STRESSES FROM PRESSURE, RADIAL, AND MOMENT LOADS IN CYLINDERT0CYLIN9ER VESSEL BY A FINITE PLATE METHOD


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1 STRESSES FROM PRESSURE, RADIAL, AND MOMENT LOADS IN CYLINDERT0CYLIN9ER VESSEL BY A FINITE PLATE METHOD NUCLEAR EQUIPMENT DIVISION BABCOCK & WILCOX COMPANY *"" AKRON, OHIO»M I»I tut te U M awn UWMWI*. *>jitet\ ** UMM Sam M tta (MM SUM 1<»«I auw^i t i ^ M MMMIIMHH, mm «i f ti MMii«fMyMnKi,tifcn,im MM iw«b4,«iiimmrh MW «Mt'WWM MX Authors: S. J. Brown M. E. Fox PXSStred by.babcock and Mtlcox Company under Subcontract No. 547HF2125S5BM for Westinghouse Electric Corporation and the IK S. Energy Research and Development Administration Contracts and EY76C for the Clinch River Breeder Reactor Plant Project. August 1977 DISTRIBUTION OFTW* CTI"""^
2 t ;i nuuctioii A structural problem that has received continued intrnist and drvclopiwut w r itic late several decades Is the determination of stresses In two normally intersecti«p cylindrical shells) subjected to Internal pressure and external lending. This type of structure in tnanifent In mnnwnys, handholes, view ports, nozcleplping attachments, etc. Over the yenrs moot analytical developments have been orl«nt<ii.*d toward the solution of the Internal pressure loading condition. This Is probably attributed to the fact th.it It vas Cell that structural difficulty from external loading could be easily deiilgned out by adjustments in pipe support or flexibility; many attachments did not receive external loads, and an analytical treatment of this problem Is considerably more difficult than the Internal pressure problem. In nuclear pressure vessels the external loading of the vessel through the attachment is encountered in thermal interaction, seismic loading, and various postulated rupture or failure mcchanismu. Historically, the solution of the internal pressure problem had itc beginnings in a 194? publication of a theoretical solution of stresses about a circular hole in a cyliiitk'i' by Lurlc (1]. In the following years this particular problem was treated by numerous authors, Withum (2), F.ringen et >>1 (3, 41, Van Dyke (5), anil tokkcrkerkrr J6J. utils.itip. various methods. Vliillc these solutions gave seme indication or hounds on the strokes in two intersecting cylinders, the Influence of the attachment, vhich is of interest, va% not Included. The effect of the cylindrical attachment vas theoretically treated by Riodclbach (7] in 1961, Erlntjen and Suhbi [6], Green and Zerna [9], Hoff [10], Crlnp.cn ct al In], Maye cad Gringcn (12], and In by Edmondson [13, 14). This work contributes sijjnjfjeantly to the evaluation of cylindertocyltnder intersections, however. Us disadvantages: lay In the fact that any effects produced by reinforcement or contouring at the juncture can only be approximated. Th*<i problem has boon treated In various numerical ways, however, the most papular arc the finite elemer.% and numerical shell formulations. These methods cover a variety of axltynntctrlc geometric and displacement, axlsymmctrtc geometric and asymmetric displacement, and complete 3D formulations. A dltcusslon of the finite clement technique, vhich underwent development in the 1950*8, way be found in texts by Callaghcr {15J, Zicnkieuicr (16), anil nutnercv ethers. Some examples of the early numerical shell programs arc dcvelupcd by Frlcdrich (171, Xalnlns {IB], and Letting! (19].
3 ,." ' Tha advantage of both Methods ia tha representation of variable thickness, branch ;. _ :. points, snd eaaa in handling larga complex components. Tha shell formulations generally i < 'assume classical type shall simplifications whereas the finite element characterises KM >\ finite region by a displacement polynomial. The swat versatile numerical method of per, ' forming a structural analysis of a cyllndertocyllnder components is by the 3D finite element method, since tha «ayaawtrlc geometric and stress nature can fee most accurately idealised. Its «lsadvantaga is that It's solutions represent the manipulation of a vary larga system of.algebraic equations, hence la generally more expensive than tha other nu, merlcel shell or finite element methods. The axis/metric ser^trie aed displacement assumption for the numerical shell snd finite element methods was tha earliest approach used to evaluate, cyllndertocyllnder structures and has been the moat popular. It haa ibeen found :o be a poor approximation of tha stress at a discrete location in tha juncture fro* internal pressure loading, however, it la adequate In representing some circinfcrenmlly averaged stress value. There are various waya in which the host cylinder is usually represented. There are a hemisphere with twice the radius of the cylinder and a hemisphere with its radius equal to the cylinder with a doubled internal pressure. A study by Rodabaugh 120' discusses the limitations of»uth approximation Methods, and ineluaes ssat u»»»mnt I' of tna uaa of the tvxisymmetric geometric and asymmetric loading approximation uclng Kalnlns '' program. In a paper t>y Brown,(21] this latter approximation, the representation of a finite region about tha attchment by a e^llndartoplate, is shown to provide results that ; are quite good and considerably cheaper than tha 3D geometric finite elamtnt method for 0*J3*l s In insutncea where it is desirable to have the full three dimensional geometric \ aaseaament of atraeaee in the cylindertocylinder attachment, it hat been shown in (21] j that t.ha finite plate method (FRO is a readily constructed and inexpensive preliminary! test prior to a 30 finite.element ot 'experimental evaluation of tha structure. It waa tha i relatively good rasulta and simplicity that resulted inapplying the method, TOM, to piping > loads and (as we shall briefly discuss in this paper) thermal loads. j Tha firat aignlfleant work uaed to addrea* tha externally loaded cylindertocylinder >' waa a aariea of papers published in the 1950' by Bljlasrd I22» 23, 24, 25, 26]. In these i well known And widely used series of articles, Bijlaard considered the problem of a eylln.! drical shell loaded by distributed momenta and forcea over some finite region of the cylinder. Although the interactive effect'of an attachment ia not implicitly accounted for. the. reeulte have been ahown to be relatively good in the host cylinder adjacent to the attsch
4 ' swnt. The durability of Bijlaard'e work U attested to by the fact that It Is still widely 1 have the acme limitation aa the theoretical praaaura aolutlona; namely, Inability to deai! wed to assess rectangular and circuits attachments to cylindrical shells. Improvement*, swdlflcationa, tabulation followed in papera by Xeapnor (27], aijlaard 128], Wlchman et al 139), Dodge (30], and Sellara (31). In 1963 Hanaberry and Jonea (32] gave a theoretical treatment of the in plane Moment loading of two intersecting cylinders, where the moment 1* appllod to the nottle terminus. In the Instance of reinforcement of the Interaeetlon, we nibs sccurately the streaaea within the juncture. In the caae of the numerical aethode cited previously, the 3D finite element method is particularly suitable to deaerlba the asymmetric nature of the problem with reinforcement; hotuver, Che computer coat la usually significantly Increased for the aotcle terminus ' subjected to force and moment loading versus a simple internal pressure load. The cost increase la a result of the increase in the mesh site needed to idealize the nonsyametric loads and rssulting boundary conditions. The axleyamatrlc geometric and displacement formulations cannot reasonably approximate the external Moment and transverse load cases; however, by ualng a Fourier aeries expansion (or displacement, an asyametric loading vlth an axlsyamctrlc geometric assumption offare a possible approximation method. Slseuaaion o the Internal Pressure Problem In reference [Sol ' '. ' the series solution. ; ma used to satisfy the equation Za(r)^ me 1 where ^^s] is a stress function satisfying the equilibrium and compatibility equations when conaldering the in plane loading of an annular plate (aee figure 1) iy uatng the following boundary conditions (aee figure 1) at r»b
5 j A simple two term expansion' Is chosen which results In a uniform biaxial stress lit the plot* and la everywhere equivalent to the atresa In tha cylinder aa a approachea sero. The wknowr. hoop membrane stress6gt at r«a of tht annular paate ta vrl ten aa a function Tr CYLINDER THICKNESS'* P~INTERWL PRESSURE Ttom Van tyke'a solution of atreasea In a hole in a cylinder (see figure 2) gf fa WHEREjfr v«were fcble to obtain with aquation (o) / AND ^ These functions at* sraphlcally ahown In fltura 3. With P.KAJ3 tha outer radius of the plate and the»f*br«ne loads «ay be determined. ' 'In a manner similar to the Inplane load paraaeterc, It vas shown for the bending hoop atress In tha plate at/*"a that 9t'r{p.G.9.p). For bendini, the equilibrium eqn. takes tha form of (2), where 0 la transvarse dlaplacement. From Van Dyke'a solution of tha banding atresa at the cutout In the cylinder and aquation (?) ve obtain '. P~F**tjS) AND G=tf%8) <»> which are plotted in figure 3..A comparison of tie maximum atraln in the HS hostl* [33, 34), which appeared In reference [21], is repeated In Table 1. G Upon calculating^(*&#), tha parametersa,)* t P and are determined, k geometric axlsymmetrlc program with Fourier displacement functions la utilised to Idealise the NS nottle attached to a finite plate with the calculated leads at r»b. An Illustration of the H5 nossle It provided in figure 4. Te.Me I JJM US»WK.E.tK4t*Midlt iulti IITO t.t MI 4» a FJt. (Nt H Mi tomcft ) 1550.«iM
6 Forces on a Moir.lc Attachacnt Zc is of Intereat to determine if the (FPH) csn ba extended to the evaluation of tresses in the cylindertocylinder Juncture resulting froa axial force silled at the nottle teralnus. Of course, we would want to utlllte a coaaon finite eleaent andel used for pressure losds, such aa the N5. following a slailar approach presented in [21] for Internal pressure, a consider the series solution of Bljlaard 122] for a unlfora pressure applied over the rortangulac area *Ct (see figure 5) where C,*Ct.. th* solutions of the fora.. t /5 V C 9 Y /ft ] ^0 AND YR0 t F>axlal Eoice, 1 length of cylinder Let's write the values Atf In M s paper %.. find <«) i» as* f Tabulations and pleta of data ofaf*and/tf*nay b«found in ret. fl8, 19!  He now deteralne the in plane force per length (R) t r«b shown in figure 7. vh«re Oh <? = <LXJ3Z {N$N*) n d <j pressure on plate over *r**7fc (figure 6} V1 We note titat^ffn is essentially an adjustment In XtitjS for the area of application of loading* In the circular plate solution we apply a pressure (q) to a circular area equivalent to the icf area. The ament/tytat the canter of the circular plate subjected to a unlfor* pressure at ro and V**5T~ "*" %L &&Z& at r>b la deterained as Using eqn. (12) and (13) we can write (17) (20)
7 I. i. Equations (20) and (21).are Illustrated la figure «. aaentaon a Wossle Attached In ref. [22],. BlJlaard praaenta the aerlea solution of longitudinal and circuafcrontiai aoasnta (M^ andm^) applied on a cylinder. The aoasnts are effectuated by a pressure distribution over the fttjfcfxzcgon the cylinder (figure 5). In the manner of the ; axial solution, we will use Bljlaard'e aolutlon to di teralne the aaxlaua internal aoaenta : and forcea (Af* «nd/v«o to be lkposed upon the circular plate. This la done In order to :' calculate the external loads ori the circular plate. Dp. until thla point we have been deal ; ing with aclf equilibrating loada applied to the plate. For an external bending aoaent '. applied to the plate, vet will have equilibrium froa equal and opposite aoaenta via a prcsaure distribution overtftf^nd a tranaverae ahear distribution along 77' >(flgure 9).. ' For this type of loading, our series solutions will consist of odd hsraonica. To be ;' consistent with the philosophy of an Inexpensive approximation aethod, we will consider only the caae otc&iiffuhenm*/. The maxima Internal aoaenta and forcea In the cylinder occur : at Ci or C2 1 depending upon the application ofm^ ora(r. As an exaaple, a longitudinal i aoaacjc applied on *Cf would reault In sxlnua awaenta and forcea Mfjjand/ /J) *t b Gi.O,O) 1 The s itessec at X*b, Y* t are found to be usually negligible or an order of magnitude lower. A circumferential external aoaent at the *Cj erea results In the maxima internnl aoaent : and force.0wa,>iu ) *tx*b,y* Ct, The tranaverae ahear G" on the plate (figure 9? is deter alnad aa where ^ivawmla Illustrated in figure 9. where^«ijc nay solve.  (longitudinal and clrcuaferentlal}, C t b)  Taa aeabrane force A b at f*cin the circular plate (figure 10) la deteralned by considering the 1st odd tern of cqn. (1) satisfying eqn. (2) with the condition* {V j"*o O
8 , {hence Equating Ato to/tf< aninp we awy solve OR Ar* yl an (2S) Ex Pla of Axial Force and ttowint toadlnt. The analytical and experlacntal evaluation of aodel #3 (see figure 11) teated at Oak Mdge la compared to an FPM analysis usingb*w finite eloaent prograa MAS (363 The curvaia 0.11O \*0.4, b */. /93/M * F*SOOlb..Me *800iNib.. Xc Z4.5*j3i*0. //. The forces at rb for each load la calculated as follows: a) for T B (radial force) X30 lb/»<?lan V (vertical force) »COO0 lb/radlan») for Me ," R  i6o0tt& Ib/radlan ) v. 284 OOH& lb/radian Tfffet m *60^int9lb/r«dian c) for Mt. / H GD56 lb/radian f  S20fi93?lb/radian TOfB  699Ain& lb/radlan, The critical atresses in the cylinder adjacent to the nozzle, which vert obtained by atrain, 3D plate finite elements idealization, and the FFM *re compared in Table II. Table II bpttimmul mtrx. llu. MM. Straw (KSiitnm r MC ML n.e J 30.0 tsjt 11.4 UA m Apprax. Computer Tim* CDC 7600 SOOMC Conclusion! la this paper a slaple technique (FRO la presented to analyse atreanee in cylinder! *
9 '  I ' toeylindtr juncture*. Since the appraaeh vacs ahallow ahell fowulatlons (by Van Cjkc and Mjluard) and a throe tew series expansion plate Cowulatlon, the janjc of applicability ia rouahly Halted **(itft)hq,&/ft p.s and curvature 1., It i» fait that tha valua of tha Mthod la Ita accuracy* cconoay, atyi ease in andclint a atructura which falls within tha rantt of applicability. Anochar appealing feature of the atbod is that ita aiapliatie approach ol wparpoaitton of results pewits an aasy rttanalon tolnelu^a additional loads net traatad. For thosa «acbanlcal lcadlnga not dovalopcd, it ia fait thtt their effect can either ba accounted for by the awchanism discussed or by lapla eelculatlons. CeneraUy, tha stresses resu^cioc item torslonal or transverse shear ara aaall coaparad to tha loads discussed, however, these shear affects aay be lacjv.ded. Finally, in tha instance of thental atraaa within tha cyllnderto»cyliader structure, it has baan shown In.an unpublished atudy by Brown that tha ffk yields very food results for tha xanse of curvatures discussed. Acfcnowledtjaanta Illppraclatlon ia alao expreased to J. L. Rechaer of the Kudesr Equlpaent Division of Xabeocfc i Hllcox for his support. '
10 I Reference*.. (1) Luris, *.. I., "Static* of ThinWalled Elastic Shells" (State Publishing House of Technical and Theoretical Literature, Moscow, 1947), Transl. Of Withum, D., "On the Shear Stress in a Cylindrical Shall with * Circular Opening," Ingenieur  Archln 26, 435 (1958) (3) Kline, L. V., Dixon, R. C, Jordan, M. F., and Erijen, A. C, "Strcsse» in Pressurlxed Cylindrical Shell with Circular Cutout," General Technology Corporation Technical Report Mo. 31 (Aug. 1962). (ace also Report 32, 33, 34, 35). (4) Erlngen, A. C, Negtidl, A. X., and Thiel, C. C, "State of Stress in a Circular Cylindrical Shell with a Circular Bole," Welding Research Council Bulletin Mo. 102 (Jan. 1965). (5) Van Dyke, P., "Stresses About a Circular Hole In a Cylindrical Shell," AIAA Journal, Vol. 3, Ho. 9, 1965, pp (6) Lekkerkerker, J. G., "Stress Concentration Around Circular Holes In Cylindrical Shells" Proceedings'of the Eleventh International Congress of Applied Mechanics, Springer Verlag, Berlin, 1964, pp B. (7) Reldelbach, W., "The State of Stress at the Perpendicular Intersection of Two Right Circular Tubes," Ingr.  Arch., Vol. 30, 1961, pp (8) Erlngen, A.' C. and Suhubl, E. S., "Stress Distribution at Two Normally Intersecting Cylindrical Shells," Nuclear Struc. Engineering, 2: , (9) Green, A. E. and Z«rc», W., "Theoretical Elasticity," Oxford Univ. Press, (10) 8off, N. J., "Boundary  Value Problems of the ThinWalled Circular Cylinder," Journal of Applied Mechanics, Dec. 1954, pp (ll)'erlngan, A. C, Naghdl, A. X., Mahaood, S. S., Thlel, C. C., and Arlaan, T., "Stress Concentrations In Two Normally Intersecting Cylindrical Shells Subjected to Internal Pressure," Welding Research Council Bulletin Ho. 139, April (12) Kaye, R. F. and Eringen, A. C. "Rirther Analysis of Tvo Normally Intersecting Cylindrical Sheila Subjected to Internal Pressure," Nuclear Engineering Design, Vol. 12, 1970, pp ' (13) Edaondson, A. J., "Stress Analysis of a Radial Noulc Attached to a Cylindrical Shell Under Internal Pressure," 08NLTK (14) Ibid  ASMS paper Ho. 74PVP48, (15) Gallagher, R. H., "Finite Element Analysis  Fundamentals," PrenticeHall Inc., Englewood Cliffs, Hew Jeraey, (16) Zlenklcvicz, 0. C, "The Finite Element Method in Engineering Science," McGrawHill, London, 1971, (17) Friedrich, C. H., "Seal Shell2, A Computer Program for the Stress Analysis of a Thick Shell of Revolution with Axlsynnetric Pressures, Temperatures and Distributed Loads," HAPDTK398. Weatlnghouse Bettis Atonic Power Laboratory, Pitts., Pa., (18) Xalnins, A., "Analysis of Shells of Revolution Subjected to Symmetrical and Konsya etrical Loads, ASHE J. of Applied Mechanics, Sept (19) Lcatlngl, 3. F. and Brown, S. J., "Comparison of The Numerical Integration Technique and The Finite Element Method In The Analyais of ThinShell Structure", 2nd SM1KT. Serlln, 1973.
11 loaabauah, E. C., "Applicability of Axlsyaaetrlc Geometry Analytic Methods to Nozzles IK Cylindrical Shells with Internal Pressure Loading," Battciic Report 1178,. MM. subcontract Ho. 3131, Columbus, Ohio. July Brown>«. J., "A Finite Plate Method to Sclve CyllndcrtoCyllmder Structures Subjected tos«mrnal Pressure," ASMS paper Mo. 76PVPC, Journal of Pressure Vessel and Piping. (22) Bljlsard, i. P., "StresWa From Local Loadlnga in Cylindrical Pressure Vessels," ASKE paper Mo. 54PIT7, Sept. IHi, (23) Ujlaart, P. P., "Stresses From Radlsl Loads in Cylindrical Pressure Vessels," Veld Ing Journal Research)Supplement 33, (24) Bijlaard, P. P., "Stresses From Radial Loads and External Moments in Cylindrical v Pressure Vessels," Welding Journal, 34, H5S. (25) Bljlaard, P.. P., "Additional Data on Stresses In Cylindrical Shells Voder Local Load Ing," Ibid, Mo. SO. May 1959:. ' . (26) Bijlaard,' P. P., "On the Effect of Tangential Loads on Cylindrical and Spherical Shells," PVRC, Welding Research Council. (27) Kenpner, J., et al "Tables and Curves for Deformations, and Stresses in Circular  Cylindrical Sheila trader Localised Loadings}" 3. Aero. Se. M.24, (28) Bijlaard, P. Pi and Cranch, E. T., "Interpretive Commentary or the. Application of Theory to Experimental.Results for Streeaes and Deflections Due to Local Loads on Cylindrical Shells," WRC Bulletin He. 60, (29) Vichman, K. R., Hopper, A. C;, and Merahon, J. L., "Local Stresses In Spherical and Cylindrical Shells due to External Loadings," WRC Bulletin 107, 1965., 1 (30) Dodge, W. C., "Secondard Stress Indices for Integral Structural Attachments to Straight Pipe," WRC Bulletin 198, (31) Sellara, F., "A Note on the Correlation of Photoelastic and Still Model Data for ; Motzle Connections iii Cylindrical Shells," WRC, PVRC. (32) Bansberry, J. V. and Jones, S., "A Theoretical Study of the Elastic Brtovlor of Two Normally Intersecting Cylindrical Shells," ASMS paper Mo WA/PVP1, 196B. (33) Van Caapen, D. H., Kroon, J. P., Xoopman, X. B. C., and Lstsko, D. G. B., "The MottletoFlat Plate Approach in the Stress. Concentration Problem of XozzletoCylinder.Intersections', "1stInternational Conf. on Struct. Mech. On Xeactor Tech., Vol. 4, Part G, Berlin, Germany,Sept (34) Van Campen, D. B., and Spasa, B. A. C. M.,' "On the Streap Distribution In Mozzleto '' ' Cylinder Connections for Small Diameter Ratios," Nuclear Engineering and Design, Vol. 21, " " " (35) Cwaltney, P.. C., Bolt, S. %., Corum, J. M., Brysch, 3. V., "Theoretical and Experimental Stress Analyals of ORKI, ThinShell CyllndertoCyllnder Model 3," OWL Report 5020, June (36) ALAS  Asymmetric Loading Axisymactrlc SolidUsers Manual, The Babcock t Vilcox. Company Report 91383, Barberton, Ohio (1971). t*i\umiis.:* /»**. r
12 OPtlom 1. Uniform biaxial loading idealisation. 2. Circular cylinder 1 with circular cutout Unit* plat* paraaatera for dlmiwion (X), biaxial loading ( 4), vertical loadinj* (P/p and o/pi varaua tba curvatur* paraawtar (0). 4. K'S notcle 5. Circular cylindar unit only loaded ever, a rectangular area (2Ci x 2C 2 ). 6. Circular plata uniformly loaded over a circular area (Utc 1 ). 7. Circular plate inpiane load paraa«t«ra for axial force (r). 8. Circular plate transverae load paraaetera for axial force (F). 9. Circular plate tranaverse load paraaeters for aoaents (Me or th.). 10. Circular plata inpiane load parameters for atmenta CHc or HU 11. Oak Ridge teat aodtl #3. i._
13 J
14 V N
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16 DIMENSIONS INCHES A B C D E F G H 1 J K L M N mm
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19
20 J 8" 1
21 fc. J 00 I
22 0 COS 2 H II fc 0. Z * w 21^ / I V o N
23 H thickness Model.Ma^rDirnerisions No  ^T~T 3 ' J. Thickness
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