of embedment zone with sand considered to be ineffective The CBI Model used in the analysis of the reduced. thickness Oyster Creek embedment zone is described in the figure shown below. Complete fixity is assumed at elevation 8'-11 l'/4. Sand springs are assumed to be ineffective. The attenuation of the thermal gradient in the meridional direction is assumed to be completed within the empty,/281 F'mbgd e zone, that is,tge temperature distribution is 175 at elev. 12'J and 60 F at elev. 8'-11 1/4. The embedmenf Sone is analyz'ed by use of the Kalnins Shgli of Revolution Computer Code. Complete fixgtybegins at 36 from g vertical axis. The model continues to 67 Boundary loads at elev. 23'-6 7/8 are taken from the original design. 9 S* *to B.ma o.v 4%drinnt Qsa A/q o I Myse Mrs. sea m Detr94 ]%v. LI L 4'/n *"lfM
- s*
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- 6 so SURECT OFFICE REFERENCE NO.
Oy. rte-6esw Embeda.& O A" E~*c " " * ' * " M4//47 MADE BY CHKD BY MADE BY CHKD BY SHT$0F. &lfU *ktdks40 keC/FC1/ Q g Inaffna+,.e fand *i.%dMf^}[ fy
- TE
- ^'E Pvueens in USA GO M REV SEP M
n .e.- Membrane Stress Intensity ts-Surface Stress Intensity (Inside) + . Surface Stress Intensity (Outside) n o a. tn { o s y D e s n 6 1 s e 3 o d g,. j _g g 0 t g j e (* I, elev.12!.3 ^ r s 4 2 4 1 4 b .d t s N I -A b O 9 l ^ 4 tlk 4 4 a 3 3 l Seele i~= n scst s+r*es,ano1 /*:Jd"4*w/e*p44 l CIRCUMFERENTIAL STRESS RLONG MERIDIAN OYSTER CREEK EMBEDMENT, CASE 1: P=35 T(MRX)=281 THICK =0.70 W/0 SAND MAXIMR 13620. 14265. 15245. l t BWSJECT MADE sv CMKD sy gnAngg ag, 0 f.de Cr.eee Ekjedmee& Jsg rdA NG/W1 E Aelvre~ Reduced 7h suose offs lk n*fl]u ..a 2J e.
=e'-
m m. _g,; ,g.a.%,, m,e,- ,,m2,_._..o ,_._er.mwe.._,_w.---.e >=<i------
l ll .4.- Membrane ~ 8 tress Intensity e - Surface Stress Intensity (Inside) o + . Surface Stress Intensity (Outside) o u .. z: ~.. h li ( 'k si i j l 4 k.. I i r 4 q-. o- . k l r s 4-elav. It '-j k 4 } k ) t i I f 4-b 6 g q t ( e4.e d'//$) A 4 4 3 3 l Seela l's M Kst s+r*es,inel l /* - Jo " 4*w /eapd4 l HERIDIANAL STRESS ALONG HERIDIAN OYSTER CREEK EH8EDNENT, CASE 1: P=35 T(NAX)=281 THICK =0.70 W/0 SAND HAXINA 43722. 8821. 26330. 5Y.hr. Co-eee Ek6no(meat-DYg" '&h " y}"jjf.{** A~imr~ na n, anae impfik auik ,, 2.2.,
~ II II ,4. . Membrane. Stress Intensity Q - Surface Stress Intensity (Inside) o + , Surface Stress Intensity (Outside) I ll .h::.- u k' k k n "4 s j 1 .1 N i Y, s .N ~ m ~ Q .s ]9 . d*v 12L3 ~ s s a g N 4 T -Ai a / tler. 8 '-!! N y 4 3 3 3 Se.js /* # q SW es, W f=. 34 8 ges leap 44 STRESS INTENSITIES RLONG MERIDIAN OYSTER CREEK EMBEDMENT, CASE 1: P=35 T(MRX)=281 THICK =0.70 W/0 SAND MAXIMR 3060S. 7S99. 18431. CHARG E MO. sveJECT esADE SV CMKD ST o f3far. Co-eee Ekboolment J3E TJA ' NGI/47 Amlym ~ A b d 7"bwk:n*Jr 'thbhat Iths
- *' l-O *"
o
.w u . a w.:.. ..~. .a.ww.aw .a a - a. ~s -._- - ~ ~. a a..... fs~) i e ~ o .A - Membrane Stress Intensity 5 - Surface S n i + .Burface Stress Intensitnside). y (Outside) t i r s 2' i i N t h .,s u g ) s{ i } s .5 t d, y (, I -1 g ~5 3 i 4 s f dI .h U u l-i s yg - f!sL]E!j 4 D 0 2 1 i 4 ele, v 3 N' v. d -11 M 3 8 Sc.Je /*'s 2 so gst /*: 36 "em lesyd4 s+rw es,lasi OYSTER CREEK EMBEDMENT,CIRCUHFERENTIAL STRE DIAN MAXIMA 21828. CASE 2: 22635. P=62 T(MAX)=175 23721. THICK =0.70 H/0 SAND iv s s r - 0 Ystar. Co-eee Ekb&ntent I 44ADE SY -AMlvrir* Arf-yd 7"bt*40s.Q' CMMO av L Jsg 7pp j j r hgg\\ th$h\\ p q t \\
._....u. _ __c. ~w._.#..wm._a - -u _ _... z_..__m m s.__ _. m... m. MW i r ti ,4,- Membrane Stress Intensity O - Surface Stress Intensity (Inside) 1 + .Burface Stress Intensity (Outsid ) n t e i t S,, m 5 '8 i / ~ o h .;. o t s y 4 D d m j j . p ..C w-O .h i 'Y, s s S N g p.J 3 x -g eier: 12 !.3 x 4 O 9 j, ..a .I = t elays!.p 4 4 2 2-s s Seele I*= M Wst
- ** s j": 36 " e m /**10 MERIDIANAL STRESS ALONG MERIDIAN OYSTER CREEK EMBEDMENT, CASE 2: P=62 TINAX)=175 HAXINA 53897.
16944. 28734. THICK =0.70 W/0 SAND WWDJEET - -. b (O*CtW $ h0CfRftMh 2P 'MADE SV CMMO py ).$$ "{}& Anelvrir~ &4ud Thsse:nesy thj'ghe hg& --- -- ?
_.-~-__....-~~~,_,,.s;---~~~' s ,, _ -so~" ~" b \\ 1 .& - Membrane Stress Intensityn'- S ? '\\ + .80rface Stress I t n ensity (Outside) e) =1. \\ y L + k t ~ I Q t t s, \\ l e l z f i s i f I .s y q w t M S (3 t g 'N f h... s b. l., {.. .. e/<r.12'3 0 J .( 4 h p ~ [Qt r% 1 s t q t \\\\. \\ ~ ~ E, 4 4 2 $<ek / ** M 3 e/er. d '-114. Kst s j':3e"nowle*1 4 sMes, W 4 OYSTER CREEK EMBEOMENTSTRESS INT (RXIMR ERIDIAN CRSE 2: 37728. 12257 P=62 T IMRX) =175 114006. THICK =0.70 ILL W/0 SRND Q}(ftaj. Cneg Ehbeeferse&i maos av t cuo l nts. Ae>lud 7"bt*4C4 ~ J3E av hi y
- ntA j
40' ontg o4ff~~ j L-
- =
-- ;it/%fgg;it/seAng\\ zi ewuo lL-i i
m m.a
- m. m
..,e.. m ~ m. - c. z_.u _e _-ww. _ m m._.. l ~ Conclusions The preceeding analysis indicates that the reduced thickness section of the containment vessel shell located in the sand transition zone will meet the allowable stress criteria as prescribed in the original applicable code, i.e. ASME VIII, 1962 Edition and Come Cases 1270N-5, 1271N.and 1272N-5. A review of the stress plots shown on sheets 1.7, 1.8, 1.11 and 1.12 for the case in which:t.he sand is operative and pages 2.1, 2.2, 2.4 and 2.5 for the c~ase in which the sand is inoperative shows that the . +: local membrane stress and surface stresses are less than their - 'I respective allowables. The stress intensity plots shown on pages 1.9, 1.12, 2.3 and 2.6 show surface stress intensities less than a the allowable as described in the 1986 Edition of ASME III. These same plots indicate that the local membrane stresses are less than the allowable of 28875, however, the length over which the local membrane stress intensity exceeds 21175 pai exceeds 1.0 frf. Since the limits placed on the definition of local membrane stress did not exist at the time the vessel was constructed, it O" is inappropriate to assign that criteria to the vessel as part of this analysis. t F
- l 4
1 SUBJECT OFFICE REFERENCE NO. { g ggg REVISION jf MADE BY CHKD BY MADE BY CHKD BY M F*[NWM N/eMdW WA Jgs-SH M oF_ QATE DATE DATE DATE It/ts/gg tz/g(;,, printed in USA 00 M ftEV SEPM -._.m, ~ _, -...
,g ,w.... .u.~... -.--.-, ;. -.. ~. ua u~ iza - :. ~ .. ~ t. I 1 I l r
- O $*,
= +. U Appendix A CBI Computer Code Description Kalnina Shells of Revolution Program i f I I l I i l l t I S
yxa .u..a...
- .:a...:.. _.
u:..:~-~.~ _.. a..- -- l 3 RAW .'Enlains ~ Sise11s. df.Revolutfon ' Program IThe Shells of Revolution ' Program is the. Chicago Bridge. & i l Tron Company ' Program 7-81.
- The program calculates..the
~ stresses and displacements in thin walled alastic shells Mf revolution when subjecte8 to' static edge," surface and/or ! emperature. loads' with arbitrary distribution over the t surface of the' shell.. The gebmetry of' the shell must 'be isymmetric, 'but the shape of the median is arbitrary., It is .possible t,o include up to three branch shells' with 'the main' ~ lall in a. single model. ' In additiion, the shell wail may s ,cyns.tst of fo0r layers 'of different orthotropic materials, ?
- 1 and the thickness of each layer and the elastic properties
,.of each, layer may vary: alodg' thi iiiidiaii; ~~ y_--- '~ The 7-81 program numerically integrates the eight ordinary first order differential equations of thin shell theory l derived by H. Reissner. The equations are derived such that the eigh.t variables are chosen which appear on the boundaries of the axially symmetric shell so-that the entire problem can be expressed in these fundamental variables. Chicago Bridge & Iron Company has extensive'ly revised clie. , j. .'Kalnins Program. The program has been altered such that a 4 x 4 force-displacement relation can bit used as a boundary, condition as an alternative' to the usual procedure I l. of specifying forces or displacements. This force-displace-
- I ment relation can be used' to describe the forces at the boundary in terms of displacements, at the boundary, or the e
displacements at the boundary in terms of forces or'some compatible combination of the two. In this manner, it is possible to study the behavior of a large complex structure. SUBJECT OFFICE REFERENCE NO. ( cyff.,. 4%g Imfsgm.,rc O die Sw REVISION gj Assetyrkr -2=al~ *J TArwrser Snrk!Or_ eff.s. 4 - % w,.,,, gg om om om Pnneed 6n usa GO N REV SEP N ~. -.... -
- x.. . z.. ..u..:..;.. .. w. 2 1.a. ' ~Lww.u .i.w:. -.u.a ( b l ( It is.also possible to' introduce a " spring matrix" at the - end of any part of the stress model. This matrix saast be expressed in the form, force = spring matrix x displacement. In.this manner it is possible to model the restraint of - . the sand cu'shion in the transition zone 'at the point of embedment. In addition to the above changes, the Kalnins. Program has been modified to increase the size *of the problem. that can.be conside. red and to improve the accuracy of the ' solution / y e.
- . N d
O \\ SUBJECT OFFICE REFERENCE NO. ( Oyyder. c>esu. E<,,feda,4-OdS*8** " ' " ' ' ' " MM MADE BY CHKD BY MADE BY CHKD BY Arnoffir R*d & % icecosart snTA7.oF-m Aff.,d,3 A -he heg g DATE DATE DATE Prtneed in USA 00 N REV SEP W i
. =. . - ~. = .x m Ana'ysis of She. s of Revolutio F J t !P ~ Subjected to Symmetrical and ,~ Nonsymmstrical loads' i
- 4. mms A,.s.ienepr.in.or of sne neertne end Are ed sene=e' a
n . v.se m The boundary ealue problem of defernation of a reiationally symmetric shellis sJsted in How nom, cona, terms of a new system ofprst-order ordinary diferentialequations suhiek een be deviend 24m.ASME for eny consistent linear bending deery of shells. The dependent seriables contained in nis systene of equations are Mese genentitles anhich appear in ne naturel boundary o eenditions en a rotationntly symmetrie edge e/ e shell of reeetution. A numerieel' method of solution which eembines ne edesninges of both the direct integration and de fnite-diference epproach is dentopatfor ne analysis of rotationally symmetric shells. This method sliminates ne less of securney encountered in de usaselapplication of ne direct integration approach to ne analysis of shells. For ne purpese ofillustration, stresses and displacements of a pressurised terms are celestated and detailed numerieel results are presented. K 5 } l na shell of revolution is an important structural tion for an ellipsoidal shell of revolution by both the Anite difer-element, and the literature devoted to its analysis is extensive. 'ence and the Rungo-Kutta method; and Penny (6), Radkowski, With regard to ameymmetric deformation, various methods have et al. (7), and Sepeteeki, et al. [8] utilise the Anite diference been employed to obtain solutions of the bending theory of shells technique. A number'of additional references which deal with of revolution by means of the H. Reasoner Meisener equations. 4he solution of the H. Reissner.Meissner equathr.e can be found For saample, Naghdi and DeSilva (1]* use asymptotic integra-la the papero cited. tion; Imhmann (2), Mans (3), Klingbeil (4), amploy a direct *
- For problems of bending la the absenes of axial.,-. 3,a numericaliategration approach; Galletly, et al. [6} And the solu-reduction of the governing equations of arbitrary elsalle of revolu-
, ~p tion to.a system of four second order diferential equations in-4
- b '.'=)
/ 8 National Scisase Foundation Grant No. sagst, Report No: s. volving four unknowns has been carried out by Budiansky and July. it es. Radkowski (9). A snethod for obtaining the solution of these 8 Numbere in brachete deelsnate Referesess at and of paper. equations le given in [9] whleh is an extension of that employed Pneented at the Summer Conference of the Applied Meehanles la M ""d (8)' hells of revoluhon are foundinpapersby W% trutadts of ~- -dio Division Boulder.Colo.iJune 9-11.19H.of Tas Aassaicam Socesvr 7 or Mscuawicas.Ewoiwscas. deformation of s Discusair,n of this paper should be madressed to the Editorial De. berg and Bogdasot [10}, where a system o( 8:sterder diferential partment. ASME, United Enstneerins Center. 346 East 47th 8treet. equations for coalcal shells la derived, and by Steele [11) and New York, N. Y.10017, and wCl be aesepted until October 10,1964. Discussion received after the eloeing date wSI be sotaraed. Mane. Schile (12), where solutions of eartain types are eoesidered by . acript received by A8ME' Applied Meehaales Division. July 31.1963. sneans of asynnptoticintegration. j Peper No. 44-APM-as. Among the papers which essplay aunnerient analye,s, two dif. n Noisienstatufe d,8,f = esordinates of a po(at of ment of middle surfeen j',( )., '- derivative with respect to 1 ehell A.,$e = angle of rotation of nor, any coordinate e' = distance measured from snal as = order of system of equa-an arbitrary origia. pe, pov y - eomponents of==el==t-tions ' along meridian la calourfaceloade ? M'= aumber of eegments
- positive direction of (
se,me = components of -t.. s = ladependent
- variable, e
8e, te, n = unit vectore tangent to. of ourface loade either 4 ore j coordinate curves (see F. Te, Yr = temperature increment ' s, = end point of segment Fig.1) and temperature so-p(s) -(si,1) matrix,fundamen- . %,Re = principal radil of eurva-outtants ' talvariables ture of middle surface K, Fe, Nu = membrane stress sueult
- J(s) = (si,m) matrix, eoefE-L r = distance of a point on ants elents of difactial middle. surface fress Ke,Me,Mee = moment roeuttants equatione t
amis of symmetry Qe, Qe = transverse shear result,
- A(s) = '(m,1) matrix, aonho.
E = Young'ennedulue anta mogeneous coeSicients a = Poisson'e ratio
- ,0 = efectivo shear resultaata
. Y(s) = (m.m)snatrix,'T ~ A = thickness of shell / =' 1/4 + sin (/r ous solutions er = eoeDicient of thennat es. U = 1/4 + y sin (/r , 2(a) = (sm,1) snatrix, nonho-
- = 1/4 - ein (/r mogeneous solutions pansion D 'n K48/[12(1 - y')]
a = lateger, designating oth C = (m,1) matrix, arMtrary f K = XA/(1 - r ) Fourier component enestanta s i . es,, ene, w ====pa===ts of displaeo- $== length factor-I = unitanstriz \\ g3 l e
yw. a -.... u. a.;_ m ..m,,,,._.m__,s._ ,,,, c. L. og,,,,, ,,,, y = be a l ferent methode af solution of the bou.dary-value problem of rJven from which the apprepna long of the wate can l defIrm-tima cf she!!e must be recognised; i.e., the direct integra-estim;ted easily. tion [2-6] and the *! rite Jf'moce approach [H). While the In the application of this method to the analysis of rotationally - direct integration approach ; se certain important advantages, it symmetrie shelle, the boundary-value_prgbjern.le foUnuleted.in t l O also has a serious diendvantage; Lo., when b length of h sheH terms of Aret. order ordinary daRerential equations.,For thle l CD isincreased,alose of accuracyinvariably results. Thisphenomo. purpose iiEtIEg wiflithe equations of thilinear classical bend-l non was clearly pointed out in [8]. The lose of accuracy does not lag th_eory[6[shellelnishidthe thermal.eNects are in_sluded, first f result fre.n accumulative errore in integration, but it is esused by
- a. system of.equagang.ia_dirived in tha farm of e_ight partial dif.
the subtraction of almost equal numbeis in the process of deter.'ferential equatione involving eight unkn' owns in such a manner ~ l mination of the unknown boundary values. It foDown that for thakthe system of eaustione containe.ng_ derivatives of the ma-every act of geometric and material parameters of the shell there terial parametere, thickness, or principal radii of curvature. The is a critical length beyond which the solution loses all accuracy. absence of b derivat.lves in the eseNidents of the diferential The advantage of the finite <li#erence approach over direct inte-equatione permits the esiculation of the eseRielente at a, point gration is that it can avoid auch a lose _of accuracy. It le con-without regard to the values of the shell parameters at preceding cluded from (8] that if the solution of the system of algebrais or following points. 'Dggynaeumine menarability_with reaneet ta equations, which result from the Gn!Le-diNerence equatione, le }he iridependent variables, the desired evetam af =Mht b=' order obtained by means of Gaussian elimination, then no loss of so-ordmary diRerential equations is obtained which h-ather =ith - curacy is experienced if the length of the shell is increased. de counaary conditione on twa edges of the.k.n --nwa a This paper is concerned with the general problem of deforma-two point boundary-value problem. The derived system of tion of thin, elastic shelle of revolution, symmetrically or non-equa(ions is appucsole so rotationally symmetric shelle with-symmetrically loaded, and with the development of a neef.r.sl arbitrary meridional variations (including discontinuities) in method of its solution, which emolove the direct integration tech. Young's modulus, Poisson's ratio, radii of curvature, thickness. j nique, but eliminates the loss of_p.reuracy owine to the len th of an# acefficient of thermd expansion. While such a systern of 9 thi eneu. 2ne method developed here is applisableje_uy two-eq;.tions le derived in this paper only for one version of the p6ini~lioundary.value problem which is goyerned_within an in. classical theory of shelle,it can be derived in the same way for all terval by a system of m first-order linear ordinary _diNerential other consistent linear bending theories of shelfe, including those eq3tions together with~lii72%EA='v conditions predribed at which account for the dynarnic efecte, transverse shear deforma-each end of the interval.__Jt is shown that the boundary-value tion, nonhomogeneity, and anisotropy. problem of a rotationally symmetric shdl can be stated in this Finally, with b use of the method and b equatione given in form for any consistent linear bending theory of shelle in terms this paper, stresses and displacereente are eniculated in a thin-of those quantities which appear in the natural boundary condi. walled torus sub}ected to internal. pressure. The solution shows tions on a rotationaUy symmetric edge. that the meridional membrane stress la almost identical to that i i The method of this paper ofere definite advantages over the ' Predicted by membrane theory, but that h bending stresses Anite<liserence approach. The main advantages are: (aQ_h. era for a relatively thin lorus may not be negligible. can be applied conveniently to a larne system of first-order dif. / (Eential equations, and (6) it permite an automatie selection of Gt0!Illlfy 3Dil Basic Equatlsas an_ dptirnum~etep eife ofintegration at each step.accorshna to.the The pitia d a point d a ahu d myolution la given by b 4, desired accuracy of the soluti_on. Tbt.first point means tha;,il}p ' codih Q eund dong b triphtM unit veckm W " equations. of the theory of shelle of revolution, characterised in n, respectively, as shown in Fig.1. The shape of the shellis de- .. terms of Aret. order differential equations. can he.htegrated krmined by specifying b two principd radii d curvatum 4 ~~ , directly, and further reduction of the equations to a smaller num-Rid b middk utfm as factione d 4. Instead of Re,~it is ber of unknowns le not necessary. The seco_n_d point seems to be conveniet k use h dishnee r from a point en the middle eur- , f great importance if a truly aeneral method is desired whicKia face to the sexis; from Fig.114 foDown that
- o expected to hold for arbitary !=da._=h=" _monfigurations,. thick-nees, and so on. With the finite <lifonnes approach, a meaning-r = Resin (
(1) 'i ful e priori estimate of the step eise le often difficult, if not irn-possible, especia!!y when rapid changes and disoc ntinuities in the egenmaung ar q ddh erface Wen W = rM shell parameters are encountered. If a pudictor earrector direct 4 lategradon approach is employed with the method of this paper, / then h step rise can be selected aute:naucally at each etep 7 which ensures a prescribed accurney of the solution and optiraum efEciencyIa b calculation. The method giwn in this paper can be divided into two parte: d$ l (a) Dimet integration of a + 1 initial value problems over pro. R* 'N selected segmente of the totalinterval, and (6) the use of Gaus-K [~ ~ elan elimination for the solutic n of the resulting system of rastrix -. equations. The Aret part of this method is a generalisation of '"*. Q _ that which is employed over the whole intervalin [3-5). Here, f* however, the initial value probleme are deAned over segments of / /* l tl totalinterval, the lengths of which am within the range of the R / ( spplicability of the direct integration approach. After b initial %gM 7 value problems are integrated over these segmente, continuity 4.j conditions on all variablem am written at the endpointe of the / segmente, and by constitute a simultaneous systern of linear / matrix equatione. This system of matriz equations la then solved f directly by means of Gaussian elimination. The resultis that the y O] %., direct integration method is employed and at h name time there is na loss of accuracy because the lengths of the segmente are X selected in such a way that the solutions of the laitial value pr m_,ept.u.no,.mm es.vedent pamm.t.r. ,t. t
- e. e eas..,se.e.ee A-4 s
~.-.r 7 _,. 1 % 7. w.- A~.-- - - - + " - - - 3. 3 4 k (9c) lf. [ g,._'g4(**)% d4
- w = Nee = (1 - a)Keee a
- 8 " E(*8 + '4) -41 + a)aDTi -
(10s) (2) -[ } g., Re, - r 1+/g, M,.'p(g + ese) - (1 + F)aDTs (106) t 1*,.. i. M e - M.e,- (1 - e)Dr, (10e) g The following analysie mquires frequent di8erentiation of r (or Re)
- I with respect to (, and it is eenvenient to express thh derintive. mi t r i--- -t relations
by the Codaseirelation 1 d, = = - (we + w ees d + e eiE +) ' (tie) - - A eos ( (a) M 1 The f;'---tcomponentsof themiddlesurfaceof theehen e..g(w.4+w) . (116)~ and the rotations of W normal are de8aed by the expression of the displacement vector U of the form seu = '3 (we - w ese (; +*g me4 - (lle) u - (w + TA.)i. + (= + TA.)i,.+ = (=) The shell'is' subjected to t$g mechanical load vector p, which is se = 1 (Ae., + g, aos () (12a) measured as fores per unit area ~of the middle surface and written F .h A and the moment voeter m, which is measured as anement per unit . "* " 5 4 (126) p'- p + pA + p. (u) area and given by ase,- (A , se aos () + g pe, (1se) m - -m t, + m te (4c) e e y,Lth_n,ference to Fig.1, equadons (4) nrve the purpose for O " ~~1 'd + ala (
- 8 -
.(138) establishier the positive directions of_the components of the a Hi placement and mechanicalload vectore. 3 g The temperature datnbution in 1h321 eaused by some ther. g mal loads is accounted for in the usual manner by means of the $.= 4v4+4we (136) f', M I*'*8'*"*E 88"P*8"" *I* *I O' I*"" .I, The positive duections of the hress~ resultants la the horegoing a equations are the same as the corresponding stresses on the edge ~ F-4 ,,,, #,8) - (1 , f((,8. fMf " ~ of the shell. The de8aitions of the stress resultants are found in (5) Ital. j. 1 . The order of the system of equations (6)-(13)is eight with re. spect to (, and tonsequently it is possible to redues (6H13) to sht h serendal equaMons which lawlw eight un. T4,8) = 12 i ffT(,8,f)df (56) knowns. If the eight unknowns an those quantities which enter A; 4.I., Into the natural boundary conditions at the edge ( = eenet, then 6 the boundary.value problem of a rotationally symantrie shell enn The derivation of a new set of equatkras carried out in te out be completely stated ja terms of these unknowns. For this Reissner [la). When refernd to arbitrary shaus of revolution,esction is based on a linear cl I ~ the governing systein of egntions of (la] esa be writtan in the of aguatione and the fundamatal wiables, respectinly.
- foBowing form. Equations of equahbrium:
Deflystion of Fenslamental set of Equations - Nu +#* N.e., + s a (Nu + 0:sia ( + rpe o (es) Aeoordm to the elamical Geory of shelk, no quutities which 'sppear in the natural boundary conditions on a rotationally sym. l ~ ~ metric edge of a shell of revolution include the efective shear re. N=4 + -N f+ (#, - N,) oos ( +, Oe + ry, = 0 (66) sultante Nand 0 de8ned by Q. 4-SW::,::- Gee + '- 0 + 0. eos ( - Ne sin 4 - '- N. + rp = 0 (7) '# " #" +
- 8
( *"I Re r y gf,[, 1 Mu + "- Moe., + 2 aos d'Mg - rGe + rm = 0 ' (8s) O
- O* + 7 8**
(10) e ^* J J-4 ow e.n A Mt. ~ Thus, the fundamental variables, which are consistent with the ._ X*.e +g, M.4 + (M. - Me) see 4 - re + rm. - 0 (s6) theory of (13), a,e the four generalised displacements w, ev. and the four generalised forces Q, N., N, and M .In the derivation of the fundamental equations,it is more con. gga,e g,,;.,,3.go e: vm.ient to employ the distance e, measured along the meridian of Ne - K(se + pe,) - (1 + y)aKT. (Os) the sheU, rather than the angular eoordinate (. Ifowever, after the equations are derived, the probleen aan again be easily , #, = K(s + 're) - (1 + r)erT. (96) formulated in tenne of ( by means of the maatan e A-f i
- y
.wa aa;.aw r -M d ; - X- ,...a. = - -. a - - ~ - ~ .~ e 3)(aft- , 4 ws e - a 5 A8 a pnhainary step, it is amesary to espam Fe, Me, Mee la yeos4 y terms of the findannatal vanables. From (9s)it feuows that % * " - E8 - W -
- d 1 -,.
1 i.- Ne - rN. + K (= sia 4 + we + w eos() +K-N. + a(1 + a)T. (s26) r = - eK(1 - e')T. (16) 1/ LDsin24 "d -- (1 yLDI sin di/%d and from (los) est g,. gr ,e. M. +,1 - r (- 1 .ein dw..,..) < ses & (1 LDN ein 4). stD sin e ~ g(1 - p)K [\\g. LD ein 4j/g - eD(1 - p*)Ts (14) 8 e Kr* Eliminatloa of me 'and w.e. from egmune (12s) leads to en espmo. aioa forMeein solum Mee - LD 1 -, s$u + s eoe 4 - A -,* ,,g,4 yee,4A r r' r wa sr ,, ;j. ;. r ,D,1. g,,,gg 4,)7, gg,,) ~' + Kw ees d - /w, + LD sia 4# (17) .c K r ' ~ y , -#(I + ') y - sLDsee'+,y
- t wher.
r* w 1 + (1 +,)Kr slad wi(1 - r)"** 1LDI *- ~ '. L.. 1 + ein. 4 n.'. e." a r .r w r* K + (1 +,)K ain d w '1 -,.1-LDN eon' d l In the derintion of the four equations of the fundamental set whichinvolve the derivatives of the steens resultante wierespect - .(1 + y) K ela ( + b(14,p)ela ( 48* to e, the use of(14)is essential. Ehmination of Q,from(es)and -," w.e g (ae)byesenne ef(les) leads to . - D(1 - y)eos + (1 + r + SL)#pe + UN,.- r #pe A ,. g g.
- ,. = Keos (Xe, s e_os,3 -- 1 Nu i,
r' r' "
- O - P ' f ""
i' + Mu - pe 'I' # me (1s) N'8 - e r r. Sunilarly, elimiastion of Qe frium (7) and (Se) Sives - a(1 - p')f K ela ( fe -
- DTsa, (32e)
Od " - em (M*d - eos#4 + ela 4#8 ~ y,,.(1 -,) ",4 LD/h+(1+a)Kela( 1 re r w + j,;,-# p, Fue - F - *; mu (19) 1 -- r ~ g ,,f _-)LD/8 w 3 4 ,4 Solving (66)from#pthereresulte + (1 - >>,;# 4 Gr.D/K + (1 + y)K) =4 + /m,,, A.e 1-x, - -- N., + - Jude, i e r .. 8, Q - (1 - y) eos $, - 1 1 LDIeia 4 8.e 1 e K + ese $ (#, - N,) - 1 Q pe (30) e r Kr t i r Ed - p, - a(1 - n')K ,s ( F. (sv) en anditfollowefrom(86)that M - - - Meu + (Me - Me) + 0 - sie (21)
- .e." 1 - a "RI.Dess' 4 r
r = sin (.** accessary, #w and Q, wem ah=taated wie the use of - (1 + p)K ela ( + (1 + a)D w.e g The fundamental set of equtione eoesiste of (18)-(21), where
- e, Ms, Meecan be uplaced directlyla terms of the fundamental.
- (1 - F) eos 4 ({LD/K + (1 + r)K]wa variables by means of (15)-(17), and four additional equations lavolving the derivauves of w, w, w, #ewith suspect to e, which e are obtained from (136),(114),(116),(126), respectively. Finally, * + 1 - p "EUI'"9 - II + ') [ + D sin'() 4', reg [ 1 / p re the system of eight diferealial equatione that governe the deformation of a she!! st avvolution ena be espressed la teruse of _.g(g._,)" 4 (1+y) .LN ##4d #,4 the eight fundamental variables and writtaa se . r* e r e 4 . A-4
a >. a..w.--~..~ w ~- - 9". -w.~ ~ -"g- -- n-"-- .f-, x.x ~ ap.e i e y2 _g,.j y . ~ y,, - p ~ m,. _ eo,u e,(22 m. ~' s r ( (m), are esa dulmed la e. form ' - wv ~ i a f ,.]
- + =0 -- c') f (Kr., + b re)(22a) '
(w,u,,A,) - (v.,u
- ,.) *
(2s.) e M - (1 ' n)D ]* (1 +, + 24)w e + LD/ (N., M.,0) - (N, X, 0.) (28) 1 1 l + DO - a).o. -(1 + a) sin d I". - #L us., [e,, x) - (, y,} \\e f (2g,3 4'*' +D1 - y "('1 + *) ene' d - 24 g
- + 0 2LD ein. N.e The dependent aa Staats with subearipte a on h right.
g gg hand side of (25) are governed by a ayetem of equations which le - (1 - a) eos 4M, - si, - a(1 - y')D ri (224) Is thia,'can be wriths as one'4 obtained from (22) and, after using the assumpties that the shell F . -i r i. 3 Equations (22), (14), and (15) to (17) determine all unknown 'a.s " "~ u, - $, variables azeept Oe which was be found from (Se) and writtea la g, (26s) the form j.*
- u
- -t/w, ,,,, 4 4,, u c-u Oe - 1 Me., + M u + 2 n dX.,-F me (23) - r F .g + p N + a(1 + s)rm (266), 3 By esiculating Mu from (17) and making use of (16),it is poesi. ble to exprese Q, directly in terms of the fundamental variables.. " " *D sin 24""
- ar This expression is lengthy and contains deriutives with respect to Kra r
. e of the shellp.rametere. Since Q,does not enterinto say bound. 2Dn sia 4, + (1 - 7)K ary conditions on the edge s = coast,it is preferable to calculate 2 p g (gg,) Qe as the last uaknown directly from (23). The derivative of Me
- e Kr' enn be easily obtalaed by numerien! diferentiation.
The procedure for the derintion of an equivalent set of equa. p,,,,. 8*',, y en ein (,,, * ,, eos ( y tions for other linear elassical theories of isotropic shells is identi. r' r* r . O cal to that sina before. For geners! anisotropic and/or moo.. ~. '-' homogeneous shells of revolution'with rotationally symmetrio g + 3,. + a(1 + y)rm (' 26d) K properties, the fundamental set of equations is derived la the anme way as (22) except that (9) sad (10) must be replaced by the appropriate strees.etrain relations given, for exaraple, by Am. - 0== g _, ((1 + *)n'D bartsumyan !!4). Otherwise, the derivation is straightforward. p For the improved theory of shek, such as the one given by Naghdi + 2n'D eoe 4.+ (g + y)gr s;,i 4},, n e 's[15), la which the eKeets of tra.---S deformation are ecounad for, the fouowing ten fundamental variable are re. + 0 - r),,,,4 O + a)Kelm ( e ' ~ -N u quired: v, sw e,# #e,0,,#, Nee,3/.,Mee. Sinosnow04and m ce appear in (13), the.li.ntnation of Q, from (6e), (7), (8s), is done by means of (las). The required equations for the derive. U ~ rh (g +,)g I,; 4 + (3 +,)g,g,4 m tiyas of the ger.oralised forces are obtained directly fresa the_5,e
- r*
r' ,g ~.
- equations of equilibrium (6), (7), (8). The romalaing Sve equa.
..+ nT1'- r)(3 + y)D,,g, 4,. ~,,,40. + UK, ,tions are derived by following a procedure similar to that of the < .x
- nDsin24N. + pn* M - P. V a
- me.
Fundaniental Equations for separable Solutions .g,, For shelle of revolution which sonsist of sornplete latitud; - ou - p') K ela ( rm + D *.7 (20s) circles, & surface loads are periodio with rupeet to # with a meriod of 2r. and they can be assumed to be of the form NJ n) so 4 *0 +,)K sie ( n'/D w.. (p.7,m.)-(P+.,F me) '" ~ . Me) ( +3 ,-0 +,)K ess' d +,7 = e DJ8 u,. (r., r.) - (r., r ) h" (2a) . U - **)*K =8 4,,, _ 41 -,) (,e, m i - (, me.i {*,,$} - p. - n - ) -
- u.
u. e (=.) m ~ s.,s where the variables with subscripts n depend only on e, and each
- In the dwivation of the erstern of equatione (s)-(13) h assurop.
ponentla a g.e of n in (24) can be regarded as oni Fourier co.ro.enerall outier series expans .'3. where A denotes the salainnum priecipal radius of eurvature. lategral valu tion is made that b shellis susciently thin. so ht 1 + AV1228 - a IM anme appsesimation is med w stain b h egemien surface loads. fresa (32). A-7
-- w;,.m ~ - m A. __ i y... .~..-.m._..,.-_.s=--w,:,.---- .y g g-. ~ P** ~ (I - )Eene ( T (2@ The double (Isas again correspon to top or bottorn trigo-l nometric function employed in (24), (25), and (27). l The remainder of this paper is concerned with the solution of ~ the system of equatione (26), subject to the boundary conditions N. - *n(1 - r) "(1 + a)D 7 ein ( + (1 + F)E e. ( n' nn w g,,gg m,, p g gg,g gg g, gg g g,g, g ggpgg, e aion of the loads in Fourier series, the solution to (26) is obtained , 1 - n')nK cos 4 n'(1 - P')K ( g,, g,gg,g yg _p,,,gg g gg; g r' - superimposed to form a Fourier series expansion for the unknown N variables.
- nD coed (1+,)g 4-n A
) g._, Reduction to laitial Valus Pfeblems 2 eos d e n."- y, - y. This section is eonearned with the sedueden of a two point r r -. .a boundary vdue problem governed by - dp - ats)ycsi + a<s) d* m '" ",,"
- u. - pe.
(ses) e me. 4 a(1 -- p') - [KT., + D
- T/\\(26g) to a series e,f laidal-value problems. In (29s), y(s) matrix which npresents si unknown functices; s le the inde-
-..r \\ r pendent variable; A(s) denotes the (m m) ooefficient matriz; ~ ., ~ - 'Me = n'(1 -U)(3 + a)D *"' w. - n' JDu.* and R(s)is the (si,1) matrix of the non'. = terms. The ~# e r r* ' elements of A(s) and B(s) are given piecewise continuous func. c tions of s. The object is to determine y(s)in the interval a $ s n . 7,. 'i, (1+a)g,4
- nD aos(
-K 'w. 6 subject to m boundary.condidons stated la terms of linear E eombinations of y(e) and y(6)la the form a , +D 1 - r [(1 + a) coe* ( + 2n']4 + Q 85. D da 4N. F.y(e) + F.y(6) - G (296) k gr, - (1 - r) eos 4Me.-m,-a(!-r,3D Ti. (264) - where F., F. an (m, wi) matrices and G is an (m,1) matrix, which eos4 mhahhm M M & h4 mhu d b ~~ problem. It shogld be clear that the governing system of equa. The double signe in (2G) correspond to the top or bottom trigono. tions (26) derived in the preceding section is stated in the form of metric function employed in (24) and (25). (29e), and that the appropriate boundary conditions for a shell of The quantities which are not included in h funnt.mantal revolution aan be expressed la the form of (296)., y, variables can be expressed by means of separation of variables by Int the complete solu (ion of,(20s) be written as ,n
- - - y/
y(*) = Y(s)C + K(s) (30) ,~- ' (Ne, Me, Qe).= (No., Me., Oe.) eos ad. (27e) .g when the (m,1) matrix C represents s' arbitrary constants, and s Yis) is an (m, m) and 2(s) an (m,1) matrix whictrare deined as (Nee,'Mee, Qe) = (Nee., Mee., Oe.) g,,, (276) the' -. - - - --_ and particular solutions of(20s)in the form dY(s)-= A(s)Y(s) (31e{ I where the e dependent aaahtaats with subscripts a most entisfy g, a set of equations obtained from equations (14)-(17) and (23) in i l the form ggg,) g,, = J(s)3(s) + 3(s) (316) i -
- s = 'yNo. + (1 - y') f (w. sta ( + 39. eos ( sk nse.)
The initial conditions for determining'Y(s) and 2(s) are l - a(1 -,e)Kre, (28s) Y(a)=I (32e) D fn l' Me. = FMe + (1 - r') -- y' v. +. 4 so'
- z(e) = 0 (326) ein(e
- a(1 - p*)DT (286) Evaluation of (30) at s = a leads at once,in view of (32e,6),'to when He the unit matrix. " l An r C = y(s),and then(30)at = 6 enn be written as 1-F 2neos4
- sea " D
- 88.
- n/ue y(6) = Y(6)y(s) + Z(6)
(33) 2r r D sin Together with (296), equation (33) co'ostitutes a system of 2n + K eos due. V 2n4 +g ,(N. (28e) linear algebraic equations from which the 2n unknowns, and y(6), are determined. Ones y(e) is known, the solution at any value of a is obtained from (30) provided that the values of Oe. - V " MA + #se + 2 eos 4Mee. + me. (28d) Y(s) and Z(s) at that pardeular s are stored. This completes the reduction of a two point boundary value problem deined by (29) to si + 1 laitial-value probleme given by (31,32). l \\- - No = #. g,4Mee. (28e) As stated in the introduction, the solution for shells obtained by means of this procedure suffers a complete loss of securacy at some erideal length of the interval. The noson for thh phe. nomenon enn be a darly freen (33). When b initial value g, _ g,, *ry,,,. <sy) problema deined by (31,32) are solved with the use of the equa-A.a
...-.-~..~...-~-m - ~ ~ ~~ ~ " ~^'- ~ ' - ~ " " " ' " * " " * - " - 1, #"" , i.. psiss nurErunwn. 2,( 4) = 0 q j [( (3Gd) assus5 Requiring continuity of all elements of y(s) at h pointa s, e S, Se .Sg - f .'2,3,.. M + 1, the following #-matrix equations are ob-tanedfrom(as): i p.... y(w) = Y.(sm)y(z.) + 2d:d) (37) i
- =.a.,,,,
where f = 1,2,.. )t.".' Equations (37) involve M + 1 unknown (m,1) matrices: y(1),7 = i,2,...M + 1. However,if the Emmassaat-- 4 quantities prescribed at the edges of the shell are the fundamental N X x; variables, bn & total number of unko' owns is reduced by m, be-Xe X, X,,, cause m/2 elements of y(ri) and m/2 elements of p(rua) are 71s. 3 N'es.nen f., dM.i .e e.e known. The same is true if 6 boundary conditions are stated
- s.o.,v.s i.e. e.e.*a'*
la terms of linear combinations of the fundamental variablse in the form of (296). In this caos, y(: ) and y(zua) should be premulti- . plied by noneingular (m, m) transformation matrices F and Fu.a.. tions (26) for shella of revolution,it is observed that the elements respectively, so that h elements of h products contain W of Y(s) and 2(a) increase in magnitude in such a way that if 6 quantities prescribed at each edp. After eliminating y(s ) and length is incrused by any factor n, then these solutions increase e i in magnitude approximately exponentially with m. y(sue) from (37) by means of these products, it is concluded that (37) will retain its form if, after integration and before sub. Consider, for example,We axisymmetric esse when the defor-stitution into (37), Ya(si) is poetmultiplied by ri, while -8 mation in the shell is caused by'aume prescribed edge conditions at Yu(sun) and En(sua) m promultiplied by fu.a. In h
- = a, say,byM,(a) = 1 and Ne(s) = Q(a) = 0. Itis reasonable following. It will be agarded ht this transformation is carried fi to expect that 6 cornsponding solutions at a = b become sma!!er out and that y(si) and y(suu) contain among their elemente those and smaller when the laterval (a, b) is increased in length. The quantities which are prescribed at z = si and z,=
connection between y(6) and y(o)is givei by b matrix equation lively. (33) with the following magnitudes of the dements: y(6)4 mall, Thus for all boundary conditions in the form of (2961, the eye. Y(b)-large, y(a)-unity. Clearly, the only way that b matrix product of (33) can giva small values of y(6)is that a number of tem of X matrix equations (37) Involves exactly knowns, and formally it can be solved by any method which is signi6 cant digita of b largevalues of Y(6)subtractout. When applicable to a large number of equatnne.
- However, h succes b length of b laterval is incrosaed, Y(6) lacrosse, while. of the procedure given in this paper lies in the application of y(6) decrease, and invariably all accuracy is lost at some critical length because allsigni6 cant digits of Y(6)in (33)anlost. This Caussian elimination directly on the matrix equ First a rearransment of elements is performed. Since those simple example serves sa an illustration for h loss of accuracy m/2 elements of y(ri) and y(suu) which m known thr
-*n encountered in the analysis of she!!s if the foregoing reduction technique la employed, boundary conditions can be any m/2 of 6 m4!ementa, it is . ~. N,- necessary to rearrange h rows of y( ) and y(suu) so ht b A convenient length factor, de6aed by known elements are sepmted from W unknown elements It is ?. '
- = f[3(1 - P))%/(RA)%
(34) assumed here that b first m/2 elements of y(z), denoted by yd:J, m known and that the last m/2 elements, denoted by where Ile the length of the meridian of b shall and Ris a min!r w(:J, m unknown. On the other hand, yi( su) are h un-mum radius of curvature, can be used for an approximate esti-known and y(zw )are the known elements of y(suw). Since mata of b criticallength of W shell. If b solutions Y(s) and h order of the variables in h column matrix y(s)is arbitrary, Z(s) an obtained with a sis digit accuracy, then the foregoing it should be emphasued that this separation of elements does not - procedure gives good resulta in the rany # $ 3 - 5. lavolve any restriction on & boundary conditions, and that any However, the loss of accuracy of W solution can be avoided and natural boundary condition in b form of (296) can be pres shells of revolution with much larger values of A can be analyud at each edge. The separation is achieved by a simple by means of b direct integration techr.Ique if h multisegment snent of the onlumns of Yi(4) and b rows of Yu(sua) and method given in b next section is employed. En(su ) after integrating h initial-value problems de6ned by (36) to the ends of the segments S and Ju and multiplying by i Multisegment Method cf Ililegfation ' 'ad '"" " d ia 'h* '*8 158-I4t the shell be divided into M.eegments (denoted by 3,, where Once it is establimbed which parts of y(z.) and y(su ) are-d = 1,2,..., M) of arbitrary length in each of which 4 $ 3. known, the continuity conditions (37) are uswritten as a parti-tiened sr.atrixproductof b form Denets b eoordinates of the ends of h segmente by a = z., .where the left. hand edy of the shellis at = si and b right. "yd w T ~Y.MasaY.Y Nw)~
- d:J + *2,4%)*
hand edge is at : = rua, a shown in Fig. 2. In analogy to (30) y " - ~ ~ ~ the solution *in the totalinterval s $ s $ sua nowcan be writteE "I'(** "
- d}-
- #'}-
- '5*'dI-u (38)
V(s) = Yds)y(ad +Kds) (35) so that mch of b equatio'ns (37) turns into a pair of equations, given by where Yds) and Z,(s) denote the matrices corresponding to Y(s) and 2(a)la each segment 3,(x, $ a $ sm) and m given by Y.Tw)Fd:d + YeTam)yr(z,) - ydsta) = -Z,5(sm) (39) gyj,) Y. Nam)vds.) + Y.Tzw)Fd:d - ydsm) = -Z,Mrs ) - A(s)Y,(s) (36o) e ds The result is a simultaneous system of 2M linar matrix equa. r tions, in which h known coefficienta Y r(z,a) and K,r(sm) are Yd J=I (366). (m/2, m/2) and (m/2,1) matrices, respectively, and b un-
- I knowne y,(ad sre (m/2,1) ma. trices. Since yt(z ) and yfzua)are (36c) known, then an exa@ 2M unknowns: yi(s.), with i = 2,3,..
= 4(s)2/s)+ A(s) ds X + 1, and yd:J. with i = 1,2,.. M, A-9
. u.. w..~
- u. w :.w a m.a ~
-~.a a - - -ma p,ya
- . t e
By E of Caussia elimination, b erstem of equations (39) Is Erst brought to deform } e "K -I 0 0 0 ' O ~ "y,(s ) " As " ~ i t.; O Ci -I 0 0 0 ri(s ) - B. 0 0 K. -I O' 0 p (s.) A. (** 0 0 0 C, -I 0 y,(s ) B, ~ a 0* 0 0 0 Kar ~! y,(s ) Au a 0 0 0 0 0 Cn..yi(sn.i) En. when the d6te ladicate the triangularised equations (39) with 4 on the dell. Such loads introduce diseoshities in the solu. i = 3,4,..., M - L The (m/2, m/2) matrices K., C. an de6aed tion for the eormsponding stress resultante, and they esa be repre by seated at every s, by na (m,1) discontinuity matrix which is Ki = Y t (41e) simply added to the matrix K,(su) on the right. hand side of (37). i s. Ci = Y 'K -8 (416). This feature is of great value if shell jointe are considered. Any i i discontinuity, eihr la geometry oc la loads, is easily handled by and for i = 2,3,.. M requiring that the end point of a segment coincides with the loca. g,..yl+ y,scu- .(4ge) tion of the dieeontinuity. Since integration is restarted at the J -- a beginning of each segment, the precise efect of the discontinuity is _ C, - tY/f Y *C i-8)K.-* ...(41d) obtained. The program outpute all fundamental variables at a The (m/2, I) matrices Al, R, are gives by number of desind pointe within each segment, and it also com-e6 Ai = -K ' - Y 'ya(s.) (42e) putes the values of y(s4) twice; once from (43) and thea7 tom. i (35). If a eartain number of eigni6 cant Agures of these values ',~ R - -K * - Y *y (s.) - Y,a*K -'A (426) maten, then the sentinuity conditions are knownto be entis6ed to i and for ( = 2,3,.. M - 1 the same number of Agures. In this way, a conveniset errer esti-J, = -K.' - Y,'Cw-'Bw mak dbenlution is hind fwenry me. (42c) E, - -K/ - Y/Cw-'#w - (Y.8 + Y/C4.e-8)K.-'A, (424). x3mple: Pfessuf!Ied Tofus In this section es stresses and displacements are determined la FinaHy, for the Mah segment a complete torus subjected to a constaat internal pmesure. It is An - -Kn' - Yn'Cn-a-88n. (42e) well known that the solution of this problem, when obtained by uneans of thelinear membrane theory of shells, has a discontinuity ,...N. 8,r = pe(sn ) - Kn* -- Ya*Cn.e-8Bn. In the displacement Sold. It has been shown by Jordan (16) And ' '. i. ~~ L - (Yn* + Yn' Car-i-8)Kn-'A, (4af) by Sandere ud IJeplas !!77 that a assisfactory solution with re-gard to the Ap1=.===at Sold for a sulliciently thin shell esa be For brevity, la place of ' /(s ) and K,8(s4 ), the symbols Y/ obtained if b maahamar membrane theory of deusis employ Y and X/ han been used. Subsequently, Rossner [18] established Aounds on eartain ?* Dy inanne of(41)'and(42), the asknowns of(39)are obtained by parameters which show when b acalinear saembrane and whea y,(,,..)'- C,-is, (43e) the Un=r bending thwry is appucable. It sums wwewhHe to sin hem the solution for a,. -. torus as pudicted by the p(s ) = En-'lyi(sne.) + Ani (436) liner bendingtheory. n sad for i - I' 2, * N - I The gametry of es um is eewa in rig. 3. me usand to the quantities employed in equations (26), the two acessaary i >(zw.444) = Cn-a-'lys(sna+a) + 2,-el (43c) pammetmo fora torus are sina as e ) pe(sn.4) = K n 4-'ly,(sn a.a) F M ] ' (43d) Me = 6 (eds) ~. It should be noted that (41)-(43) must be evaluated la sucesosion, r - e + 4 ela ( (446) because each equation involves the result obtalmed by the preced-l lag equation. Beems of symmetry with respect to the pim II, Fig. 3, the Once all the unknowne y(s.) are found, the fundamental variables are determined frcm (33) at any value of: at which the [g,ggg, 4 i i solutions Y,(s) and K(s) are escred during b integration of the 0 4 laitipl.value problems of (36). The integration of (36) esa be h atoomplished by means of any of the standard direct integration /- methode. / On the basis of the eyeiem of equations (26) given la an earlier / g) section and the method of solution developed in the last two see-f tions, the author has prepared a computer program
- which hae g/$=270*
been applied to many shell configurations having large values of # X 4 X and successfully tested against known moults. One example of a I e/ pressurised torus with # = 57 is presented in the next section. \\ 4 l The program admite arbitrary meridional variations, including discontinuit:es, in all shell parametere. It also admite ring loade ~~ in the form of prescribed values of N,, Me, N, or Q at any value af @\\ (
- TI,e proerem was written and' an eniculations were entri3d out by the auther en the IBM 700 eemputer at the Yale Computer Cenier. The direct intesration of (36) is perferrned by ineene of the Adams predictor. corrector mothed. which selects an optimum step aise at every ansp asserdies to a pensessted assumsgr.
Fis.3 See= eery of serve essendered le seemple i A-lo - - ~ - ~ ~ ~ - -,,. - - - - - ~ - - - ~,- ~ -.-~ ~~~ ~ - -,~ --r
u G.... %. a.c -.i G. ^ ^ u..:L...? h., R. ui-. -- " '-~ - - - G ' A A A L febse 1 Sereues and dieplesomenes of a preusessed eerms ph/3 = e,ses, s/6 - 1 e.8 X 108 0.005 (een/E) X IC' - - 0.05 (w/6) X 10" 0.05 0.02 0.005 0.02 0.005 f ~ 90 1.001 -0.0G3 - 0.031 - 0.016 1.249 1.284 1.208 108 1.613 - 0.188 - 0.093 - 0.019 1.261 1.315 1.328 i 120 1.650 - 0.886 - 0.123 - 0.030 1.359 1.303 1.427 144 1.720 -1.015 ' -1.378 - 0.010 2.820 2.580 2.150 - 0.008 - 0.020 . 786 1.507 1.625 IG2 1.832 **' 0.895 171
- 1.006
'1.002 0.168 -0.G05 3.467 3.403 s3.297 ISO 1.900 ' 3.089 " "* 2.277 1.482 3.904 4.334 4.815 164.5 2.042' 3.800 ' 3.035 1.0G8 4.150 4.576 5.248 160 2.104'
- 4.270 3.119 1..*C0 4.208 4.637 5.151 103.5 2.174 4.17s 2.aso 0.630 4.lso 4.500 4.e03 103 2.254 3.610 1.589
- 0.274 3.908 4.221 4.162 4 216 2.042 - 0.587 - 0.957 = 0.079 2.652 2.627 2.481 234 3.168 -1.245 - 0.201 . - 0.006 1.273 1.200 1.260 252 3.730 - 0.717 - 0.344 - 0.077 0.416 0.417 0.414 270 3 997 -0.'824 - 0.331 - 0.081 *0.103 0.101 0.100. w/b XId' '5 h/b .i ; *. - Q005 A O.02 4 / QO5 r 3 7 h/b= 0.05 n t 2 O f i l ~ Q02 % L'8 t l s q h 0.005 % j { n g '.O m F ~ / "r m y ~ ~ ~ f -2 e ISO' 90' Pie. 8 Nerinel displosement w versos ( ebewone defesumed seetles Ple. 4 Meridional headles strees aos et esser eher verses enesedlemal eessdioeco ( 7 Table 2 as a.-- ene,qdesel headme essess andonesidieseleseenbsene ~ ~ -
- inieg,su. of 6 i.iua.voue,r w.ms is on,ried t f,om d =
.0 to d = 270, and the bounda,y oonditio.s at hess end,eint.
- /*
As a.? a.00s are we = #6 = Q = 0. For the purpose of esenpar: son with the (e dk X los .he .053 .082 2 resuha of (16) and [17), the load parameter is abosoa as p6/KA (, X 108 0.427 0.312 0.197 I = 0.002 and s/6 = 1.5. 100 en/ae ) 20.8 . 15.0 9.4 The numerical values of W normal displ===aat, mendsonal membrane stress a = #e/A, and meridional bending stress - e e. = 6Afe/48 at f - A/2 for a pressurised torus are shown la i. e
- Table 1 and in Fiss. 4 and 5. These results were taken from the It is of signi6eanes to note that even for the thlehn== ratie output of the computer program prepared for an arbitrary shell of A/6 - 0.005, which for many applications would be regarded as revolution after prescribing the geometric parameters as given by ornall, the maximum bending stress is about 10 percent of the (44). The meridional membrane stress distribution agrees very membrane stress at h same point. Such efects of bending la a well with that obtained in [17] by means of the membrane bory torus were previously noted by Qark (19), and they are also la of shells and it shows only a small variation with A/6. The de. agreement with h statement made by Goldenseiser [20) that formed shapes of W cross section of h torus shown in Fig. 5 for when the middle surface touches a closed-plane curve, which la a three values of A/6 are in qualitative agreement with those givio torus corresponds to 4 = 180', then la the vicialty of this curve in [16) and (17), but their quantitative agreement osanot be ex. bending stresses should be expected and the assesbrane theory is 4
pected beesume h values of A/6 used la this example are outside not applicable.- a the range where W bending efecte art negligible. This is con. The boundary layer shown la Fig. 4 la also la agreement with Armed by the examination of the bending stresses shown in Fig. 4. the conclusions r==ehart la (18] to the efect that when y and p The manimum value of a.occun at ( = 189* for A/6 - 0.05 and given by e at 4 = 184.5* for A/6 = 0.005, which are also the points of i mazianum normal displacement and survature as seen la Fig. 5. ss = !!2(1 r'))'/46/s)(6/A) The companoon of the membrane and the smanianum bending j stress at various values of A/6 is shown la Table 2. p = 12(1 - s'Xp/KX6/n): A-ll
p - ..a ,~. m. _ _ _. _ _. _ m,. _ l- _AAW _ 3
- 9 are large compared to unity, then a boundary layer in the neigh. Revolution by Finite Differened Method." Journal e/ MacAanical
} borhood of 4 = 150* should be anticipated. For the present Emaineerine Science. vol. 3.1961. pp. 3@-377. 7 P. P. Radkowski, R. M. Davis, and M. R. Bolduo. " Numerical example, p rangce from 44 to 440 and p from 9 to 874. However ^ Analysis of Equations of Thin Shelia of Revolutico." American s since p to the only, load parameter of the problem, the solutions Rocket Saciety Journal, vol. 32.1902. pp. 30-41. shown in Fige. 4 and 5 are proportional to p. and the boundary 8 W. K. Sepetoski, C. E. Pearson. I. W. Dingwell, and A. W. layer remains unaffceted if p alone is varied. Of course, for very Adkins. "A Digital Computer Program for the General Asially Sym. metric Thin.Shell Problem." Journal, or ArrLato atacasawice, vol. Isrge values of p the deformation of the torus may exceed the
- 29. Taa ws. ASal E. vol. 84. Se ries E.19G2. gip. 655-661.
limits of a Imcar theory which according to (18] rea.ttict p to the 9 B. Budiansky and P. P. Radkowski. " Numerical Analysis of rarige p C p'A. Unsymmetrical Dending of Shells of Revolution." AIAA Journal, vol. 1.1963. pp.1833-1842. ,1 to J. E. Goldberg and J. I. Bordanolf. "Statio and Dynamie ACIDOWI2d! DUD!$ Analysis of Nonuniform Conical Shells under Symmetrical and Un. Th.:e research hat been supported by the h*st.ional Science symmetrical Conditions." Proceedinas e/1Ae Shth Symposium en Aallistic MisaGe and Acrespace TecAnelear. Academie Preas. New Foundation Grant /23022. Many ideas leading to.thla p.per ye,i, u. v., vol.1. toot, pp. slo-sse, originated from the consult!ng work perfoamed by the author for 11 C. II. Steete. " Shells of Revolution With Edge leads of Rapid the Unite The Circumfmntial variation." Joumwas. or Arriazo afscuawice, vol. author m. d Technology Center, Sunnyvale. CaHfornia. ehee to thank the staff of the Applied Mechanica De.
- 29. Taawe. AS11E. vol. 84. Series E.1962, pp. 701-707.
12 R. D. Schile. "Asymptotio Solution of Nonahallow Shelle of partment of UTC for many illuminating discussione conceratog Revolution Subjected to Noneymmetrie Imads."Journale/sAs Aere-thle subject. ,J '. apace Sciences, vol. 20.1962. pp.1375-1379. ~ 13 E. Reissner. "A New Derivation of the Equatione for the k jIIOOOOI Deformation of Elsatic Shella." Amerioen Jevenale/MatArmatice, vol. 63,1941 pp.177-184. 14 8. A. Ambartsumyan.
- Theory o'
.ninotropio Shells" (in 1 P. M. Nsghdi and C. N. DeSilva. " Deformation of Elastle Russian). Coeudararsenneye fadoteratie phile.MatematicAse&sildrero-Ellipsoidal Shells of Revolution." Proceedinae */ sAs Second U. B. lary. Moscow. USS R.1961. p. 91. Kational Cesareas e/ A pplied NecAasics.1954 pp. 333-343. 15 P. M. Naghdi, "On de 'Iteory of Thin Elastic Shelle." 2 W.14hmann. "Beitrag sur InLesration der Reiasner.Meinsner. Quarterly e/ A pplied MatAematics. vol. 14.1957. pp. 369-380. achen Schalentleichung for Behalter unter konstantam Innerdruck." 16 P. F. Jordan. "8 tresses and Deformations of the Thin. Walled Inaenieur.A rcAis. vol. O.1935, pp. 338-346. Pressurised Torus." Journal e/ she Aereepace Sciences, vol. 29.1962 3 II. Mcas. "Ein Integrationsverfahren for die Basechnung der rp.213-225. Biegespannungen scheensymmetriacher Schalen unter achsensym. 17 J.L.Sandere.Jr and A.Liepins." Toroidal Membrane Under metrischer Belastung." Incenieuri.ArtAie, vol. 19.1941. pp.103-117 Internal Pressure," AIA A Jeurnal. vol.1.1963. pp. 2105-2 t t o. 255-270. 18 E. Reissner. "On Stroaaea and Deformations in Toroidal SheIIe 4 E. KlingbeII. "Zur Theorie der Rotationsschalen vom Stand. of Circular Cross SecMon Which Are Acted Upon by Uniform Normal punkt numerischer Rechnungen."laaenieur.ArcAie, vol 27,1959. pp. Pressure." Quarterly V applied Machematica, vol. 21.1963, pp.177-242-249. 187. 5 C. D. Calletly. W. T. Kyner. and C. E. Moller. " Numerical 19 R. A. Clark."On the Theory of Thin Elsatle Toroidal Shelle." ...m Methods and the Bendin-of Ellipsoidal Shella."Journale/iAs Seeisty Journal e/ Mathematice and PAyeice, vol. 29.1950. pp.146-178. e-' g/ladustrial and Applied MalAematics. vol. 9.1961. pp. 489-813. 20 A. L. Goldenvolser. TAsery e/ Elassie TAin Aha!!a. Pergamoa ' ^ - - ' 6 R. E. Penny. "Symmetrio Bandans of the General Shell of Press. New York. N.Y 1961.p.480. i Reprinted from Septe:nber 1964 issue of the Journal of Applied Mechanicse ,{. 8. t e q
- e. mss #
t A-12
..: ~.. ~ - _.i w._.a a _. 2 w. m y....~ i a a w On Free nd Fernd Vikah,ou of h ,,A g/g l {\\. Rotationally Syimactric layered Sl; ells ig@ I_ I, I j A. KALNINSi a, t
- se,.v,,e, < ie xote i. io ge we.mc. -aw w..ia,ei.
_ s.. 2J
- ~ ' *
- I 1, i l
qf sotationally synernetric timin elastic sheth givesi in tais earlier - ggg ~fff payere, t e, elssi the shell war consists of any museber of g p se !wre.f 6.tmpic.c.raoimpie terint. m Sing.hi.h , h&getCE SURFACs ,aust be anadet8 to arensist for eurb progmeties are enanned N to the stree-strain rehit*ione, sliich, in turn, affect oedy the re-afielents of theeight Gr 4. order feindsmentaldiffetcuttalt<giantions Fig.1 Elenwse of oben.weg seesissing of leyew of ebb,ary gld. news derivedt8 for the suhttien of certain initial-vahic problems. Tlw . E 3,8g(J(s=8 - s4) I Ce l (sm - as) ] multleegment method given in the previous papers for the 1 (s * - s,')g (Se. enslysis d free er forced vihrstion er static deformation reassins K, 8-t 8 esactly the some if the derivatives d the funrlamental varia!>1et. tD,j derived in this Note, are employalin h initial-vahie intes ation. [,In y h 8 f " '" h-(8u%' + 8"'**%+8* 3l h,(rm, - s ) - In h pienent fonnulation, the shellis deficcd by.means of any W ( senvenient, notationally synsmetrie, sentinuous reference airface, co -em r .a. s,mmeta,bo,mung.ua.c. in . m n [ y,,] war are loested in'ith' respect to the refereire surface in any , _,,) 4 [ y,,}. _ { (3, g e 4 g g q({,(, J arbitency manner. This feature is entremely useful in practical p, s.: }(,,e ,,ey ,,S. applicatinns, beeanse the reference surface can be elmers as b. simplest sudece of the shell and is not ruf ricted to the " middle" ' surface er to a special ese which is deteenined from Llw elutie where* Dmp"M#-. 3,. 3,g g _ p,e,) Using & notalloa of the peedom par' r,* h al.ti..ns e 4 bet' ween the (<lependent segerable' solutions of the stre.tv-An = soE.At - e,pe) = p,Fedt - e,se) l altants and the strain awasures Int a layered orktrnpic she!! 3,,. g,g g ,;,,,) [I see cebta*msert froen Ambertsunipn and, after adding the tem-S e,e Gap perature tenas, they enn be written la the foran Ke. = Cmde. + Cues. + Ku e. + Kim + N.To. + NuE:. Wlodex (denotcs the delayerbounded by Gw weediasta s. r and 34, Fig.1; e., av and K., E, are b eoef5cicols of Gerau A g,.# V. e>pahsion and Young's moduli is the 4 and i directions, respo ~
- . #e, = Ces. + Caes. + Ku,. + Ku e.
Lively; y,; en are the correspor, ding Po'.sson's reties; e.nd Geels 0 n c e 4 g,p,, 4 g,7,, g33) shear moduhrs. For en isotropic byn d he sheksB,e, = c = e, Es v Ze - K, Fe 'm og a F, and Gee e I/2(I + F). 'M F. '=' (C. + Kei sin 4/rXe. + mi ) ' ***2Perature ruultants re,and fa.are deAped by 7 (Km +. Am sin 4/rXs. + us..isin.4/r) (ter r., = (ra.s.,e, - r si)As e -- an). .M ' N I D.n + D e N K.4 + K e. r = (re. - rs.)As.* -- sh) (* + K.Te. + r T (Sa) =bere Ts.(d) and Fse(4) are prescribed temperatore distribi tiore of h nth harmonic en b apper (s =. s. ) and leen
- + +8~+K+ +x -
+ 1 re. + rnr.. m .(s. sa su,ra,e a h. sun.,s,,e uve>y m oba e,= i a a se.,e desig.si. e, b. onice a e,es,e co rf. "8 "'" I' Mao. = Dufs. + us. sin 4/r) + Kafe. + in ) (2e) relating these strain measures to the dieplacement ooinpoose wlwee the suheeript a denotes tlw oth Fourier harmonic, aml, !s of the reference surfs,ee esa be,found in the previous paper.s yi b distance n.easured along the eneridian of the reference surf.,, convenience, we have de$aed l - of the shett. The elastic parameters Cg, Kg. D,s. K, amt T,. .It should be anted ha en'euantitles deinet peevissiety of occurring in (1) arul (2) are de&nad for a shell conei. ting of m e.,pect to the " middle" surfaes (such na & ecordinates e. 8; at ~ layees, Fig.1,in the forat placement co.uponent, e ee. er radit ed eve..ii.e. Jte, av; en,t even eieneare.) are new deruied la the esma war eith esapset en t [seetate professer. Depertenent of mehantes. Lehirh Uni-soference airface. The coinhete slees b eernial of the refreet I wereity hthlehem. Fe. Fernwelr. Departnant of F.ncineering and surfue is hern denoted her s. ApplieJ Srienes. Yefe Unlaretty. New ltaven. Cann. Mern. AMIE.
- 88. A. Anderlauntian. T&reer e/ Asiseerspic SAdle. W.V e A;Kalniaa. " Ast dreie of Shells of Revolution Suhjerte<t la Sa rn-Tkhnical Trvnelation F Itt, Washington. D. C., May, IM4. pp.
I metrient and Neaspanistriot La.ide." henne. wr Arnise Mc-
- ad 44.
esaarten, vol. 31. Ta sirs. ASit E. vol. 86. Strica E.1964. pp. 467 476. e ang,,,,,,,y,,,eeriuitinew (12.23. p. 44. ~ Jeeeest s/}nine,= Free vihrstion of itetationally Synimetric Shelta."sha Aseverfre! Smed,ry e/ Aswrire, s ei. 2n ,3,n, ;,,gg gy,,, 4,3;ggy,,,,,,,,,,p,;,,,gg g,,,,,y, A, g. ,3 g,,,,,,ned in (4) At the truneserne therma? emidvetivity in th taas. end Ti. ev.ults if the teengersteiro T. = A' + afi.'In e eimed i &faneseript seceived by A311E Applied met. antes Dienn.un. time sih tarer, and the twoultanta are determined to empdring eier l y, blarek 4.1983; Anal draft, Star 28. tous. g;,,,;ty of temtwrature and W Sus of heat eeroes tise bwunding se t. faces of in era. +e g A-B
- .l
.__ :..&. m
- v m.
u.. m. e = Sree.- % ~ - Theenblationof thedrilestivesof theTmalsierntalvarinIJr _ , g _ %,i,47, starts with the erdentiue of (3) frien given Au', Ba', An', R.', c o ', e/, and ts (i = 1,2,.... so) et o ogw4 rad niese of (. h s It shonid he noted that N,/, c/, se ene turistrary (even dia. meing knowa vebre of r, # tad the fuivl.: mental varishire,(8) 3
- 3 continuous) fimetions ad 4. lad they most be r'aistant sith re.
e,e evalunt,J in pig,culi,n, starting sith (.*w) avid emling sich , pert to the circumferentini evvedinate 8. Con cepiently, the '(St). Equ.itions (5) can he trenerritml directly an emirecutive elav le.paranneters een he snede tempeisture skyendent for. FOllTitAN statemente in a eoenputer pmgenen beesuse every e .mi,y,nmetric, but not norwynometrie, temper.dnre variatinn. quantity ocestring in se equal, Inn has hues dsdirud by a preelee8. le the lategratian of the initir.1-value pn,hlems dermal pre ' espreerion.' qiis ply, it is ame,aary to ea'cuhte et a given point tie vahws m,c,gna: Inns (5)nte applicable to the analhis of thesacaJy - of he Erst derivatives of tise,eight fundamental variabice- {r. state regesias of an arbitrary reite:E..imRy synametric sher to harmontrally seri11 sting enefm, iJge, and/or thermal lenJs. Q. w, N,, #4., M, me K.) when the verlablem sl.cmetres To and the static Jcrormation,*se simply set lu (8) m - E ar, known. IJaing (1), (2),*as.d (3) and the strairslieplacemrut For a shcIl of revohi ice, spinning slewt its axis of symisetry wieb f a-f equilibrium equations (mith the traiohtory inertia terms sashbr velocity D, all the inads and v s.e set equal to zero, ee-ded) from the previous papee,' the calculation of the deriva. a.pt that res ese be arranged in the fonosing orJer: pe = uG's toe 4. ree - rO's sin 4
- n. === /r + v ens 4/r + r. sin 4/r (k)
For free. vibration prut, lese:.a en leads are sheent (p. = pc = Rs. = mw Jr + ev. sin 4/r (66) ~ pe. - T - Ti. - 0), and then (5) can he used sith the niethod ~ gi'*n peMy for & muinath d ik natural %e ,,, '.,p,f, 4 4;,,, (f, (g) and mode shapes of any layered sheit of revolution. .~, ~ a.' = - mut Jr w e 4 4/r (ad) j; ir. - --Saw. ess 4/r' + av.Jr%) Ackn.wtedsment This recankhas been supported by the Natisinal scienne + (ess $/rX1/% - 2 sia $/r)mn - 2n$c/r (k) Ornat No.28try. -m 2,,= c.D. K es. = (1/A )l(N - KnTm - KnTm - Cuen - Ka e.)Du r ~ - -(Ar - m T - p.T.- K e. - Du e.)r ) (s ) r n e ~ . s - (1/A )l(M - NaT - F T. -Kee. - base.)C. -(Ne - KmTu - Rari. - Caen - T.9.)Kul (84) Ko. = Cne. + Cure + Kn e + Knee. + ReTu-e e + NmTm (ii) _ F. - Me. = As,. + 4 ee. + Kn e + Keen + N.T e e 3 +T.T. (W) r- ,i,. -, Je = C. + 2r.sia 4/r + am(sin $/r)s (at) ~ 'se = (t/5)(#. - (C. + K.sta 4/r)e. - - (K. + D. sin 4/r)r.) (ar) i No. = K.de. + %) + Da(s. + % sia 4/r) (a=) ~* (s) . u, = w J4 - A. m = a "-- sJ4 (b) 1 e l.. i ~ l ; A.,=n,. ..(sp)
- c., = -nNJr + att/% + sin p/r)AreeJr + ye. ees (/r
- N eos 4/r - CJN - pe - pw'v (s,) Afe, = Me. ees 4/r - 2nMe,Jr - Me ens 4/r + Q.,(Sr) l' . O = at.Ar.ces e/r' - 0.ece $/r + Ke. sin 4/r u + #4J% + nWedr' - p. - pe's. (h) K, = (1/4 - sin $/r)Me eos 4/r - 2N. eos 4/r + nNsdr + nye. ela $/r' - pe. - pw've. (5t) %rbere the density permmeter a has been deAned by ~ ~ m '. p = E /(eni - se) ~ s s=t in terine of the mass Ienaity of p'4r the ithlayer. The ilefmition of other spnbols can be foiind elscwhere.8.8 4 A-14 a
u -.~ ~ ~ ~ -.ue..w:n a.' "-- = -l " M " " ~-- .~ i -h} L Appendix B __,.C ::. - Containment Vessel Drywell Configuration and Stress Summaries .L* from i Original Design Report Oyster Creek Nuclear Plant .-_m.
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-(,,__- --. DISTRIBUTION for Meeting Summary Dated: January 14, 1987 Facility: Dyster Creek Nuclear Generating Station Doc le NDr Local PDR' BWD1 Reading JZwolinski JDonohew CJamerson OGC-BETH(InfoOnly) EJordan BGrimes i ACRS (10) Oyster Creek File RBerne o Glainas BDLiaw RWHouston PCortland RHermann RBlough, RI l JStrosnider, RI WBateman, RI TRotella JStang RGilbert BTurov11n l HConrad i EAdensam WButler DMuller GHolahan CPTan OPA
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