ECCS - 125 - Buckling of Steel Shells, European Design Recommendations, Eurocode 3, Part 1-6, 5th Edition - OCR

ECCS TC8 TWG 8.4 Shells

Buckling of Steel Shells European Design Recommendations

5" Edition. 2008

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ECCS CECM

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All sc£lions thai arc repealed word-for-word from EN 1993-1-6 nrc marked with a thick \"crtical line on the left hand side.

BIIl'kling ofSrcel Slr1'1/5· European De.fig" ReCO",mell(!arlOlI.f

Fifth

ChUPll

• PART

Buckling of Steel Shells European Design Recommendations

10

"1" 125,5 1• edition. 2008

Published by: FCCS - European Convention for Constructional Steelwork publ icutiQn~ 'tI ~1r..'Clco"~lruct.com www,slcc\construct,colU All rights I'CSCned. No parts of this publication may be reproduced, stored in a retrieval systcm, or trnn!omilled in any foml or by any means. electronic, m(.'Chanical, photocopying, recording or otherwisc, without thc prior pcnnission of the copyright O\\'ner. l-res fbsumes no liability \\ith respect to the usc for any application of the material and Infonnalion conluined in this publicntion. Copyright

2008 ECCS

ISBN : 92-9147-000-92

European Con\'cntion for Constructiollul Steelwork

Legal Ocp.: 282703 08 Primed in Mu1tioomp Ldn. Mcm Martins. Ponugal Photo co\'cr credits: Prof. J. Michael Roller

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-------------------------------= Contents Stllbility ofSlccl Shells: European Design Rccommendations: Fiflh Edition 2008 PART I: General recommcndations for design against buckling Cha]Her Tille

2

•

Introduction

II

Shell buckling behaviour and design concepts

21

Scope. con ... entions. definitions. units. symbols and sign con ... entions Modelling of the shell

33 47

Materiul assumptions

57

Geometric tolemnees and i~rfcctions

69

Rules for Ihe plu3tie limit slUle assessment

83

Rules for the buckling limil stllle assessment using de3ign by global numerical analysis

95

Rules for Ihe buckling limit siale assessment u... ing l>tress design

147

PART II: Recommendations for plirlicular shells

:lriC\ 01 system. or recording or oth-

erial and infonna-

C hapter

Title

10

Cylindricul shelll> of constant wall thickness under general loading

167

II

Cylindrical shell ... of slepwise vllriable wlillthiekncss

217

12

Cylindriclll shell ... under wind loading

237

IJ

Conical shell3 nnd truncated conical shells under general loading

259

14 15

Liquid filled conical shells supported from below

281

Spherical shells under unifonn external pressure

16 17

Cylindrical 3hclls with ring stiffener.; under eKtemal pressure

309 319 331

IR

Cylindriclil shells wi th longitudmal stiffener.; under meridional eompn.:ssion

353

I.

Conical shell,; with longitudinal stiffeners under meridional compression

20

Saddle or ring supportl..-cl cylindrical shells

365 371

Toriconical and torispherieal shells under unifonn e:(.'Cts, the succe.ssor to thc 4th Edition, taking o\cr thc rolc of fonnal regulation of dc!.ign of mctal shcJls against buckling. Its scope was far greater thun that of the 4l11 ldilion. co\'cring othcr failure modes apan from buckling. extended to include compUllllional treatmcnts for shclls, and having a vcry strong and c1car struclure 10 pemtit application to shclls um!..:r 011 loading and stress conditions.. II olso borrowed greally from DIN 18800 Pan 4 (1990). "hich had becn dc\clopcd for Gcnnon-s.peaking countrics in thc Intc 1980s. The Eurocodc "as required 10 h:l\c a fonnat. :.lylc. notalion and Icnnlllology thllt is compatible wilh the remaindcr oflhc [urocodc s.tandnrds (EN 1990 to E 1999). Thi!>lcd to a number ofchangc!>III fonnal and tcmlino\ogy reloli\'C to trnditional shcll dcs.ign fonnulations, and thcse are all adopted into the 5110 Edition of the Recommcndation!>. The 5110 Edition quotcs. cl'tlensi\ ely from the I·urocooc EN 1993-1 -6 (2007) and IS complctcly compatible \\ith that standard. I-Io\\c\cr.thc Furocodc hos no commentary. so thc meaning. limitations and origins of Illany mles are not ulwllys clear. This 51h Edition pro\ ides un cxtcnsivc commcntary on thc cxisting rulcs relating to buckling in the Eurocodc. but cxtcnds far beyond It III gl\ ing rccommendauons. c'tpan!>ions.. ad\ icc ond wamlllgs. cxplanntions and cxamples. all of which !>hould gl\e thc u!>Cr con~idernbly more IIlsight and confidcnce in applying thc mles of EN 1993-1 -6. Struct urc oftil c document Thi~ 5111 Edition is divided inlo t"o parts. Pan I sets oul the basic infomllltiOIl and gcncrol proccdures required to undcrtakc all shcll buckling calculution'i IIccording to EN 1993·1·6. It dcscribci.

the methodology and conccptual principlcs for numcrical analy!>is. cithcr to dcri\c the basic data that Can be u~ in a ~trnightforward buckling design by hand calculation. or to replacc parts of this calculation wilh numcrical as!>C, .. mcnts, or to carry out a buckling de~ign that is completely based on IIUtnCriclIlassessmcnt.

5

Bucklillg o/Steel Sllells - Europeall Desigll Recomme"tllIli01I$

Pan II sets outthc dctailed infomullion fOf hand calcUlation rrocedures when a shell of a p..1nicular geomctry is being designed for a p..1niculllr loading condition. Many well-pro"cn engineering formulas. emplricnl data and simplified rules extrncted from numerical p..1rnmetric studies ha\'c been included in this p..1n . In panicular. Ilan It contains nldiculty updmcd "cr..ions of the rules sct OUI in the 4th Edition of the FCCS Recommendations.

An

T he Dra ning Cumminee ECCS T WG tt4 In the extcnded period since the 4'" Edition ",as published in 1988. thc mcmbership of the dmfting eommiuee has seen sc\ ernl changes. Chamntlllship of the commillee \~n); first with Dr Lars Snlllueisen. wus thcn lukcn o"cr by Prof. Itcrbcn Schmidt. und finally passed 10 Prof. J. Michael ROUL,.. The cITons of cach of these chairmen to progress the ",ork towards its final fonn are here ad:no\~ lcdged. The good \~ork of Ihe secrcturics to the committee during this period. Prof. Marios Chryss3luhopoulos. Prof. Carlo Poggi. Prof. Wcrncr Guggenberger and Prof. Spyros Kammano!> is ulso gr.ucfulty acknowledged. Past members of the committee "'ho contribu!niiU) Rules for Ihe buckling limit statc IISSL"Ssmcnt Schmidt. Rotter. using design by global numericnlllll!lly~il> (KunIlTlanos. Schneider. Guggenbergcr. Vnn lucre, Doerich. Ilol;!) Rules for the buckling limit Slate nssessment Schmidt. Rolter. Greiner usi nt' stress desi !II Karomanos. J lolst) Cylindricul shells of constant wall thickncss Schmidt. Rotter under genernl loading (Kurnmanos. Lm\3m) now chaMS bv POl!lll and Vanlaere) Cylindrical shells ofstepwisc variable wall Greiner. Roller thickness Karamanos. Docrich. Schmidt) Cylindrical shells under wind loading Greiner. Guggenberger. Schneider Schmidt. Rouer. Marcinowskil Conical shells and lrUncalL'd conical shells un- Greiner. Poggi der generolloading (Schmidt. Lugae. Vnnlaere) Flow chuMS bV Po~~i and Vanlacrcl Liquid-filted conical shells supponcd from Lagae. Guggenberger. Vanlaere below Sphcrical shells under unifonn external pres\Vunderlic~) ,ure Knromanos Toriconical and torispherical shelts under uni- Wunderlich fonn external and intemul nrcssure Cylindrical shells with ring stiffeners undcr Schmidt, Greiner external nrcssurc

7

BI/ckling of Steel SheHf - £tlrolX'(lIIlk.f;g" Recormnemk,tiolJS

18 19

20

Cylindriclll shells with longitudinal stiffcncrs undcr lIleridional compression Coniclli shclls with longitudinal stiffeners under meridional compression Saddlc or ring supported cylindrical shells

Schmidt. Salllueison Rotter) Chryssnnthoupolos. SP.1b'110Ii Krupka Schmidt. RoHer. Kllnamanos)

C losure

1 he eommiuee Furopl.-an

Proposed 6'· Edition The members of the committee are .. ery aware that simplc advice on many eritic..-ally important pmcticlll problems has not been gi\en in the Sill Edition. The origmal plan was to include many more chapters. 1I0\\e\ cr. in the interests of completing the publication in II rellsonablc time. these chapters were omitted. The following subjects are expected to be treated in a focuS(.-d manner in a forth· coming 6th I::dition. • GMNIA allIllysis and interpretntion nd"iee for complex load cases • Cylinders with cut-outs • Cylinders on local supports at the base • Cylinders on local bracket.; and engaged columns • Bending lind truns\ersc shear In long cylinders • Bendmg nnd trnnsverse shear in short cylinders • Cylinder-conejunetions in chimneys. towers and masts • Cone-cylinder junctions in .;il(~ and mnks • Shallow coniclli roofs and caves stiffeners • Corruguted stlfTened and unstiffened cylinders for silos Disclaimer The opinions expressed in these Recommendations nrc those of the authors and mcmbers of thc droning commitlc..'C nnd arc not necessarily those of the ECCS. Every effort has been made to ensure aceurney in these Rl."commendmions. but the publisher. the ECCS and the authors cannot accept responsibility for any loss, damage or othcr consequence resulting from the usc of this infonnation. Anyone makmg usc of the infonnation or muterinl comained in these Rccommendations. in wholc or in part. docs so at his or her own risk and assumes any and all liability from .. uch use.

8

1M. Rouer m.rotterfil ed.ac,uJ.. Edinburgh July

Prefil(.'e

I ~. Srag noli

Karumanosj

C l os u r~

The committee hopes tlUlI the reader will find much useful infonnalion In this Sill Edition of the European Recommendntions. will forgive any errors in the document, und '" ill provide feedback 10

the commiUL'C on any issues that should be reconsidered. J.M. Rolter

itically important pructo mdude many more

m.roncr(tl cd.llc.uk Edinburgh July 2008

nable lime. these chapUSI.."'d

~

manner in a forth-

and members of the

s. but the publisher. the If'

other consequence re-

)mmendations. in whole 1m ~uch UloC.

9

/",rodllctioll

I

Introduction

1.1

T he purpose of these Recommendations

These Recommendrllions are intendt.-d to provide thc designer of a metal shcJI stnleturc with un extended guide to the design of the shell against Inilure by buckling. TIley represent, on the one hand, the Sill Edition of the ;ntcmat;onnlly widely-used EeeS Recommendations on Buckling of Steet Shells. of which four edition), \\cre publishcd betwf.:en 1980 and 1988. On the other hand, they go much funhcr and provide thc designer with a far more eomprehcnsi\'e packuge of infonn:lIion than the previous editions, as outlined heI'C3ficr. These Recommendations arc fuily eompalible with the European Standard for the Strength and Sl3bility ofShclts, (ENV 1993-1-6, 1999 and EN 1993-1 -6,2(07). and repeat many of the OIles Ihat arc gi\'cn in that standard. A few small differences are noted below in 1.3.10. HO\\c\t."f. these Recommendations provide greatly extended nIles to cover a wide range of problems that arc beyond the scope of the European Standard. pcnlliuing hand calculations to be used for lI\any problems Ih31 h:l\e been explored by rcsearch ill\cstigations. oot y,hich ha\e not yet been codified into a standard. In addition. these Recommendations provide a full commentary, both on the rules gi\ocn in the European Standard. and on the additional nlk-s ghcn hcre. Wherc\er it is thought to be helpful. the commcntary includcs worked exalllph.-s. Because the European Standard is not penllilled to include cxplanatory or educational materinl. users of that standard will find the cxplanations nnd amplifications given in these Recommend:lIions of great assistance whcn they are undcnaking the design ofshclls against buckling. These Recommendations arc limitcd to considenltions of fuilurc by buckling. Thc additionul covcmge of lhc European Stllndnrd is described in 1.3 below. This document should always be used in conjunction with EN 1993-1-6 (2007).

1.2

How to usc this documcnt for dcs ign calculation s

1.2. 1

General

These Recommendations are divided into two pans. Pan I selS out the basic infonllation and general procedures required to undcnake all shell buckling calculations according; 10 EN 1993-1-6. This pan also describes the methodology and conceptual principll:s for numcrical analysis. eilher to derive Ihe b.1sic data that can be used in a strnightforv. ard buckling design by hand calculation. or to replacc parts of this calculation with numerical nssessmems. or 10 catry out a buckling design that is completely based on numerical assessmcnt. Part 11 sets out the delailed infom13tion that muSI be adoptcd illlo Ihe genernl procedure whcn a shcll of a particular geometry is being designed for a particular loading condition. Many wellproven engineering ronnulas. empirical data and si mplified rules cxtnlcted from numcrical paramctric studies arc included in this pan. III particular. Part 11 contains radically updated vcrsions of the n Iles sel out in thc 4'" Edition ofthc ECCS Recommcndations.

II

BlicklinK o/S{('('I Shells· Europe'", Desig" RI.'CO",,,,endatiollS

Mcml shcll SlnlClures are used in many different applications. leading to a great varicty of shell dl."Signs. The layout of this document has been chosen 10 try to ensure that all dcsigns are conducted to a common mcthodology and wi th upproximately unifonn reliability. Thc layout pennils additional information on specific new problems to be quickly assimilated into the shell design procedure. and new research findings to be adoptcd mpidly. 1.2.2

Procedure for a shell buckling asscssmcnt

When the geometry. muterial of cunstnlction. loadin},!. lind boundary conditions for the shell hale been dcfined. the Recommendations pro\·ide a set of calculations thut penni! the design to be verified agains tlhe limit Mate of buckling. TIlrcc approaches are aVlliiable: hand calculation. mixed hand and computcr calcuhuion. and fully numerical calculation. Hund clilcuintion mcthods (see 3.2. 1) for Ihc buckling u.o.scssmcnt of shclls have been successful o\cr a long period. For thi)o reason. the eonccptunl frnmc",ork uscd in hand calculutions fonns both the basis 011 \\hich numerical evaluations are adopted into the design procl."SS. and the interpretation that is put on each kind of cvaluation. The hand calculation process (Fig. 1.1) begins with detennination of the key rc)oiSlances: the plaslic limit load and the linear elastic critical load (A). These t\\O loads are used to detennine the shell rclnti\·c slendemC!>s (13). which gmems the assessment of the relative importance of clastic and pla)olic behaviour. Next. thc gcometry. 10:Id Ctlse a nd filbrie:llion quality Icud to no USSC:>~I1ICnt ufthe charactcristic impcrft.'Ction (C). "hich is used to detemline the sensitivity of the clastic buckling strength to both imperfections and geometric nonlineanty (0). The geometry and load case a re used 10 detemline the foml that thc clastic.plastic interaction should takc (El. These itcms arc then combined to detemline the characteristic clnstic'pla)ot1c buckling resistllllcC (Fl. Finally. a plirtitli factor (safcty margin) is llpplicd to the chamcteristic resistancc to obtain the design value of thc elastic-plastic buckling resistance (G). The hand calculation procedure is indicated in Fig. 1.1. taken from Roller (2002a).

a nil!

Where replaced bYj buckling a] perfonnL-d. may bc rcpl Thc dcsig chaptCl"o in (l'lnicular same- app clliculation

j

If thc pani the dcsignCl1 approaches I design may The dc)oign (LA) "ith a: colla)SC ana geometric 1 Where onl asses5n'W!nt load" nnd "C critical \ al critical bud impcrfcctio~

buckling ~ three buckli

12

1"lrotIIlCl;on

'Cat \llricty of shcll signs are conducted "'he layout penmts 110 the shell dcsign

Geometry

Elastic

Plastic limit resistance

Material Load case

s for the shell have it the design to be , calculation. mixed

I\e been successful tulations fonns both id tnc interpretation

istances: the plnstic detcnnine the shell ance of elastic and III assessment of the the clastic buckling ~ load ca:,c are used lese items nrc then ). Finally. n p.1rtial design value of the

,).

Boundary conditions Imperfection amplitude Quality of fabrication

Partial "safety" factor

Hustie imperfr.:ct buekling resistance Elulltic-plu!>tic intemction Chnmcteristic clastic-plastic buckling resistance

Design clastic-plastic buckling resistance

"-igurc 1.1: I-land l:l11culation process or I'. N 1993-1-6 and

th~

[I

@)

Rl'Conuncnliations

When: u numcrical analysis is pcrrom,,:d, some or nil or the Pltr1S or thc abo\'c process may be replacl't! by a more precise assessment rrom the numerical resulls. Where a rully nonlinear buckling analysis on an appropriately modelled complete shell with geometric imperrcctions is pcrfomled. the entire hand calculation process. with thc exception o f the partial "safety" factors. may be replac(..'t! by the numerienl calculation. The designer should first check to sec if hill pllrtieulur shdl buckling case is covered by one of the chapters in Part II. Eaeh clmptcr in Purt 11 gives hund calculation aS5Cssment procedures for n particular shell buckling geometry and load case which are rclmively easy to usc. TIley follow the sume approach as outlined abovc. The methodology and concepllllll principles of the hand cnlculation appro.1ch are outlined in Part 1orthesc Rl'Commcndations. If the particular shell buckling case being designed is nOi amongst those described in Pan II. or if the designer docs nOi wish to usc those p.1niculnr ntles. Part I otTers a wider mnge of alternative approaches based on numcrical analyses in line with EN 1993- 1·6. These numcrical approaches to design may use onc of se"eml altcrnJti\-e procedun.:s, depending on which analyses arc undcnakcn. The design ca1cul3lions may be based on a linear elastic analysis (LA): or clse on a lincar clastic (LA) '" ith a linear bifurcution analysis (LOA): or else on a linear bifurcation (LBA) and a plastic oolla.)SC analysis (MNA): or else on a comprchcnshc fully nonlincar global analysis including geometric imperfcctions (GMNIA). Whcre only linear clastic numcriealnnnlyses (LA) or hand calculations are used. the shell buckling 3SS(."SSmcnt is usually pcrfonned by stress design. The two kcy n.'s istances of the "plastic limit lood" and "elnslic criticallo.1d" are cxprc!>!>l't! in tenn!> ofstrcsscs. i.c. the yield stress and thc clastic critical values of the three buckling-relevant membrane stress components. which are tem'K,.'t! the critical buckling str(.."SSl.'S. The evaluations that follow (the shell relative slendcmess. the clastic imperfl'Ction reduction factor. the elastic-plastic intentction. the ehllntcteristic clastic-plastic buckling resistance and the design buckling rc!>i~tanee) lire nil pcrfonned sepamtcly for each of the three buckling-relevant membmnc stress components. Only at the very end or the assessment

13

Bllcklinc ojStC'C'/ Sh(!l1~ - £lIrtJ/Jellll De~'igll Rl'Comll/en,'C .'C"C·oC "C.,_ _____________

procedure are these three components re-combin,:d by means of a buckling intemction check. The procedure is explained in more detuil in Chnptcr 9 of these R,:commendations.

\~ hole calculation

Where linear clastic numcrical anulyses fonn the mam basis of the design. the designer may choose to usc the LIlNMNA procedure. which imohl"S a global IIncar analysis of the structure. with a bifureation analysis (LIlA) to detenninc Ihc 10\'oest eigenvulue ","hieh represents the elastic critical load of the structure. Thc pillstic limit load may be fommlly ~-d using a !'o mall dIsplacement theory elastic-plastic analysis (MNA) or else this limit lood may be conservatively estimatcd from thc l"l"Sults of the linear elastic analysis. In the luller casco the result is likely to be vcry cOIl.'>Cr" mive whcn high loeul bending cffccts or high locul stress concentmtions ure prescnt in thc structure (for example. udjacent to locul suppons), but it involves less computational clTon and m;IY thereforc be seen as a desirable dl'Sign procedure. In this CIISC. the "elastic critical lood" und thc " plustic limit load" are c\aluated in tcnns of multiplying fuctors R. r und Rpi on Ihe applied load sct. Thc characteristic elastic plastic buckling resistance is then directl y evaluated using thcse t\'oO values to obtain u global slenderness. and inferring appropriutc \alues of the clastic imperfection reduction factor and the ehL"lic-plastic internction factors on the basis of the type of stress condition and buckling mode that the critical mode or plastic collapse mechanism displays. This aspect, in which these key \alues must be inferred. means that the method is. 31 present. only good for \ery experienced designers or for situations in which the Stress dL':Sign appronch leads to great conscrvutism. Il o\\cver. as this method is uSl-d more. un increasing understand ing will devclop on all optimal methodology for assessing these parameters under dilTerent conditions. und it is hoped that this method can be c.'(ploitcd in the futu re to enable complcx struetufCS to be designed using numerical analyses without having to fCl>On to the very onerous procedure o f a fully nonlinear GMN IA analysis. When a!>M.~illg the II hcll buckling n..':Sistance by global numerical GMNIA anulysis. the imperfections and the material nonlinearity ha\c to be included explicilly into the modelling of the shell. and thc analysis has to be perfonned laking full ueCOUlIl o f Ihe changes in geometry caused by the loods and. where Ilpproprialc. changes in the lood directions cuust.-d by thL"SC changes in geomctry (geometricully nonlinear analysis). The result can finally be used to lind the ehanacteristic elaslic-plnstic buckling resistance of thc she ll . I lo\\-c\'cr. this type of analysis is difficult to perfonn and is only successful if the analyst has great experiencc und physical insight into the buck ling behaviour of shells. Nc"enheless, it is increasingly favoured by many design engineers for large cxpensive and complex structures. nlC nccess:uy rules have thcrefore been devclopl-d for the European Standard EN 1993- 1-6. and they arc ,..onscqucntly included in these Recommendations (Section 8.3).

1.2.3

Location of reluted information

'" I'I2)1 l)culltChcr

A",,,h,,·.j

i Supplementary

In EN 1993-1-\. Ihe relali ve slenderness of a column or beam is given the nOI:1Iion i. The bar on this !>ymbol has ils origin In Ihe usc of A. to mean Ihc slenderness mtio (Ur for II column). Ilo"e"'er. EN 1991-1-1 no longer uses A. 10 signify Ihe slenderness of II column or beam. except for the sIX'Cial cnse of II battened column. For shells. the distlllclion bet" een slenderness (not defined) and relathe slenderness. 1II1he sense implied abo\'e. has no meaning. It Iherefore seems approprillte to drop the mther precious detail of Il bar, and usc the notation of A. to mean the relati ... e slenderness

(l- JRpI . R" ). 1.3.10.5

Aeeidenlal and unintended ccecnlricilles in lolcrllllce limits

In EN 1993. eccentricities that should be limited urc defined as "unmtended" cccemricitles. I low ever. these ceeemricities were previously tenned "accidental". and although Ihis correction was made to a late ... ers;on of EN 1993-1-6 before publicalion. the change was unfonunately omilK'd from the published \ ersion. This eorr(.."(tion is mcluded in thc~e Recommendations. 1.3. t 0.6

Greiner, R. (1997) eonsidemtion ..... Spon. London. pp Greiner, R., Rotter, steel , h"lll "ru'''"'~'' j

Pelersen. Ch. (

": rrors in cqultllons Structures:

A \ery s mall number of errors have been detected in the equations in EN 1993- 1-6 si nce ils publication. Some of these errors were detected before publication and corrected. bUI the change was unfonunately omitted from the published \ersion. WhlTC\er nn error is known to exist in the equations of EN 1993-1·6 (e.g. in Eq. 8.24), it is corrected in these Recommendations. Wherever it is found that a discrepancy exists, these Recommendations shou ld therefore be regarded as more

18

RI ~ k.

Rotter. J.M. I'mmework of the Hiologicnl

S""",",c'l

''''rolll/cllO"

'n thc hmit \tatc is reachcd) i~ buckling or plalllicity fllilure!. lillg between sc\cral different \alues o\cr 11 tillite pan of thc lancc" is therefore difficult to

reliablc than EN 1993- 1-6. It muy be noted thut csscluially thc SIlme dr.tning team wus responsiblc for both dOCUlllents.

1.4

Bibliography for C hapt er I

as 5500 (1996) "Unfired Fusioll WeldL-d Pressure Vesscls··. British Standards Institut ion. London. )r many years thereafter. this LTCh and prncticc community. )Iication of EN 1993-1-6. that ,lead a note is inscrtl.-d in EN he !.ymbol for the reference r to amid misunderstunding. it c'pcrt field of shell :.tructure

. (Ib \\ell as hU\lng potential ammcndations. quantity as a "resistance ratio" ance" of the structural system.

CRCJ ( 1971) " lIandbook of Structuntl Stllbility". Column Research Council of Jllpan. CorOIlIl. Tokyo. DASI-Ri 013 (1980) " Beulsicherheitsnachweisc filr Schn1cn". DASt-Riehtlinie 013. Deutscher Ausschuss ftlrStahlbau. Kal il. 1980. DASt-Ri 017 (1992) "BculsieherheitSllaclmcis fuer Schalen - spezic1le Faile". DASt-Richtlinie 0 17. Deutscher Ausschuss rur Stahlb,1u. Kaln. DIN 18 800 ( 1990) "Stahlbautell: StlibiliUltsfillle. Schalenbculcn". DIN 18800 Part 4. DcutschL'S Institut filr Nonnung. Berlin. No\ember.

lces (1988) European Recommendations for SIL'CI Construction: Buckling of Shel ls. 4th Edition. European Con\ention for Constructional Steeh\ ork. Brussels. I: N 1990 (2003) Eurocode: Basis of strueturnl design. CEN. Brussels. ENV 1993-1-6 ( 1999) Eurocode 3: Design of steel Slnlctures. Part 1.6: Gener.tl rulL"S Supplementary rules for the strength and stability of shell structures. CEN, Brussels.

:n the nOlatlon 1.. The bar on , (I~ r for:l column). Ilowe\'er. or beam. e,cept for the special mess (not definL-d) and reilltive seems appropriate to drop the ~3n the relative slendemess

ImilS -unintended" eeccntnellLes. )d although this correction was ~c was unfortunately omiued mmcndations.

EN 1993- 1-6 (2007) Euroc()(k: 3: Des ign of steel stnlclurcs. Pllrt 1.6: Strength and stability of shel l structures, CEN. Brussels. Greiner. R. (1997) .. /\ concept for the classification of steel containments due to safety considerations". Containment StnJcturcs: Risk. 5.1fely and reliability, (Ed. B. Simpson) E & FN Spoil. London. pp 65-76. Greiner. R.. Rotter, 1M. and Schmidt. II. (1998) "The new c urocode on strength and stability of steel shell structures", Proc .. Nordic Steel Structures Conference. 15-16 Sept. Vol. I. Jullien. IF. (00.)(1991) " BucklingorShell Structures. on Land, in the Sc3 and in thc Air". Elsevier Applied Science, LondonlNew York. Petersen. Ch. ( 1982): "Statik und StabiliUit der Baukonstruktionen" ("Strellgth alld SflIhilily ojC;d/ £"gil/(willg Sin/eli/res "). 2nd Edition. Friedr. Vie\loeg & Sohn. Bmunsch\\cig/Wiesbaden. Rotler. J.M. (1997) "Challenges for the future in the design of bulk solid storages", Containment Structures: Risk. sufety and reliability. (Ed . B. Simpson) E & FN Spon, London. pp 11-34.

~ns

in EN 1993- 1-6 since its and com.-cted. but the change error is known to exist in the ccommendatiolls. Wherever it hercfore be regarded liS more

Rolter, J.M. (2002a) "Shell Buckling and Collapse Analysis ror Structuml Design: The New I-rolmework of the European Stllndard". in New Appro.1chcs to Structural Mechanics. Shells and Biological Structures. Eds I-I R. Drew and S. Pellegrino. Celebnnion volume for the 60th binhdllY

19

B"ddi"g Of Steel Shells· EUM/X'(If/ 1J

'trie, (e)

l-- ¢,(-- - l i ncar prcbuckHng

symmetric

ormally exhibit stable symmctric nstable symmetric bifurcations, "Cd. for c:tbuckling condition progrcssi\ely. "ith a limit load representing the maximum load, The bcha\iour then resembles the snap-through behaviour displayed by a perfect structure (Fig. 2.1). By contrast ...... herc the perfect structure ha!! a stable post-buckling path (Fig. 2.3a and 2.3b). the efTect of geometric imperfcctions is less significant. as sho\\11 in Fig. 2.5b with c> 0 or in Fig. 2,5c wLlh either c < 0 or c> O. A fastcr growth of deflections is obscn cd as the bifurcution load of the perfect system is IIpproached, but the equilibrium path remains stable and continuously rising.

25

,--------

Blldding ofSleel SheHf - Europeall

~igll

Rt..'Co"''''elfdaliolrS

----------------~rlection

fonn ... ca,h prci.iI"ion!> of the _ a......:.. ~ments of the tum.

Load

The form of g,ometric Ix .. imply ddincd. a., the foml teriou!> 10 the .. trength ill. !>Irength unoer .... un es from lime to • independent of the mea~urcmenl.. tll mean!> of 3\ aiding

or "o",d,", .•"

Dcncclion Load

c=o

Load

l\iftP;:&ct)':t

PIl~IIJ1)Cfroct ~'-o-?"i

0 __-,W;.!!b

In de ..ign strength }:l.'Ollletry before

(c:..)_.Jl'----:w;Eb

rigun' 2.S: The em.oct or geometric unperrl.octions ror. (II) !>Illip-through mstllbllllY 10 II perrcct

~Iructurc.

(b)-

tor

tI.

and the

I

I

'''''''''''''1

(d) blrurcallon Insl.lIblhtlcs In II perrect structure

2.4

Accounting for imperfections in des ign against bu ckling

2.4. 1

T he efTl.'t:1 or imperret:cionll on buckling llirengch

Mo"t shell buckling problems di'iplllY unstnble post-buckling puth:i wilh rc:ipcct to somc pos!>iblc imperfection paltems. so the em.'Ct of gl.'Ollletric imperfections lIlust be understood lind quantified. This !>itumion has led to the developmcnt of imperfcction sensitivity theories. particularly in relation to shells thai di.,p[ay compound unstable bifurcation .... "iuch a.ionll of the bnsis of the O"r... the clastic criticul buckling stress of the perfect ~hcll. UnfonunalcJy. for blfur· catlng shells the traditional definitIon of Ojr omitted the effects of geometric nonlinearity in thc

respect to some possible J~t be understood and quantified. )' theories. particularly in relmion axially compressed cylinder and Ie reduction in the 10.1d carrying

prcbuckling p.'lth. whilst for shell .. exhibIting snap·through the delinition of O'r r ncce!>sllfily includ,-"lime shell nplc. a cylinder under axial com· om. whilst the same shell under 1aK'T" sources of imperfection may ely lillie allention from research·

III the fonn and amplitude of the I weaken. the ~tructurc. and \ cry IJSth abo\'c that aloSOCiated with lin a .... ide mnge of difTercnt im·

The imperfection senslthil) flletor a ln EN 1993·1.{;

Although the value of a has traditionally been related to the lower of the two potential perfect shell bueLling stresses (elastic critical bifurclltion buckling strc!)!, and snap-through buckling stress). thIS prt.'SCnll> a logical difficulty when a shell can easily pass from one fonn of buckling to the other. Since lillap-throu!!h buckling in a perfcct .. hell is caused only by geometric nonlinearity. the role of geol1'lCtrie nonlinearity falls within the (\'aluatton of O'('r for these shells. BUI for shelilo in which bifureation is the reference buckling mode of the perff.'Ct structure. the efTL'Ct of geometric nonlin· earity is placed within the \alue of a. Mon.'O'er, the linear bifurcation buckling lo.1d can alwaylo be detennined if compressive strcssc-s exist within the shell. but a lollap-through load only occurs in .. p..'Cial geometries under lopccial load cases. TIlus. for a completely geneml description. it i\> m~t stmightforward to place all cfTccl.s of ehllllgc of gcometry into the fuctor a. nnd to usc only the lin· ear elastic bifurcation load liS the reference ... alue for detemlining O'er This delinition has been adopted within EN 1993-1-6. Itnd i~ follow,-"Calc structu(C\. The clastic imperfection factor a is selected so thut a high percentage ofexpcrimentul test results (typically 9511 0) ha\e u buckling load abo\c the valuL'S predicted by the dL"Sign method.

It should be rcmemben..'d Ih3t Yoherc the elaslic imperfection factor a is used in connection with bifurcation buckling. il accounts for two separate phenomrma: nonlinearity in thc prcbuck ling p.1th and geomelric imperfections. llL-elluse two :.cramle elTeclS are invohed in this single parumeter. it is not easy to exploit numerical an3ly!lCS in design if thc) takc only one of the:.c elTecls illto account. Thus. the predictions of a linear bifurcution a nlllysis lhat includes geometric imperfections (LBIA). and those of a geometrically nonlinear analysis of the perfL'Ct clastic shell (GNA). arc difficult to assimilate mto the dL"Sign process. Apan from the UloC of a numerical LA analysis to obtain an L"quivlliem stress stllte corresponding to that found III hand calcu lutions, the EUroPCllll stllndard EI\ 1993·1-6 pennits only tYoO procedures for the usc of numerical allulysilo. In Ihe first. the lmear clastiC bifurcation entiealload and the plastic Illnil load may be detcrmined numerically. with the rcnlainder of the e\aluation following the procedures of hand calculation. In the second. a full materially and geometrically nonlinear analysis that explicitly inc1udL":o geometric imperfL'i:tions in the gL'Ometrical definition of the stnlcturc may be us..:d to assess the eharueteristie load directly. lIowe\er, Yo here the laller procedure is used. it is 3 requirement thaI the linear clastic bifureation erilicallood. the plastic reference load. and the geometrically and materiully nonlinear fllilure load of the perfect shell should 1111 be detenllinL'd as pan of the cvaluation procedure. so Ih31 the user may be eertuin of the slcnderness of the system and th3tthe effectivc reduclion factor is appropriately low. 2.4,4

Impcrfeclionsllnd tolerances

Because the ~trcngths of shells display a high sen... itivity 10 geometric imperfcetions. any dL"Sign method should c1eurly identify the maximum allowable an'fJlitude or imperfection associa ted with the design strength. The amplnudes so identified can then. in theory. be translated inlo tolerances that should be mct afterwards m the fabrication process. Tolerances define the limning Illllximum ... alues of measurcmC1lts Ihut can be made rclllli\cly ealoily on a final constructed shell. The proce...s of devising methods of measurement that renecl Ihe rich range of geometric imperfcctions thaI may be considered in the design process is \ery for from straightforward. The Eurocode (EN 1993-1-6) defines a 13rgc sct of dilTercnttolerancc mea ~urcs, many of which are conducted by u'sing a stick placL'd against the shell \\ all. Other new fonns of tolerance measurement deservc in\'cstigntion. and there is an urgent need to find improved methods ofrclati ng the imperfection fonns thai arc chosen in design t:alcululions 10 lolemncc measurements on Ihe conlolructed shell. The tolerances required in pmetlcal construction must e\ idently be chosen to relate to common fabrication and manufacturing proccsses. It should therefore be IlOK'd that care must be cI(ercised when

28

de .. ign methods :lre the le.. t "]lI!clmcn~ ing proce". The form and III 'm,,"""d~ im]ll!rfcction Cct elru.tic shell (GNA). arc dim-

k'Tlt \tress state eOlTCllponding to 1-6 pennit:. only two procedun.... rcation critical load lind the plu.!!of the c\'alualion following the d geometrically nonlinear unalylrical definition of the lItnleturc here thc IUller procedure is ullcd. IC plastic referenec lo.1d. nnd the !hell should all be detennincd as hc slcnderness of the sy:.tcm nnd

ytric impcrfL"Ctions. any design !~f Imperfection a.ssocilllcd with be translated IOtO toler,IOCCurcmcnt that renect the rich ~Cioign process is \cry fur from of diffcrent tolemnce mC3!>UTCS, !:hell wall. Othcr new fonns of nl'Cd 10 find impro\l-d methods bons 10 tolerance mca!>urelllenb

.hoscn to relate 10 common fab'lut care must be cxerci!tL'(] when

29

BUl'klmg o/Sleel S}",'ll\ - Em'OfH!llll Ocsign RL'l'OI111f/(mdafitJIu

(11)"":1~~~~;::~1 'I

[C(,S Europo.:an r

p lastic col lapse

\

I ...,linger, \1 IOd York.

clastic buckling (pcrrt..'Ct s hdl)

It'

e lastic buck ling (imperfect s hell)

c urve

~:::;t=====:;j:======:plastic collapse

e la ... tic-plaslic buck li ng

___!S~lldeniess

19]6)

J..

daslic buckling

Figure 2.6: TYPical cu'""c orbuckhng load factor HI ogU,"51slcndcrUl.~ A for a shell Where the material ~tress-strnin cUl'\oe cannot be simply reprcscnlcd by lin ideal elastic-plaslic behaviour, shell buckling design recommendations s till gencrully limit the number of parumeters charactcri!>ing the material to only two, so thut the chlstie modulu!> E lmd a "proof we!>!;"'!;, are used ith \ ery simi lar design infonnalion. For consistency, the proor stress should be taken as the stn."SS at which a plastic strain of 0.2 0 0 has dc\elopcd in a lensilc test. Funher mfonnation on nonlincar material behaviour and its trealment in the design process may be found in Section 5.3. VI

2.6

Summary or behaviour in relation to shell buckling design

All rational !thcll buckling design procedures should be b.1scd on: • thc stability behaviour or the perfect !>hell: • Ihe imperfection ~n!>ilivity appropriate to the geometry. lo.1ding and bound:lry conditions con!'>idered: • the tolerunees thac are imposed on imperrcctions: • the interaction between clastic buckling and yielding. These rUetors arc alllilkell illlo account in the design procedure thut is descrilx.'d in Chupters 8 and 9 or these RI..'Commendutions. based on the procedures or EN 1993-1-6.

2.7

8ibliography for C hapter 2

Brush. D.O. lind Ahnroth. B.D . (1976) "Buckling of Bal'h. Plntl..'S and Shells", McGruw-llili. NcVl

Yo",

Budian~ky, B. (1976) (Fd.) Bl/cHillg u/Slnlcillres, Proc .. ItJfAM SYlllpo!lium 1974. lIanard Un i\crsity. Springcr. NcVl York. Bushncll. D. (1985) Compllierird hllckJi fIg O/w(I·.ri!. o/.vI1('l/s. Dordn.'Cht. Calladinc. C.R. (1983) Tlu'oryo/Shdl SIn/cIl/res. Cambridge Unj\'crsity Pre!tS. Cmnbridge. Donnell. L.H. (1976) Beams. Plafe.v and Shel/s. McGruw 11111. New York.

30

HUgge. \\

Koiter. W.T. sity. 'etherland~. Pctc.....cn. C (lQIC!) .\'1 'tCh\\ cig. kng. J,(j. and Roller. J ThoIHfhOn.l\tT and England. Timo:o.hcnko. S.P and York Yamaki ..... (IQS·1) Et Science Puhh h..-r;. \m Teng. J.G. (1996) "8u, \'iews. Vol. 4Q, "0. 4

Shell bllck/illg behaviour Qnd design co",:eplf

ECCS (1988) European Rccommcndallons ror Stccl Construction: Buckling or Shells. 4th cdllion. European Con\cntion rorCon.\truetional Stcch\ork.Brussels. b~linger. M. and Geier. 13. (1975) Poslhm:kJillg Be/wI·jour of SrnlClllrl'S. ClSM. Springer. Ncw

·errect shdl)

;Iing (imperfcct shell)

York. nOggc, W. (1973) Srn'!>'S(!S m Shcll.\. 2nd cdn. Springer.Verlag. Bcrlin (1st Edition in Gcntmn.

1936) __ ~s::'''':,;:demcss A. )

Koiler. W.T. (1945) "On the SllIbility or Ha.!>l1c 1:":luilibrium". (in Dutch) PhD Thesis. Delft Uni\crsity. Ncthcrlands. Petersen. C. (1982) Sralik I4IIlI SllIhJli/llI/ tier BaIlAolI!>lrllcliollell. 2.... cdition. Vicweg. Bmunschwcig.

M1 by an ideal clasllc-plastic belimit the number or pammctl'fS f: and a "proor strcss",h. arc used I'C!;5 should be takcn as the stress :urthcr mronnation on nonlinear )lind in Section 5.3.

ling design

g and boundary condi tions con-

is described in Chapters !:! lind 9 ).

jnd Shells". McGru .... -11i11. New

r.>mposium 1974. lIarrard Un i-

lecht.

r..ity Press. Cambridge.

"orl.:.

Teng. lG. and ROller. J.M. (eds) (2004) BlId/illK of Thill Mewl Shells. Spon. London. Thompson. J.M.T. and IIunt. G.W. (l-dS) (19M3) Co/fap.f{·. Cambridge Unhersity Press. Cambridge. England. Timoshenko. S.P. and Gerc. J.M. (1961) Tlu.'Ory' of £IcH/ic Stahi/il)'. 2nd edn. McGmw-11i11. New York. Yamaki. N. (1984) £I(I5lic Siabilill' ofCimllar Cylilldril'lll SIIt·lls. North Iiolland. Elsevier Applied Scienec Publishers. Amsterdam. Teng. lG. ( 1996) "Buckling or thin lohclls: reccnt IIdvanees lind trends". Applied Meehllnies Reo \,ie .... s. Vol. 49. No.4 April. pp 263-274.

Scope. conW!l/IiotlS. d(:jiniticmr. ""ils..'i)'mhol'i and .~ign rom-elllions

----------~~--

3

Scope, conventions, definitions, units, symbols and sign conventions

3.1

Scope of EN 1993-1-6

3.1.1

General

The scope of EN 1993-1·6 is wider than these Recommendations in that it deals with a tOlal of four different limit states: plastic limit. cyclic plasticity. buckling and fatigue. The following statement of scope therefore extracts only the restriellons on scope that are relevant to the buckling resistancc ofshdl struetun.'s.

3.1.2 (I) ~hell

(2)

SeOI"! limitations in EN 1993-1-6 Sectio n t.2 EN 1993-1-6 gi\'es ba;ic design rules for s teel shell Slructurcs that ha\c Ihe fonn ofa thm of revolution. EN 1993-1-6 i)o intended for usc in conjunction with EN 1993-1-1. EN 1993-1-3. EN 1993-1-4.

EN 1993·1·9 and the relevant application parts of EN 1993. "'hich include: Pan 3.1 Pon 3.2 Pan 4.1 Pan 4.2 Pun 4.3

(3) of:

for lowers and masts: for chimneys: for si los: for tunks: for pipelines.

TIlis Standard is concenll,:d with the requirements for design against the ultimate limit states plastic limit; cyclic plas ticity; buckling; fatigue.

(4)

EN 1993-1-6 defines the churnctenst1c and design values of the resistancc of the structure.

(5) O\'er.dl equilibrium of the structure (sliding. uplifting. overturning) is not included 10 EN 1993-1 -6. but is trcmcd in EN 1993- 1- 1. Special considerations for specific applications arc included in the relevant application p:arts of EN 1993 . (6) The provisions in EN 1993-1-6 apply to :axisymmetric shells and associated circular or annu· lar plates and to beam section rings and winger stiffeners where they fonn part of the complctc structure. Gcneral procedures for computer calculations of all shell fomls are coveR:'tl. Detailed expR."Ssions for the hand calculation of unstiffcned cylinders and cones are given in the Annexes. (7) Cylindrical and conical pancls are not explicitly covered in EN 1993·1·6. However. the pro\'isions can be applic:able if the appropriate boundary conditions are duly taken into account.

))

..

~---.-...

Budding o(St('4!J SheJJ~· EUroP('OII De.figll Recomm('"datiOllf

(8) I:N 1993·1-{; i~ intended for applicatIOn to structural engineering teel J.hell structures. How. e\er. it!> pro\isions can be applied to other metallic shells pro\ided that the appropriate material propcniCl! arc duly taken into account.

3.2.2

(9) The provisions of EN 1993-1·6 arc mtended to be applied within the temperature runge defined in the rcle"anl EN' 1993 application pans. The maximum temperature is rc.stricted so thnt the influence ofen.oep can be neglccted if high temperature ere!.'P elTects are not eO\iercd by the relevant application pan.

tlk! fcn.xJ by the ~I

(10) The provisions in EN 1993-1-6 apply to stnlcturcs that snli~fy the brittle fracture provisions given in EN 199]- 1- 10. (II) The provisions of [:N 1993-1.6 apply 10 slIUetuml de~ign under uctions that can be tremed as quasi·!oIutic In nalure. (12) In EN 1993·1-6. it is assumed that both wind loadmg and bulk solids flow can, in general. be tn.~.llcd as quasi-Malic aelions.

(13) Dynamic elTects should be Inken into account according to the relevant apphcnlion pan of EN' 1993. including the consequences for fatigue. Ilowe\cr, the stress resultants arising from dynamIc behaviour arc treated 10 this pan as quasi.sUltic. (13) The proviSIons in this Stand3rd apply to structures that are constructed in aceord:mce with EN'1090-2.

A calculation that

c\ aIUali("'":;'":~·~~:1

3.3 3.3.1

Structural

3.3.1,1 shell A 'itruclure or a ittru..:lul

3.3.1.2 shell or re\ olu A ~hdl" ~ gcomctri generator Jjn~ around 11

3.3.1.3 conlplett nh.

A ~ he Jl comJlO!-Cd of 3 r (14) ThIs Standard docs not cover the aspects of leakage of stored contents.

3.3.1.4 'ihtll (15) This Standuro is intended for application to Slructures within thc followlOg limits: design IIlctaltemperuturcs within the runge -50°C 10 +300 c: radius to thickness ratios within Ihe range 20 to 5000. For the elT!.'Ct of elevated tempemtures in the runge 10000C to 300"C on matcrial propenics such as the modulus orelustieity and proportional limit. sec EN 1993- 1-2.

'i~ment

A ~h ell ur re\ nlutilln I cylinder. conical fru~tul 3.3.1.5 shtlllJanel ~hdl

NOTE: It should be notOO thal the stress design rules of this stllndnrd may be rather conservative if lIppliOOto some geometries and 101lding conditions for rclati\icly thid•• walled shells.

A n im:ollll'icle

3.2

3.3. 1.6 middle !turrllc,

Terminology used in these Recommendations

Different methodologies or analysis arc referred to in these RecommendatIons tb follo,,"s_ 3.2. 1

flllnd u lculation

A calculation that uses calculations perfomlcd on paper or in computer spreadshccts to evaluate both the stress SUIte in the shell cau'lCd by the loading and the rcsisUlnee of the structure to those loads. It is based entirely on the usc of ex is ling fommlns.

ax i ~ th rou~ h

or

11...., than 2

Th~

surface thai Ii~... III \\ here the :.hell IS "'illTe a, the middle surtace I ana l )'~i • and (".In be ecccntrieillCll Ihal Ina) 1 3.3.1.7 junction

The IlIle at " hir.:h hHI I line o r nltachmcni of a r

34

Scope. txmlY!nliolls. dejilliliollS. IInits. symbols UlKi sign cOIIIY!ntkm.f

ering stccl shell structures. Ilow. idcd that the appropriate material

within the temperature runge c.lempcruture is restricted so that the :ts are nol co\-cn..-O by thc relevant

'ify the briulc fracture provisions

3.2.2

Numerical l1nal)'sis

A calculation tllat USL'S a linite clement or similar computcr calculation 10 c.lclemlinc 'lOme or all of the evaluations n..-quircd in deilign: the stress stille eauscc.l by the loads. thc resistancc to them offen..--d by Ihe structure. and the dcfonnations of thc structure.

3.3

Definitions from E

3.3.1

Structural (orlm lind I,:eollleiry

3.3.1.1

shell

1

1993-1-6 Section 1.3

kk"'T actions that can be treated as A .. tructure or a structural component fonned from a cuned thin plate.

ilL. solidl> flo .... can. in general. be

3.3.1.2 shell orre\olutlon

, the reb ant appheallon pan of stn:ss resultanls arising from dy.

A shell \\ hose g,."omctric fonn is dcfillCd by a middle surface that is fonned by rotating a meridIonal generator line around a single axis !.hrough 2n r.ldians. The shell can be of any length. 3.3.1.3 conlplelc alls) mille' ric shell

: constructed in accordance: with

A shell compo!.Cc.I of a numhcr of pans, eaeh of .... hieh is a shell of revolution. contents. 3.3.1.4 shell scgllu'nl

the following limits: A shell of revolution in Ihe fonn of n dcfincd shell geometry with a constant wall thickness: a eyhnder. conical frustum. spherical fru!>tum, nnnulllr plate. toroidal knuckle or other form . Ie on material propcnies sllch as

3.3.1.5 shcll pan el

ard may be mthcr conservative if 1cL.-\\alled shells.

An incomplete shell ofre\'olution: the shell foml is dcfin(.-d by a rotlltion of the generator about the nis through less than 2n rudians. 3.3.1.6 middlc su rraet' llte !>urface that lies mid\\ay bct..... ccn the insidc and outside surfaces of the shell at e\cry point. \\ to cvulualc ;i~tancc of the structure 10 those

3.3.1.7 junction Thc line at which two or more shell segmenb 11l(.'Ct : it can include a stifTener. The circumfcrential linl! of attachment of Il ring stiffener to the shell may be Irentcd a.s a junction.

35

.

----_.

-.

Buckli/lg ofSIC.'t'1 Shel/s - Ellropea/l

De.~ig/l

RecomltlelUk,tiom. e.-~ die.-

3.3.1.8 stringersliffenef

3.3.2.3

A locul stiffening member that follows the meridiun of the shell. rcpn..'SCnting a generator of the shell of fC\·01ution. It is provided to increase the stability. or to assist \~ ith the introduction of local loads. It is not intended to pro\ide a primary resistance to bending enecls cuu!tOO by tnm"verse lo.1ds.

leading tll a exhau!>lcd. 3.3.2",& bucklinp,

3.3.1.9 rib A local member thm provides a primary load eanying palh for bending down the meridian of the shell. representing u genemtor or the shell of rc\ olution. It is used to transfer or distribute transverse loads by bending.

The ultimate hlllil cornprc..,ion and or .. uppon til\: appli..:d 3.3.2.5 rlttigue

3.3.1.10

ringslirrcncr The uhimale \;;m;;" '''~

A local MilTening member that passes lIround the eircumfel\.'11Ce of the shell ofrevolutiol1 ut u given point on the mcridllln. It is nommlly 1I!>!>umed to have no stiffness for defomUltions out of its 0\\ n plane (meridional di'piacements of the !>hell) but is !>tifT for defonnations in the pillne of the ring. It

3.3.3

is provided to inercase the stubility or 10 introduce loculloulh uCling in the plane of the ring.

3.3.3.1 a};ialload

3.3.1.11

Actions

bast fing

A stnlcturnl mcmber Ihat passes around the circumference of lhe shcll of revolution at the base and pro\ides a meuns of attachment of the shell to 11 foundation or other structural member. It is nceded 10

ensure thut the 113~umed boundary condition~ lire IIchie"'cd in practice.

3.3.1.12

ring bcum Of ring girdcr

A circumferential stifTener tlmt has bending stifTncss und ~trcngth bolh in the plane of Ihe shell circulur seclion and nonnallO Ihut plnne. It is a primary load cunying structural member. provided for the distribution oflocalloads into the shell. 3.3.2

3.3.3_' internal Component of th..: can \ary in ooth

Limit state'!

NOTE: For 11 shell. there ure five situations defined as ultimate limit stnte where the structure is considered to have reached itl> lo.1d bearing capacity.

Component of th..: clln \ Ilry in both the

3.3.2.1

3.3.3.S

plastic limit

The ultimllle limit state where the stnleture dc .... elops zones of yielding in a p:lIIcm such thut its ability 10 resisl increased loading is deemed 10 be exhausted. II is closely related to a small deflection theory plastic limit load or plastic collapse mechanism. 3.3.2.2 tensile rUI)1urc The ultimate limit stale whcre the shell plule experiences gross section failure due to tel13ion.

36

h~ dro\Utic

3.3.3.6 lull f,;''';"'~ Component or the prc.... ure and actlOg cont;lincd within the

Scope. cOIII'l'miollf, th'fi"ilio".f, IIl1irs, symbols and sigll col/I·ell,ioll.'

33.23

representing a genemtor or the ~t .... ith the introduction of local ng effects euultCd by tmnsversc

plasticity

The ultimate limit stme where repeated yielding is caused by cycles or loading and unlo.'lding. leading to a low eyclc fatiguc railure where the energy absorption capacity or the muteriu! is exhausted. 33.2.4

lding down the meridiun or the I tran~rer or distribute transverse

c~clie

bueklin~

lbc ultimate limit state where the slnleture suddenly loses its stability under membmne compression and/or shear. It leads either to large displaeell1ellL'i or to the structure being unable \0 suppon the applil.:d loads. 33.2.5 fatigue The ultimate limit state \\'herc many cycles or load1Tlg cause crocks to

he shell of re\ olutioll at a gi\ en ror deformations out or its OWII lIion~ In the plane of the ring. It in thl! pla~ or the ring.

33.3

dC\l,~lop

in the shell plate.

Actions

333.1 axial load rXlcmally applied 10.1din8 U>.;lin8 in Ihe: axial direclion.

ell of re\'olution at the hase and structural member. It is needed lice.

333.2 radial lond E~tcmally

applied loading octlllg 1I0nl1alio the surraee or a cylindrical shell.

3.3.3.3 internal pressure both in the plane or the shell ng structural member. provid(.:d

Componenl oflhe surrace loading aeling non11all0 the shell in the outwurd direction. Its magnitude can vary in bolh the meridional and circurnrerenll3l directions (e.g. under solids loading in a silo). 3.3.3,4 external pressure

mit state where the structure is

Component of the surface looding acting nomml to the shell ill the inward direction. Its magnitude can \lIry in both the meridional and circumrerential directions (e.g. under wind). 3.3.3.5 h)drosfatic pressure

Iding in II pattern !ouch thut its t is closely related to a small

Prc!osure \'lIrying linearly with the venieal coordmate of the shell or re\'olulion. 3.33.6 ,ull friclion load

m railure due to tension.

Component of the lourfacc loodillg ucting on the shell wall due to friction conncctl:d with internal pressure and acting in the direction or motion or the contained mnterial (e.g. when solids arc contained within the shell).

37

- - - - ._-..

3.3.3.7 locolload

the mlddic surfaec 01 nnl relate tu \ Ihr...luID

roint applied force or distributed luad IICling on a limited pan of the circumference of the shell and o\er u limited height. 3.3.3.8 plltch 100td

3.3.4.5

An

anal~!>l~

bal>i:d llll

Imear clastic matenal al.a.:ounl... fully lilr, chcc.:k. i.. indud,.;d al

Local distributed load acting nonnal to the shell. 3.3.3.9 suctiun

303A.6 ma'criall) no Unifonn net extemul pre!>!>ure due to the reduced internal prcs.sure in under 'Aind action. 3.3.3.10

II

ahell \\,;th openings or \enls

An anal)"!>b bibcd on ..mall ddla:tlOn as 'n

partia l \sC bum 3.3.4.7 geometricalh

Unifoml net external pressure due to the remo\al of stored liquids or solids from" ithin a container that i!> inadequutely \ ented. 3.3.3. 11

Iherma i llclion

Tempernture variation citlll~r do\\ n the shell meridian, or around the :.;hcll circumfcrcnce or thruu~h the !ohellthiekness. J.3A

T) pes

of a na l) sis

303.4. 1 glubul un:ll)sIS An analysi~ that includes the complete structure. mlher than mdividual struetuml pans trcmed scpamtely. 3.3.4.2 memb ral ne theUf) a nlll)sis An analysis Ihot predicls the behaviour of a thin-walled shell structure under di~tribuled 10:1ds by assumin.: that only membrane forces stltisfy eqUIlibrium with the e~ternolloods. 3.3.4.3 li nellr elastic shell ual)sis (LA) An anoly!>i!> that predicts lhe behaviour of a thin-\\ullcd ahcll structure on the basi!> of the small deflection linear ela.... tie shell bending theory, related to the perfl..'Ct geometry of the middle surface oflhe shell. 3.3.4.4 linea r elastic bifurcatio n (cigen\lIlue) ana lysis (LOA) An analysis that evulumes the linear bifurcation eigenvalue for a thin-wnlled shell structure on the basis of the amnII deflL'Ction lineur elastic shell bending thcory. rclated to Ihe perfect gcometry of

An anal) I t"Ia.-.cd on nonlin.:ar lars,.; dd1...·d A bifurcation ci!!clI\ul 3.3.4.8

I:.co m c tricall~

An anal)" ... is \.\lIh [ml"' primarily of membnme stresses. but under some conditions bending Slrcsscs may also be required to achiC\ic equilibrium.

Thc nominal m'tnt'~

3.3.5.2 S4..'Condar)' stresses Stresses inducl..-d by imemal eomp..1tibility or by compatibility with the boundary conditions. associated with impo!oed loading or imposed displacements (Iemperuture. prcslrc!>sing. settlement. shrinkage). These !>trcsscs arc not required to achie..e equilibrium bct,,-cen an internal stress state lind Ihe e~lernal loading.

TIle value of ~trc!>~ . a buckling limit·

33.6.9 The calegory of

33.6

Sl,ccial dcnnilions for buckling clilculations

3.4 3.3.6.1 criliclil buckling resislance 3.4.1 The smallest bifurcation or limit lo..1d detennined assuming Ihe idealised condilions of clastic mmerial behaviour. perfect geometry. perfcet load applicalion. perfect .!.uppon. nULlcrial isotropy and absence of rcsiduill stresses (LBA unulysis).

33.6.2 critical

bucklln~

slress

The nominal membrane stress associated with the erilical buckling resistance.

40

Bucklin~

3.4.1.1 blfurcallon A load at \\ihich

a new fonn '::~:;~~ slope of the 14 which is followed be librium path may in' modal fonn (hg. 2.:

Scope. COIII'I.!flliOlU·, dejilliliolls, ImilS, symbols and sign com>ellfiollS

--------------~-3.3.6.3 plastic r eferen ce r('Slstance

leriallaw

Shell geometry

apphcable

perfect

TIle plastic limit load, detenninL-d assuming the idcalisL-d conditions or small displacements. rigidplastic material behaviour. perfect gL-ometry. perfect load Ilpplication, perfect support and material isotropy (modclled usi ng MNA annlysis).

~

perfcct

" "

pcrrL'Ct

3.3,6.4 characteristic buckling resistance

perfect

4linear

pcrf«1

The Jo.'Id associated with buckling in the presence of inelastic matcrial behaviour. changes of geometry due to loads. the gL'Ometrieal nnd structural imperfections thai ;are inevitable in practical construction, and follower load eITeclS.

-linear

pi.'ffL'Ct

"

imperfect

-linear

Imperfect

3.3.6.5 characteristic buckling st ress The nominal mcmbr.:ane stress associated with the characteristic buckling resistance. 3.3.6.6 design buckling r('5lstance Thc design value of the buckling load. oblllincd by dividing the charncteristic buckling resistance by the partial factor for resistance. 3.3.6.7 dcsign buckling slrt.'Ss

19. This conlli,!,!!> primarily of y ;also be required to achie ... c

11le nominal membrane stress asM>Ciuled with the design buckling resistance. 3.3.6.8 kcy \'alue of till' siren

ith the boundary conditions. nure. pn.·Mre~ing. settlement. etV.een an mtemal stre,';!. state

The value of stress in a non·unifonn stress field that is u!>Cd to characterise the strcss mab"l1itudcs in a buckling limit stllte assessment. 3.3.6.9 fabrication tolerance (IUa lily class

TIIC catcgory of fabricntion tolerance requirements thllt is assumed in design. see Chapler 6.

;lcalised

condition~ of clastic

eet support. matenal isotropy

3.4

Additional definitions needed in (hese Recommendations

3.4.1

Budding hcha\ iour and phenomena

3.4.1.1

bifurcalion load

A load at which the static equilibrium defonnation mode of the structure abruptly begins to take on new form superimpost.-d on the previous dcfonllation mode. This usually leads to a change in the slope of the lo.'Id-displacement relationship, The dcfommlioll modc of the primnry equilibrium path which is followed before the bifurcation point may continue to be fo llowed. or thc secondary L'quilibrium path may inslcud be fo llowt.-d. involving the development of displacements in a diITercnt modal foml (Fig. 2.2). Following bifurcation. the stntcturc mny support cither incrensing or de-

:1

iSllmcc.

creasing loads whcn it follows thc MXondnry pmh (Fig. 2.3). For a fuller description. sec &.'Clion

2.2. 3.4.1.2 snllp-Ihrough or Ilmilioad A load at \\, hieh the changing geometry of the structure causes u condition in which incrementnl addItional dlsplacerncnts occur in the current mode of defom'l3lion \\ ithoul nny chnnge in the value oflhe appli,..d load. TIt is corre!tponds to zero tangent stiffness in that mode relati\'e 10 the applied load. The structure rcaCM'S 3 ma;timum lo..1d-carrying condition and smaller loads are carried at incT'Cnl>ing displncemcnts (Figs 2. 1 and 2.4). For a fuller description. see Section 2.2.

3.6 3.4.2

Material eonstituthe modebi

3.4.2.1 s)mmctl) in compression and tension

Forth\! 10\\ illS are

It is commonly assUlrn..'On's ratio v) and a plastic part at constant SlrCSS);_ It commonly assumes symmetry in compression and tension (sec 3.4.2.1). TIte nutteriallaw is esscntially a two-parJ.mcter model.

3.4.2.4 idrlll clastic - linur hardening This tenn is ust."cr unit circumference nunmtlto the ~ hcll; load per unit cireumference ac ting in the meridicnul direc tion: load per unit circumfercnce IIcling circumfercntially on the shell.

n/l

If con~i!otenl units nrc recorn-

rt\iS

kNlm kNmlm kllu

N

(4) lalerial in uniaxiultension mlly Poi~son' s mtlO v) and II plastic s:;ion and tenston (sec 3.4.2. 1).

ten.1I 10 uniax.ialtension may :toisson's ratio v) up 10 a yield iUrlcning modulus £ •. It com. rhe matcrialla\\ has a bilinear

kl'u tonne/m'

mm N mm'

Symbols from EN 1993-1-6 Section 1.4

(I) For the purposes of E lowing arc used. ial tcnSKm is equal to the yield no \olume change.

m kN 'm1 kN kN In

meridional stress: circumferential stress; von Miscs L"quivalent SlrcloS (can also take negative values during cyclic loading): in-plane shear stress;

Blldding olS/eei Shells - i::llrolH!tm Desig" R('Commclldd/iolls

r, .. , ra, meridional, dreumfercnlilll trnns\ crsc ... hcar stresses ussociu{(,:d with bending. (8) /I

" w p~

(9)

d L I

u.

U,

(II ) I'ropertlcs Younl!;'s I. Ion \ii'>C, r~ yield

r.

r.,.

gauge Icngth for measurement of impcrfl'Clions in circumferell liul direction: gauge length for measuremcnt of imperfections ucrms welds: limil(..-!>CS has .. Iso tx:en fC\CI"S(..-C the MIme nllcs as EN 1993- 1-6 for severnl purposes and a commentary is prodded. For modelling thc shell gcometry for analysis. the nIles are SCt OUi in Section 4.2. The dcsign values of g(.'OmctricIlJ dat3 lire sct out III &-ction 4 .3. The general rules concerning bound:try conditions arc M:t out in SL'Ction 4.4. 11IC spcciul ntk... on bud, ling-relevant bound;try condition" arc sct out in Section 4.5.

4.2

Rules of EN 1993-1-6, Section 5.2.1, on modelling of the shell geometry for analysis

(I) The shell should be repl'CM:nted by iLhould not Whe" a IXIrl of a .,·hell .HntClllre i.\ eXll'tJc:ted be subdh'ided into scp:.lmte scgtnetns for annly- frolll the II'IIOII! for design pllrposes. care IIIl1sl J;is unkss the bounffiuy conditions for each seg- he 'akell Ihal the adjlcellf .{('gmellts IIdlher (IPIllent are chosen in lIuch a way as to rcpl"CllCTlt P~I' tllle.\1x'CIC'tJ loall... 110/, prol'ide /I/I(·x/X.'cted intemctiOlls betwccn thcm in a COllllcn'uti, c re.\·lromt.l' 011 thf! wUllysed segment, manner. 'stcm with EN 1993- 1·6) arc

here they occur.

vhere. except in Ch:lptcr tI on

(4) A base ring intended to tmnl>fer locall>uppon forces inlO the ~hell ~hould nOI be sepurutL'd from the shell It suppons III an a!>scsslllent of limit 1I1ale LS3.

A INlse rillg is pcl,·ticlllor~l" impol'tOIlf ill keeping (I .dICJI bO//lJdw)' circillar, ali(I OIlt-af-roIiIltJ dl'tpllK.'CIIIl:ms al a bolllulan' N!JlIte\lrfmg~lt to It\'illl IIlf!mbmm' slresses in lite shell.

(5) Eccentricities and steps in the ~hell middle The dt:"imioll of 0 /ille of 1111'1151 III tllf! ..hell surfacc should be included m the analysis model CWI.H.',J bl' a'ftep in the mielelle .'fllr/{U:t> mdlICI'S if thcy mduce significant bending cffects as a Jignijical1f local bending. III relation 10 the e/fect rc~ult of the mcmbr.tnc !>Ire!>!> l'Cl>ultnnl!> folio\\'- of 'Well ~lIfriciti(!S on hllckling. rl.'glllllliolis ing an eccentric p:.lth. mat· he f(Jlllld ill SecliOIlS 6.3.3 and 11 .3 of IIu·sI' Recol1lmelldotioll.\.

(6) At juoctions bcl\\ccn ~hcll !>cgmenL all/dil' Ihl! rudial flijJlIl',fS of Ihe 11n«:lil)lI clt'II/('/11 (I'.g. UII elld "ill/um slilrnt'll rt'ljllir('d tu • "".\lr"illl tit'pelliis UII 'he load (or Ihel" ,\Iiqm.'ut',~ IllY! Ihat'. , rt'1t!WIfII cllUfJll''''' or SL't'liUIIS 't."JallO/ls,

Support boundary conditions should be St.'('U/IM! the .\hell is lhill, il is slIsceptible 10 wcrn.-cked to ensure that they do not cause exces- I'CI'C local belldillg if the membrane fort'e}j' ill il sive non-unifonnity of trunsmiued fon."'CS or in- arc 1/01 cO//Cel1lricaf(I' slIpportt'd by Ihl.' hollndtroduced forces that are ecccntric to the shell l try. Where Ihc bollndOl)" £."Olldilioll may pmmiddle surface. Reference shou ld be made to the ddc tI ,fli~""y ('(.'celllric SIIPIH)rt 10,. rhe memrelevant ..::N 1993 application parts for the de· hnl//(' lo,.ce ill the sltell, care sholiid be flIkelllO l'l'lIltlllte rht, l"OIISl'qlll'IICes Olll,l' re,mltillg local luilt:d applicution of this nile 10 !oilmi and tunks.

(3)

.\"'ell belldil1g. When a global numerical analysis is used. This sftlletlt('111 is dis('u,uL'(1 iI/ Ihe the boundary condition for the nonnal displace- ahOl'eoll(I). ment 11' should also be uSl:d for the circumfercntial displacement v. except where special cireum· stances make this inappropriate. (4)

4.5

McndlOllal

Mcrnhonal

JI"J'II~b

fOIallOll

, "

0

/1. 0

0

PI~

0

P.

0

.. 0

Spt.'Chd rules or EN 1993-1-6, Seclion 8.3, on buckling-relevanl boundary condi1ions

For the buckling limit state, specinl nllenlion should be paid to the boundary conditions \\hich are relcvanl to the incfement:ll displace1I",-"lIb of bU1;kling (as op(JOSl.'tI 10 pn.:-oo1;kling displacements). Examples of relevant boundary conditions are shown in Fig. 4.1, in which the COOl'S of Table 4.1 are used.

(I)

I,

Cl)lI/lIfclIl

II is I/(llIlrllllltlll tlte bolilldar), colldiliom for It sllell .,hollid be thollsltl of as IlIIcitollgellble Ix.. IlI'('f!II differl'III fK'rt.i of the sll"IIclIlral r(-'spolls(-', 1/0II"el't.'r, bet.·mue Ihe b/lckll/lg .wrc/lglh If .wm/l'timc..f INlrliclllarly sl'tlSitiw: to 'he fIXifl' tlf ,Ill! hollllt/llries. if is particlliarly importttlll Ilwl the.(e condWolIS .iltould be asseS.fed ill a COli· sl'I,'mil"l'mOl/ller. Comml'llts all tilt' "C()(/ijit'(/'" bolll/dllr)' cO/u/ilioll ("(IlIIbiIlOtioll.f lI.fed i ll Fig. 4. 1 olld (/(1;11(.'-(1 in Toh/e 4.1 aN! gil'ClI ill 4.3 (I) . /;"lIrlher disClIssioll of Ihe boundary c."(Il1diliulls

11"'0

PI'" 0

"",0

/'_",0

F"""",-, Ilo..'C

" nunnal 10

(4)).

I

;s gil't.'l1 ill Seclion 4.6.

lhe

bullhe 1.1Ul., In

l"t'rl' shorl, lit" 111'0 boll/It/un. ncrt (III cqa'i thai ('IIl/lIr~.~ ~/h of ,Ite shclI, tllder IIt('l'e nuliollal re.flrui"t UI thc elld r/ft'd Oil the mcmbrw/{' ,\II'cH /wll. 1I001'(!1't'r, Iltlf does III}/ fry' ,ftre.ujidd a.I.W)('i(lf('d lI"illl ~fflll',m Ihl'.I(' rotlliioll(li rt'Ion·Jforlhir ""·(",Il/Iim/.

BIIt.'klilfg u/Slf!el She/!!,' - Elllvpemf Design Recomllft!lfdflllmfS

Although the

Ben

OO f

Bor BC lr

a) tank withoutllochors

aboH~

clastic cnilclil \tre~\ i~ truc for bolh I siclll formulas lor I tlmt arc obtained

c) tank with anchors

b) silo withoutaochOOi

___ .L... __

lu\,

___ -1. __

I

Bel

KOf

Bcn _ _ _ J.. _ _ _

nClr

---+---

..nIN fiwn buill ....

d) open tank .... Ith anc hors "'I~ure

c) iabonuory experiment

4. 1: Schematic c:'(ampics of boundary

conditlon.~

I) section ofiotlg ring. stiffened cylinder fOl'" the huck ling limll ~In l c

4.6

EXlcnded commentary on boundary conditions

4.6. 1

Ge neral

Shell struCIUrI.'S prescnt a particular challcnge in the identification of conscrvative boundary condi· tions. Real boundary conditions on 1111 structures are never ei ther ideally free nor ideally fixed. so these 1\\0 simplifications are nlways intended to be chosen in a manner that is eithcr conservative or unimportant to thc design. 4.6.2

Historical background

In the carly dc\'clopmcnt of thc analysis of the buckling of shells. it was difficult 10 produce IIlgebrait buckling solutions for conditions in which the prebuckling stress slate was anything othcr than a unifonn mcmbranc stress stale (sec, for c:'(amplc Tirnoshenko and Gcre. 1961; FIOgge, 1973). The boundary conditions required to achieve purely membrane pre-buckling stress stale nre ralher special. and these usually involve no restrninl at thc boundlry against displacements nomlal to the shell surfnce (e.g. rudiul di ~p laccments at the end of a cylinder. nonnal displacements at the cnd of a cone). Such boundary conditions are gencrally unrealistic. bUI they do produce simple prebuckling stress states. By contrast. the buckling modes generally require radial constraint of the displaceIllCnlS at the ends ora bucklcd 70ne (otherwise the buckling strength is very low). so in the historical de\elopment of this field . different boundary conditions wcre adopted for the incremenlnl displacements in the bifureation modc from thc boundary conditions used in the prebuckling stress analysis. The fonnulas resulting from this methodology nrc referred to as the "classical" expressions.

S2

4.6.3

The eITects

Fixity genemlly l>tresscs may be culculation. Thu~ it buckling and b",'kh~ Many commcrcial value .~':~:':,.::~,::~!'~ of thc..c ~ign n.....il>tance is sumed boundar)' ling anal)':o.C\ arc nllal)'),e~. it ~hould the hund "'~"'.,,,ooj calculation. This muhts to eslabli.,h Ihl

Finally. il should be: conditions for bolh a

A more dcnulllding in Section 4.7. SuitD 4.6.4

Bounda'1 l

As notcd

the COlT ",hit..t it '" hich might rcdur..'C III

rcsi~tancc.

Where a flcxurally f boundary. \\ohich le~ an LA unal)"i, (thai n.-duction in assessc elaMic analysis Will translallonnl rcslmiT

Mcx/eJlillg o/the x/,ell

Belr

:

I Bell

c) IlInk wilh anchors ___ .L...__

_

___ .J. __ 8C2r BC2r _ _ _ J... _ _ _

---+--- -

sectIOn orlong nngsulli:ncd cylinder the buckling limil Slate

Allhough the above may appellr completely hiSlOrical and no longer relevant. il is imponant 10 recognise ilS cOlllinuing place within Ihe design o f :.hell Siructures. Both hand calculation designs lind hllnd buckling checks made using stress analyses undertaken by computer usc Ihe classical elastic critical stress condition to provide u relcrence buckling resistance. This rcferem:c value i ~ Ihen modifiL-d to account for thc effects of imperfections. geometric nonlinearity and phlSlicilY. Funhermore. most of the [ileroture on shell buck ling (e.g. Yumuki. 1984) relnlCS all other findin gs to the elustic critical stress condition. so il is entirely nPI)ropriute to retain it in all hlllld cnleulalions. This is tnle for both the results of more accumte analyses and the inlerpretation of test dalu. But the classical formulas for elaslic critical stresses nre only able 10 be wrinen as Ihe very simple equnlions Ihnl are oblained because Ihey nre based on a prebuekling siress field Ihal assumes pure l11embrnne slrcsli'-"'S. which nrc often constnnt over the shell surface. Thus the elnstic criticnl stress fomlUlas used here include the misnullch of boundary conditions Ihal lie within the classical reference fonnula~.

4.6.3

The ('fleets of boundary conditions on ass('Ssed clastic buckling slr('ngths

Fixity genemlly leads to higher induced "tresses in a prebuckling LA calculntion (these higher Slresscs may be S(."Cn as detrimental), but fixity can also cause higher buckling strengths in un LBA calculation. Thus it has long lx."Cn common praeliee in shell dl..'Sign to dislingui:.h bel .... l.."Cn the prebuckling and buckling boundary condilions.

I

f eonscrval;"·e boundary condi:lcal1y free nor idcally fixed. so Icr that is eithcr conscrvllti ve or

Muny commercial FE puekages thm muy be Uq -d for bolh a Siress cn lculmion (LA) and nn eigenvalue analysis (LBA) do nOI provide any opponun ity to ehllngc the boundary conditions from one of these calculations to the other. Care should therefore be taken to ensure that the calculated design resistance is nOi sensitive to the cho ice of boundary condilions, or to guarantee Ihul the uslo umed boundnry conditions lire fu lly rcali!>Cd in the final constnlelion. Where numericlll shell buckling analyses are undenaken using the Slime bounda ry conditions for the prebuckl ing and buck ling nnnlyses, it should also be noted that these calculations will probably not provide n perfect malch to the hand calculntions bccllUSC of the change of boundnry conditions used in the fonnu lus for hand calculution. This should be kepI in mind when numerical results nre checked IIgainst classical formulas to establish their ueeumcy. Finnlly, it should be nOioo againthnt sometimes it may be unconservative to usc the same boundary conditions for both analyses.

I was difficull to produce alge~ state "as anything other than jKt Gerc. 1961; FIOgge. 1973). s of the ooundllry condition :lIld modelling it with an appropriate elllstic clemcnt.

4.7

Example: A nat botlom tank

A nat bottom tllnk is a Iypical example in which it is necessary 10 pay sp(.'ciul allcntion to the modelling of boundary conditions whcn a numcrical buckling analysis is undertaken. Onen these tanks are built w itoout anchors into a concrete foundation to restrnm \ crtical displac~ments of the bolloll1 of the wall (or with only weak anchonng for the erection condition). Thus. BC2f is a proper boundary condition for the lower edge of the cylindrical wall in u buckling analysis (LBA). But the free mcridional displacement condition 11 :1- 0 which is required in BC2f(Fig. 4.2b) cannot be used in an LA analysis of the prcbuckling condition. as it would not resist vertical forces at nil (thus "iolnting equilibrium) (Fig. 4.2a). There is no simple solution to this anonully. In pruclical tenns. thcre are two choices a\'ailablc to the analyst. First. ifthc FEM (':olllputer packthut is used permit!': it, the meridionlll displacement boundary condition should lJc ch;mgcd rrum "restruined" (BC If) for the LA stress analysis to "free" (BC21) for the LUA eigenvalue analysis. Altcm:uively, whcre thc software docs not penni! such a boundary condition change. two scpanate calculations must be carried out. In the first calculation. the design action combination under considemtion (including the panial fuctors 7F) is analysed with the boundary condition "rcstnained" (BCll). TIlis analysis delivers the meridional reaction forccsA. along the base supponed edge (Fig. 4.2a). In Ihe second calculation, thcse reaction forces nre applied as an e)(tenlul linc 10:ld, giving an equilibrium system together with the other cXlemnllo.1ds. In this second calculation. the appropriate boundllry condition Ue2f can be uS{.-d in order 10 deJi\'Cf a COlTCCt LBA result. It may be notC detail, be n.'Cogni'-l..-d i I only linear tact or spnng calculallon~ in

analysi~, and ~~~~~

is obtained. F thai must be _pplied

Mocklling o{l~ sltclf

lv.ccn the limits defined by the I be imcsligmcd as part of the Jndary conditions lead to a con:ases for ally degree of fl\.'(.'dom :onsidcratlon shoultl be given to IX it with lin appropriate clastic

ay spL'Cial allention to the modi undcrtnken. Oftcn these tanks :al displacements of the bottom Thl&. Be2f is a proper bound19 analy~is (LBA). But the free :Fig. 4.2b) CanllOl be used in an ical focces at all (thus violating

rsl. iftht;' FEM (''Ol1l pulcr pack ,dlllon should be changed from . the LBA eigenvalue analysis. condition change. 1\\0 sep.1mtc action combination under conoundary condition "restmined" g the base supported edge (Fig. an eXlcmal line load. giving on :tond calculation. the upproprif t LilA result. It may be noted id to prevent rigid body trunsll1care to a\oid constraining the

z,w a)

b)

a,V Flgu~

4.2: McridlOlllll dlspllKrl1lCfll condItions al the Iov.~ edge of a flal boilollliank Willi WIthout ancbormg. a) rrebuckhng analysis (LA). b) bud!;n& anai»)1.s (LBA)

The first of these two methods may also be affected by another typical femure of nat bonom tanks ~ the phenomenon of uplift under linear .. tress conditions (e.p;. v.hen the empty talll. is exposed to v. ind loading). In this cnsc a simple LA analysis lhal adopted the BC I f boundory condition at the lower edge would yield some tensile reaction forees .~, at the basc v.hich cannot be resisted by the baltC detail. Instead, the tan!.. base rises (uplifts) in a limiK-d region oflhe base perimctcr. It should be recognised immediately that this is a non-linear phenomenon. and is not ca! bello in th, critical. Nc\ crthck"-· below lhc O.21!. proc tangent modulm; al bt.

Mull.>rial ussumlllHNu

· ,hil fpt·(it.11 rule

I.f

Ihal plastic

our III 10llL'r _f/ress lel'el\ IluIII fhe I/lell IIII/ferial, are lXXQIIIIXlI1ied

, ill the IIJl/gent mot/llllls. Thi~ mUlt'rial .\liO'll..,.'(S I/UII' CtlU!)'f! Wng Curl' should be' wkel/ i" redlR'eJ Wllm'" to ellSllre ,htll il 'tIlhe dC_Ii/.:"

5.3

Extended cOlllmcnhlry on nonlinear m:lterial belul\'iour

53.1

Buckling allol)5e5 using an idcal cll1Slic-pJlIstic model

The assumption that is used throughout most shell buckling analyses of metals is thm the muteri,,1 Ims an ideal clastic-plastic response, with Young's modulus £ and yield stress,,~ Symmctry is ru.~Ull1ed in uniaxial tension :lIld compression (Fig. 5.1). Thc multi-axial stress yield criterion of \-on Miscs is commonly assumed, with a nonnal now m le for plastic strai ns.

slrt'fS lewd

C:Oncerllf flreU design It~illg

-a/nllaliOIlS or LA all" LBA '!If.:lures made ofmalerio/s OIll('r I carbon 51«1. " !J/ttJu/d ht! r IlIIIM ruler gi\YII hen! ami ill In! hal!'" Oil WI idhll dllstir.:'"w/erial m'ht.1I"/OlIr, .'e~ /rul th" modulll~

/It! n!dllC't:d

It is also commonly assumed that the stmins are always increasing. since the load path 10 failure under static conditions is undefined. As re\en.al of the din..'Ction of strains (under monotonically inerca~ing loads un the ioImeture) lelld~ to nn c1l1stie modulus responsc in those parb. the structUI"C may be e.,,:pccled to become more stable if thc direction of straining re\erscs. Thus. this as~umptioll is believed to lead to a conSCf"atlve treatment of the stability of a structure under most conditions,

Since most computcr Ilnalyses of elastic pllLSlic buckling adopt this simplified model, il is always necessary to take care in intCTpI"Cting the n..'l>ults of ~ueh calculations in a manner that is L'OIlscn'uli, e \\, lIh respecl to the real Stress·strain responses of the 1113terials of construction.

oal",. '" tlw 0.10. proof ,ftres.f cnuk nil,. illd...'('(I. The nile UPl1I"0.:cimule allmo,ance for the stiffnl!SS IIh"1I11w f/reH Il"I"(" is

fy ioll if giren in 5.3.3 he/Ow, Ihe O.:r' 0 proof used Uf Ihe rc[erelll.'e ')'il'ltl sllIIple three-purClllle/er nuxlel

Tension

Il'S fonl/ull"lhal

aill clln-es gil"(,11 belall' are ~ those of Illp idclllijicd

Stmin c

Compression

ry Figu rt' 5. 1: Ideal elustic-plU!olic ~tn:~!o-wll in rcsponloC

reRlperCltllre.f, bolh Yow/g '.\ e yield stress dec/in,. for IIIO,II of(/edilll' is slightll" faster for Ihon for lhe I'jeJd Slreu,

jtwfhells al dem/('tI /timId n'rog/lise Ihe I"W.'It!WIIII

if each propt'rl\' 01

'''e de.fign Where ,!lnuted lemperalllN!.f '"t' inm/l't'd, cclll/ion shollld he ,oRlI!' meWls (,..g_ austenitic and alllmillllllllJ do dupla)' r~I' l'lIll1e red/lclivlIs il/ ,IIi.\ '01/ vn these mullen lI/f1y he

Where a ITWtcrial hus a significantly diITerent stress strain cunc from this ideal model. there arc SC\ ernl altcm:llin~ ways of devising a simple conl>Cn ath'l! clastic-plustic n.-prescntution of thc IT\3terial behaviour. as indIcated In Fig. 5.2. The representation shown in Fig. 5.2a is gencrully sufe for calculations in which plasticity dominates and the diITerence betwccn the proportional hmit and the 0.2~. proofstrcss is not great (c.g. pemaps 2oo.). The representation shown in Fig. 5,2b should be used where there is II large dilTerencc bet\\,ccn thc proponionallimit aod thc 0.2". proofstrcss. the stress le\'els in the structure: significantly exeeed the proportional limit and stability elTccl!; are critical. Ne\crthclcss. it should be noted that a stability flU lure that occurs at a mean stress Just below the 0.2"/" proof strcs~ is gcnerdlly not I.:onscnutively represented by eithcr choice, as the tangent modulus at buckling is lo\\-er thun the secant valuc shown in Fig. 5.2b.

~

4-7 (2005).

~

---

..

59

Blick/illX urS/cd Shcl/:J - Ellropcall Design Recommellt/lIl;oflS

---

-----

Conventional elastic-plastic treatment

elastic-plastic treatment

roportionall1mit

o

Proportional limit

o

0,2

0,2

a) Young's modulus ....,lIh 0.2-_ proohtrcss b) reduced modulus with 0.2-, proofsu"Css ,,'IAure 5.2: Itleal clll~tie·plll"lIe representations of nonlmear stres..~~! rllln mponse 5.3.2

Buckling a llul)ses "hcre the load path ill\ohcs rncnaluflitrain direction

Where the lood path to failure is known and somc parts of the structure may experience a rc\crsal in Ihe dlreclion of Ihe strams. Iwo effects occur. The pennancnt plastic sirnins Ihal hn\c de\c\oped arc retailll.:d. leading to residual stresses that may later influencc the beha\ iour. and the local modulus dromatically increases as the Ilunerial becomes claslie (Fig. 5.3). In genentl. this situalion Icads to an increase III Ihe buckling lood. as illustrated by the diffcrence bet""":en the langent modulus tlnd double modulus Iheories of column buckling. lIowc\cr. where Ihe slrains are considerobly reduced from the plastic condition. funher account may need to be t.'1kcn of the Bauschinger effect. which causes carly panial yielding in the re .. erscd Slntin din.-ction palh and leads 10 a n.-duction in tangent modulus belo"" Ihe elaslic valuc al stresses smaller Ihnn Ihe yield siress.

5.3.3 Where materials .... ilh under !tlre!>s dl'Sign initial mngent \lIlue al caeh point in the Where the Slrcs\C" in II modulus should be rep imc.'>Iigalion. " here no single \ ulue of Slre~~ n modulus Er at the rele' rolher unifonn membral For more complicated of the buckling reS!,tan reduced \ulue COfTC'opo Ihe Commenmry abO\ 1993- 1-6 to be conS!" conditions where the g~ 5.2b is le~~ Ihan the ;, Ilo\.. ever. '" hilS! it canf com;crvali .. c treatment In a global numerical U' from one of the followl

s!~

/1.1

,,

" " " "

\

I

It "hould be noted Ihal part or Ihcse slress slm plaslic strains III Slresc;c

Sinun dm:cuon from 1II!n:

re\~I'l>l.' o"".Ni'''.~ tn uncon..cn:lIl\C

0 -5

E" mlllal 1.1n£c.. modulu~

0.2

62

I'"n,,,,,,

a=1

{

.

Thc value of hE

\lal.:riuJ a.UllniptiOlU ~.then

Forc":::G ... (5.3)

tbticity.

q

Ec

... (5.5)

For G < cSt; (the plastic plateau)

from thc proofMrcsscs til 0.0 11.. \alue of II as

(T -.

/y

... (5.6)

For &., < c"::: c.,

/. + hE(t: - G.)

... (5.4)

... (5.7)

In which h is the rotio of strain hardening modulus 10 elastic modulus and mg a direct be:.t fit to the p..1n of

c..

h t;

.~--:~:~~~---.-:;-.- .. - - - - . -

E

... (5.8)

+ (f,.

h.J

(hE)

... (5.9)

The value of hE for con\cntional ~ tructurnl carbon steels is commonly taken to be 600 MPa . The length of thc plashc plateau is \'cry "ariable. I-'or structures with one-dimcnsional stress fields alld mad.: from structural carbon sleels. - /0 G· <

t;

< -J5t;.

. .. (5.10)

bt.u in some finite element calculations. II may be appropriatc to ha\ c no nla'ccn the "lram :lte tensile stress/" oceul"> at the

and plastic Mrain.

63

----.---- .-...

Buckling ojSfl'el Shells· EIiTupeanlJa'lgn Recommcndations

Stress

a

Ramber&-Osaood CUrve

----

~

-- --------

ultimate

______ '

~t rc!.!..

0.20. proof ... tl'D~.

00.0.

The \ alu ... of EJlI'"

0.0001 0.002 Fi~ u re

fO,l

5.5: Errors in the Rnmbcrg.Qsgood representation Ilt higher Slrt'S.'i le\-els (after Rasmussen. 2(03)

To address this problem without too much complexity. Rasmussen (2003) developed the following equutions to represent the behaviour of austenitic steels at higher plastic strains.

The stress·strain curve up to the 0.20: 0 proof stn.'SS a., - o'(U is still rcpn."IiCnlcd by Eqs 5.1 to 5.4. To cover conditions at highcr strains. the curve was extcnded 10 become thc fo llowing.

. . . (5. 11 )

E,=

1+ nE,c

and 1/1 i ... dl," I ".."",,

111 _ 1+3.5[: ) Ra!ltl1u~n

(200J)

. .. (5. 12) with 'P 0,002. 0;, is the 0.2°~ proof stress. E" is the initial tangent modulus and Ep is the tangcnt modulus at the proof stress. The strain at the uitim:J.te tcnsile strcss 0'.. is 1.;..

and. whcre m.'\:c . . "'-II)'.

u

(u

o'w

E

. O.2+IMS -

rn..-cis"':;;:::.::1

A 1110r... 5.11 \\ilh th ... c II

in

II"

"

"hich the ""''''''''9

run~

64

II/

...

\fa/erial tl.ullmplioll.f

Stress

-

~------

.

ultimalC stress. 0 "

.

0.211" proof stress. 0"

0.2~.

str:lln

Strain

ultimate Simin.

Figure 5.6: Defin1t1on of m1tial tangent modulus £ ... o.r. proofstrt"is a... uit1matt' !t'"sile stress idell·

69

Bllcklillg ofSleel Shells - ellropeall Desigll RecommendaliOI/.v

uitim:IlC limit stalCS to be considcred. additional lifh'(/ tJ.v reql/iring IHlrticlilar/y cUrli,,1 defil/ibuckling-relevant geometrical tolcrunccs have to lioll bt'cllllse lite hllcklil/g ~>Irenglh of mOllY be obscned in order to keep the geomctrieal im- .~hell gr"cllll"('S is tlCllteJy .~eltfiti\·e to \"el)' perfections \\;thin specified limits. These buck- slIIolI dt!\·iariol/s froll/ the idelll geomelry. ling-rclevunt geometrical tolernncL~ urc qUllntified in Scction 8 of EN 1993- 1-6 (and SL'Ction 6.3 of these Recommcndat;ons) or in thc relc"ant EN 1993 application pans. (3) Calculation values for the de"iations of the shell surface geometry from the nom mal geomctry. as required for gL'Onletrical imperfection uss umptions (ovcrnll imperfcctions or local imperfections) for the buckling dcsign by global GMNIA nnalysis should be derhed from the specificd gcomctrical tolernneL'S. Relevant rules arc ghen in &.'Ction 8.7 of EN 1993-1-6. Chapter 8 of thcse Recommendations and in rele"ant EN 1993 application pans.

Sec.-lion 4.1.3 or EN 1993-1-6 (4) Because the strength under the buckling limit stmc LS3 dcpends strongly on Ihe qUlllilY of cOII"lmetion. the strellgth assessment shull IlIke account of the associated requirements for fabrication tolernnces. NOTE: For this purpose. Ihree fabrica tion quality classes are SCI oul in Section 6.3 below.

Where geometrica/(l· f/of/lilleor fillite elemelll clIlclI!t"iollS are "sed Iu oblai" bllcklil/g .\irellgllts. the el/'lil'lllcm g(.'OlI/err;c imjJetfecliollS ellas·ef/ Wilhill rhe jillile efemelll motiel mllst n.:/l('C1 Ihe IHHelllial imperfi.·etioll.~ alld IOlermlC'e restriclialls oflhe filial COIISlructioll.

cI,,\~ified ""'"'''''' cronce qualit)

Three lliffe,...m qllu/iI;e.,· of COllstm(·tiOll are permillcd wilhill the c()(.lijied rilles. Ih'll 1"('51111, a rdllli l"fdy lUll r,jel"/!II(:e 'Ire"glll f.~ IIS('l/ lI·heN! tllen! i.f 1m gllam"'ee Ihal the qilillilY of colISrruelioll will he high. hili II'here Ihe design if P(lrt of (I n'pctili1'e pnx:e.,·.,·. rite t/('sigll C(l1I helUjil from 'lte ('xtm ~/rellgllt allrihmahfe ta SlnlCtlll1'.f hllill wllh righllu/erallces.

6.3.1

General

NOTE 2: The geometric lolernnces given here are those that nre known to hnve a large impact

70

~OTE, Th, 10)"",01

(3)

Rules of EN 1993-1-6, Seclion 8.4, for buckling-relevanl geomclrical toleranccs

NOTE I : The charncteristie buckhng stresses detennined hereafter include imperfections that nre based on Ihe amplitudes and fonns of geometric tolernnc~ Ihlll are expected 10 be met during execution.

(2) The should he cho~en according to Ihc 6.1 10 6.4 of scnpll0n of ea..:h c\alualion .

rdul ii,o~'~":~~h.ii~P:,:,~~~ '~~ and tl

6.3

(I) Unless specific buckling-rcle\allt geometrical tolerunee!' lire given in the relc\ ant EN 1993 application parts. the following tolerance limits should be obscn.ed if Ihe bud.ling limit slate LS3 is one ofthc ultimate limit stutes to be considered.

on Ihe ..ufel}' of the

high tolernnce. dC'o1gn. (4)

The different

m:~~;n~~~:~

treated nonn:llly (5)

sample checks on thc mcasuremcnb lion.. !olay "ilhin Mipuhll\.-d in 6.::\.1 d:ltions. Sample !thould be u",len"" , (except for ~clf \\llh Ihe operalional

(6)

Tlte to/(.mnce.~ slx'Cijiet/ here lire .f/x'Cijiml/y collcerllt'd lI'illt Ih(' sC'IIsitil'ity af ~ltell hllckJillg stre"gth~ 10 Ihe fimll mrt/ lmiplillll/e of Ihe g('Omelrictll imperfections in rhe shell .miface. Tlten! is {f l{frge liler{flure 011 litis s/lbj('cl, hili it difficllit 10~pecifY to/erimces Ih{fl make relewml mem;ure~' for all .~/lclls. rite lo/eralice.,· Ilml (".e imporlcllli c/elJel/d fJll lite .Hress state ill Ihe sltel/. hili tlte dnif/illg Projet:l Team decided tlwl it 1I'01ild he too complicated 10 make Ihe reqllired IOlertlllce delJl!IId emin.'l)' Oil II,e de.lign ~tress cOllt/itiol/. As a reslllt, O"~I' tlte dimple ImperjL'c:liOllS lire lIppliL'li lI'ilh some is~till

(7) fCCllons mel1dlltion~.

!>Iroightening. cided indi\ idually.

Geonlelril'1I110lerallCe!llllld imperf«IIons

illg parlicllftlr/l' cUr(jul dt'filli',e hltcldillg flrt>IIglh of 1II1I1II' . is (ICII/e/l' femilil'(" 10 11.'';' 'from Ihe idelll gf..'fJlIlell1',

on the safety oflhe structure,

(ICCf)Iml /x'iIlB wh'II of Ihe slrl'J.s slllie. Oil,,:,.. Ilife. tI/I .fhe/l~ tire reqllired to lIleel tI/I wIertlll("c~, el'e/l whel/ Ihef/Jecijicd wlemm'e may /101 IUII'e a gre(11 bellrillg all II/{> hllckllllg slrellglhlor lilt, fXlrtlclllur ,~"el/,

(2) The fabrication tolemnce quality clu~!!o should be choscn as Class A. Class B or Class C Dccording to lhe totcrnncc definitions in Tables 6.1 to 6.4 of these Ih,'COITl/nendution~. The description of each class relates only to the ~trength c\'aluation, NOTE: The tolerances definl..-d hcre nUlIeh those specified in the c.''l:ccution standard EN 1090. but arc set out more fully here togi\e the detail of tile rcllllionship bcl\\ccn the imperfcction amplnudes and the evaluated resistance,

8C'f'(lfI.It! lilt' shell hlleldillg rt'.... i~ta/ll·e if wry' ,Ie//,Ii/il 'c IV Ihc Cllllplilllde of Ihe geolllerri{,ltl

(3)

Each of the imperfection types should be classified sepanllcly: the 100\cst fabrication tolcronce quality class obtained, corrc!>ponding to a high tolcrance. should then gO\ em thc entire design,

"'/tell tit" lolerances relemlll 10 tI higher qualil)' doss cmlllot be (1CItierN/lor all the cliffer('l/t ~pt'Cified lolerance m(!(lSUrt'.f, Jhe qlwlill' c1clSS flJr delign ItIII,W he wke'l (I.~ Ihe lowel't dast,

(4) The different tolenmce tyIX"S may each be treated independently. and no internctions need normnlly be considered.

£(1C1t lolerallce metlSllre is dcemed 10 be II sillgIl' ,fe/xlI'me lIIellSllre.

(5) It should be cstablished by rcpre.c;cntntivc sample checks on the completed stnlcture that the measurements of the geometrical imperfections stay within the geometrical tolernnces stipu lated in 6.3,2 to 6,3.4 of thL'SC R,.'commendntions,

Ti,e ICI'III "represellfatil'e sllmple che(:ks" is IIWtl 10 illdic'(lre 111lI1 it is 1101 1Il'Ct'.UlIl)' to II/e(l.l/ll'e el'el)' /Xirt ol,he ,fllell, prol'ided 111(11 Ihe slml/)Ie ICtkell lllleqlllllely ('CI/Hl/reS II,e \\VrM jmlX!Ifi'Cliolls,

Ie lellSilil'il)' ofshell

(6) Samplc imperfection measurements should be undertaken on thc unlO:lded struClllrc (except for self \\-cight) and, "here possible. "ith the operational boundary conditions.

lerolll" Oil rltis slIbj('f:I, bill it specify lolerrlllce.f IIItll lIIake lor till ,IIIellv. Tht, wIer~al// tlt'pelld 011 'lte 1"/r(',H blllihe dniftillg Proj('{'1 Tell/II wJ/lld be 100 compliCilled 10 '1Olerallce depelltl ell/irely all vlldilioll As tI resllll, O/l{I' lite toilS are (Ipplied IIltlt ,,'ollte

The gtVlllelric illlJN!ifecliOi/s shollid be 1II1!tJ.\'IIred II'he" Ihe stnlCtllre is fmlOlJ(ktl /x'CtllI,fl! ftJllI.lillg II/a)' eith{'r increase or dct:rease tire metbllret/ umplitllde 01 imperfi'('(iollS, hili Ihe sirellgih assessment is based Oil Iltese iIllIWifec:lio"s ill tll(o fmlOlKkd slttte. Where self-lleigitl reprt!,fellls a significolII complmellf ollhe lolal 100ul Oil Ihe shell. il is 1101 pa,u ihle 10 meel Ihis colldilioll.

(7) If the mcasurements ofgcomctrical imperfections do not satisfy the geometrical tolerances strlted in 6,3.2 to 6,3,4 of these Rccommcndmions, any correction steps. such as by strnightening. should be inve~tigatL-d und dccided individually,

Allempl.f lit Hraigillening ca// uften le"d 10 more .\'eriolls illlJJetfecliolls, a ,f tl.ueued Il,fillg Ihe lIleasuremelli s),slems de.~crihed here, Siraighft'llillg (/Ctil';lie:o' oftell c/l(lIIge ol/e lorm 01 imlX!ifccli(JII illlo (IIIOIher: lite lorm rhar is mO.~1 t/(IIllagillg 10 Ihe strtmgllt de/x,"ds 011 rhe

1Ctl/{1' //cJIIlillem' jilllfl' ,'Ielllem 'e IIsed Iu uhlajll hltdlillg qllil'fl/elll gl'OlIlelril' Im/X!rfi'(" 1tlti" lite jillile eft·lltem mOl1e1 • polemial imperfi'(:liollS allli liolls oJlhe jill,,1 cmtllml'lioll.

q/wlilie.1 of cmulmaioll 1Irt! Ihe codifit'll n/ft·\, At (/ I'('sllll, rljcm.'lll'e sirellgll, is IlSeel J gllorOl//ee Ih(l/ Ihe quulill' of be Itiglt, hll' where Ille de{ig/l "ilil'(' prtX'e.\'.\', ,Itt' tlesig// C(1Il e:r:ml Slrt'IIglh (fllrihl/whl" w 'illt lighIIOll'nmct•.\,.

rch~\' anl geometrical 101-

pedjied here are ~'IH'Ciflcally b//{'klillg Jonn (flld umplilllck' 01 ,lte r!eclio/ls ill the shell .wifllce,

imlJelfi'Clioll, il is m/l/ablc 10 ha\'t! di/Tere", c1lt,ne,~ ollabrictlliOll, permitting Ihe higher

CJllality (\/llaller illllX!rj'ecliom) 10 he defiglletl /() higher Slll.'IIglhs, The lime ill(licales Ihat (Ilrhollgh sOllie illlormurioll i.~ cOlI/uilled ill lite eYeclllioll Slltlldllrd £,V 1()9(}.1 (1006) ill relalioll 10 ,~hell ,~rMlCr"res, il is gin'" ill fillh·r derail here OCCU/I,\ t! of il.\' criti· cal role illihe srrellglh '!'IVllllt/jOIl process.

Non ·:: Be fore a decision is made in fa ....our of !otmightcning to reduce geometric impcrft'Clions. 11 should be noted Ihllt this Clln cause additional residual stresses. TIle degree to which the design buckling resi~tanees arc utilised in the design ~hould also be considered.

.Hress romliliOl/s ,har lite ~ltellll'i11 C,\7H!rie"ce, fa it is "o1 tl/lI'(lYs delll' 1/1(11 Ihe (mumdedform IIHf fJrtll'i(ic- 1I MlIl'I' SIl'IICIIII'I: thtlll Ihe origiI/al. III lIdditirm. II-/,ere gl'Ollll'tricul illlpeift.->cI;OI/S tlrt! redllced h,l' straightening. lhis I/WI' illstead ;1It/lice /'('.\'itilwl .l tre,He.I' II'hil'l, cOlild film he de/e/('riom to IIt(' il/1'1!1IJ:lh of till! ,III'/ICIlIrl!, bw u-ltlch lIl'l! 11!,\'.f \';~ihle Wid II/UI'I! dilJiclii t to lIIetlsure. lli'(·i,\i(J/I.~ rl/l the lIppropritlfl! cO/mie lytlClioll

1I f)ltell doe,f 1/01 meet lite tlesiJ{1I tolera//ce usslimpliolU should Ihe/'(>(ol'e all(\' he lakell lI'illl expel't atll'ice.

6.3,2

O ul-or-roundncss lolera nee

The out-or-rour suti .. ry thc condition

(3)

Ur S LJr,nllu

The out-of-roundncss should be as~scd in tenns of the pammeter Ur (sec Fig. 6.1) gi\cn by:

(I)

u,

"

if

... (6.1)

\\here: d"'f is the l1la)(imum mcn~ured internal diamcter, dm1tl is the minimum Illcallured intern!!1 diameter. is Ihe nominal intern!!l diallieter.

tI,,,,,,.

0111 of round impetfi'(:limu h(1l'e I/ot beell Ilide~I' sflle/it'e/ fbI' 1IIMI ICHIe/ ctue,~. alld Ihey nppenr /Q he moSI SC'I'i(m.~ III/dcr l'XiL"n1(l1 pref)',fIII'e 10001;lIg 1H.'(·tl/I~e II,e IHIIII of Ih(' I'esliiting cirrlm!f"rl"'lia/ (·ompre.llicm j.~ tiel'in/eJ/rom th" ;d{Ial,

lig"ifiC(III/~I'

The ;//creu.\'('d /tl('(1/ mdi/l.\' (If ('un'tlilire IIllIy also affi'cl the resislallce ItJ lIxitlli'O/l/pre'uioll h/li-kiillg. hlllihif i~ 11,\11(1/(1' le.\.\ .\·el'ioll.\ .

\\ here Ur, ml1.~ i, Ihe parnme\er for the rei quality cla!i.~, ' OTE : ValUe!> for anee parnmetcr Lr.r, the "alionnl Anne, arc ghen in Table lI.

71,e pa/"tm/t!lel" Ur make.1 Ihe toleral/ce di/l/ell.\·iol/le.I.I,

.I't)

Ihal it cal/ he apl,/i{'(1 to till ,\be,~ of

.~hl!lI.

(2) The melbun:d intern:11 diameter from a gi....en point ~ hould be taken Ib the largest distance across the she ll from the point to any other internal point at the same axial coordinate. An appropriate number of dinlllctcrs should be measured to IdentIfy the maximum and minimum values.

nle mellmtl dtjitllid h"re if) prohah/l' Ihe .fimp(e.w wal' of d('lenllil/;I/g the (:ffi't:til'l! Olll-O!rolllulll('U of lite sltd!. II 11lL\" .m /l/C distltil·lIl1· lage."i hectl/lf)e il fni/.\' ItJ d('(c('/ some .~;gllificalll compOl/elll.v of 101l'-mQ(le drclllllferelllilll impetfet:IIOIIS. II0we\'(·r. /1O hell('r .I·i/l/ple method Ctlll \'el be aqeretl

Tlble 6

Fabricallon loil'-r.II1CC

qualuy cln!. C'~A

CI3SS B CIObSC

72

Geomf.'lricollOlertmces lIIId illlJN!rj"eCliolu'

Ihtll Ib" ,fh,,1I11'i1l e.xperiel/cf.'" c/t"ar Ihm Ihe umellded/brm ("tlcr slmc'llIrl' Ilmll 11/" origi. where g("olII~'lriclll imlN'tji."C'd hr wroiglllt'IIillK, 1111,1' IIm\' -l'sidulIl .f/rl'.Ut.'.f whid, ('(JII/~' IU 10 II/e .l/rt'II#III ()f lilt' SI"" ,· ,,.,.. leu I'isihle lIlId /l/orf.' diJIi-

.J"

\, •• •• •

b) unsymmclrical

upprOpri(1/1' ('OIlrse of (

ror malll-

n pcnnm..-d

ttCCIltntlt 2mm

pcrfl'Cl joint geometry

1'-1: co -- .."tot - c·Int :;

~: .,

:1

Ie,

!i

I' ;iC1n;1

~

I n".~

Il) uninlendcd ecccntncll), when there is no ch.,nge or plote thickness "' igurc 6.2:

1

.;

i\...perfect join gcol1¥:try

~c.,1

I' 1mn' -I

c} lota1 eccentricity (unmb) intended olfsctllt a tended plus intcnded) al cha ng~ or plllt~ thick~ "ilhout unintended eccenchange orpllltt thickness tricity

Un1lltcnd~-d

ecCCfllnclt)' IlOO mtended offset Ilt a jomt

(4) The unintended eccentricity pamlllctcr Uc The laleraflees UI!,ma.~ giwII ill Table 6.1 are based Oil long"i!Stablislted IrtNlifiOfl(l1 m/lles. should satisfy the condition: Tlte)' UIJ/X'C/f 10 be hlIsed more Oil a ql/(Ilil), of U" S U~. _. .,. (6 .S) Uppe(lrtIllC(' for lilt' shell Ihllll Oil llll)' (USt'S!>'mem of ri,e eDecr all ti,e sllell bllck/i/lg stre/lgll" where: U~.-u is the unintended eccentricity tolcrnnee parnllletcr for the relevant fitbricntion Where till illll'lml (Jimple illlfJeifl.'Climl OI.'CII'" jllSI (Ibo\'(' u joillf 1I"ilh Imil/tellded ('ccelllricity, tolernnee qUlllity clIISS. thi.~ is lite mO.~r deleleriolls a/"l'(l1/gemelll for the NOTE I : Values for the unintended eccentricity sllell (ROller alit/ Tellg, 1989). J-Jo\\,('lW, '''i~ toler:lIlce pamllleter U,..lmlf may be obtnincd from ('ollclitiull i~' c;o l 'CI'Cl1 by II,e dim/lIe wiertlllce the National Annelt. The recommended values arc mea,\/IN!nle,,1 gil'ell below,

given in Table 6,3.

The NlIIiOfWI A,,"e\' to EN 1993-1-6 "'")' spec· ify the loler(lfl(:es 10 be IIsed, Table 6.1, which NOTE 2: Intended offsets and lapped joints nre dejine,f f('COmme"ded 1'tI"le~·. is pan of Ihe treated within Chapler 10 of these Rl'Commenda1I0ll', and is 1101 striclly (KIN of Ihe sUllldartl. tions. These two ca'iCS are not treat/,.'(! us impcr. rcctions within EN 1993-1-6.

t'

Tablt' 6.3: Recommended vlllues ror unintended KCrotricit)' tolcrunces Fllbricntion tolerance ualit closs t>escnption Rcc:ommcnded \'3l u~ of U Class A Exeellenl 0, 14

J~

'mm ~ J

1'--->\

;rS,o",,"'1'

I

'''1'

rJeS

I~

I min

Ollllilll:ti i" Sdlmidt and Greiner (199.Y),

,

--' c)

h~tlnclllol

Geometric,,, IOlertmce., (lilt! imIJerft,. rolling curvature errors): deviation!! from nominnl thickness: lack of evenncss of !>uppons. b)

material imperfections. such 1lS: residual stl'CSSC!l caused by rolling. pressing. welding. straightening etc.: inhomogeneities and anisotropics.

Funher pos!!ible ncgathc ml1uenccs on the imperfect clastic-plastic buckling rL~istancc mtio RGMM1 • such as ground SClllements or Ilexibilities of COllnections or suppons. are not classed as imperfections in the sense of these prO\isions. Where appropriate. a global numerical nnalysis of the structure should include all possible relevant imperfections from the abcne list that may be significantly deleterious to thc strength of the structure.

?Iich thi;o. tolenmCl! is lIS-

79

Bllck/illg ofSleel Shell.\ - Europca" Desigll RL'CQI1II1II.!1ld(llioR\

------------------

6.... 2

Thickncss imllcrfections lind tolerances

6.S The e\alumions of EN 1993-1-6 ure based on the nominal thickness of the plnle. Where the plate is relathcly thick. this poses fc\\ problems. 1I0we\er. where \ery thin pillte construction is in\'ohed (e.g. < 3nlln). the lower limit of the thicknchlI that is Ilcceptable under production requiremcnt!> mny be below the \ulue . . required for buck ling design.

Bomscheuer. F.W,. Rundsch\\'ei!.!oniiht

DIN 18 800 (19901 No restrictions have been placed on imperfections Ihm nre local thiekJII..'SS variutions in the shell. This topic has been the subject of one known mujor !>Iudy (Gllsic et at.. 1998: Gu!>ic. 1999). but the delcteriou!> em.'Ct nppcars to be co\ered moderately \\cli by the reqU1rements rel31ing 10 ImperfectiOlb of shape.

6.4.3

Institut rur Normung.

(19Igg'8~:),~:~~:;

FCCS European C

'Vcar and ctJrrosloll Bnt!>sels.

Where the shell structure is in contact with other materinls that may either cause mechanical or chemical degradation of the material of the structure (mechanical damage as wear: chemielll damage as corrosion) proper allowance should be nlUde for the expccK"d effecl o\er the lifetime of the structure. nle commonest case of both wear and corrosion is in si los. Some silos contain rough and IIbmsive solids thut slide IIgainst the wall during discharge. clmsing considerable wenr. Although scverul examples exist of silos being so badly worn thnt holes IIppearcd in the cylindricn l walls. no failures are known due to this em..ct. and no research studies are known that ha\ e cillplored the reduction in buckling strength due to local wear. Corrosion often ari"C5 where there are eorrosi\e subSlnnccs within either a stored sol id in a silo or a stored liquid in II tank. Rules concerning wear and corrosion in si los may be !bund in EN 1993-4-1. 6.4.4

Non-uniformities of IOllding, 10 boundaries

fore~

transfer bet"ccil shell segments

IIl1d

force lransfer

No restrictions hll\ e been plnced on imperfections that arise from minor non·unifonnilies of the applied loading and support. Where the shell is subject to membrane forees from other structural clements. thL'SC may be significnntllTld cllre should be exercised to eTlloure tlmtthe 1I0n·unifonnity of forees applied to the shell boundaries is appropriately considered.

6.4.5

Residual slr('Ss('S

No special allowance has been made for rcsidunl stresses in the shell structure. A few studies of the development and consequences of residual stresses in cylindrical shell s have been conducted (Guggenberger. 1996: Rouer. 1996: 1I0lsi et al .. 1999. 2000). These gcnentll y lend \0 the conclusion that Q consistent residual stress field (onc that sntislies equilibrium lind can be present in thc shell in its final imperfL'C1 gcomctricul fonn lind docs nOI increase the amplitude of the geometric imperfcction when the shell is unloaded) is usuully slightly beneficial. in Ihat il increases the buckling rcsiSlllncc of the imperfecl shell. Some earlier studies of the problem (e.g. Bornschcuer et al.. 1983) reached a different conclusion bllt were conducted without using a consistent residual stress field .

structures. CEK

EsslinQer. M. lind Axi!tymmetrie Irrcgu\ Dowling. J.E. ilardi Esslinger, M .• Geier. concerning I!.otropie Liege. Greiner. R. (2008) .. Rotter. Feb. Greiner. R. and Yan stepped wall thiekn Measurement. Charoc Guggenbcrger. W. (\ process and coru.i~tc Workshop on Impcr C A-Silo. Lyon. F Gusic, G. (1999) " F~ d'epaisscur" (Buekli imperfections). ThCi< Frunee. Gusie G .. Limam A.• influence of loca Mechllnics Ad\'aoce!'

liolst, J.M.F.G. and I Metal Shells. cds 1.G

80

Geometric,,1 to/erCI"Cf.'s u"d imperfecfioll.\

6.S

,f thc phllc. Whcre the plate is platc construction is invoh'ed r production requircmems may

ckncss variations in the shell. /.. 1998: Gusic, 1999), but the he n:quiremenls relating to

Referenc es

Bomseheuer. F.W .. IUfner. L. and Rallllll, E.. (\983) "Zur Slabililfil eines Krciszylinders mit einer Rundschweissn5h\ unter Axialbeillstung'·. Ocr StllhlbllU. Vol. 52. I left 10. pp. 313-318. DIN 18 800 (1990) Stllhlb..1lJlen: Stnbilitlltsflll1e, Schalenbeulcn, DIN 18800 Part 4, Dcutschcs Institut fUr NOnllUng, Berlin, November. ECCS (1988) European Recommendations for Steel Construction: Buckling of Shells. 4th edition, European Convention for Constructional Stcclwork. Brussels. EN 1090-2 (2006) Dmft: Execution ofstccl structures and aluminium structures. 2006 draft. CEN. Brussels.

}' cither cause 1l1. Vol. 43. No.5. MIlY. PP HI 1-825. 1I01st. J.M.F.G .. ROllcr, J.M. and Calladinc. C.R. (1999) "Imperfections in Cylindrical Shell" re'lulting from Fabrication Misfit~". Joun131 of Enginecring Mcchanic... ASCE. Vol. 125. No. 4. April. pp 410-418. Iioist. J.M.F.G .. Rotlcr. J.M. lind Calladinc. C.R. (2000) "hnJX=rf(.'Ctions and buckling in cylindricill shclls Yo Ith con .. i~tcnt residual streioM.'S". Journal of ('onslnlctional Stcel Research. Vol. 54. pp 265282. Rolter. 1M. ( 1996) "Ela!lic anal} ..i .. to lound In Seclion fLU.

oo:r,~'h:"(~~~;~

In lenn!> with the 11

sires, .,WI". \\hiht

w'('nes. it is not possible to u..ing II linear elastic In an attempt to n.'SOh "Ulte, EI\ 1993·1-6

~;~::b::,~,i

ing and n, criterion finol 1 mtemellon that i~ !>!Ife

action. It i, slilll ~.~~~ pla .. ticity oflhc S4 a",

86

----

RlIle.\ '/or Ihe plmlic h"'il~tale all(/ plaslic referellce lomi lJ.ueument

Second principal membrane stress !\!Sultant "1

!DCe resislance Safe stress !\!Sullant

eombinaaions inside this envelope

'direct design" \\here II fonnal splacelilent nurnericalullulysis rovidcd Ihut the results call be

if. L - - , T·T

-if. nMed "ith II lower bound ap-

T.c

e attainment of first yield. but y.

First princiIXl1 membrane st!\!Ss !\!Suitanill/

C·T

if.

c.c

-If.

T TcnSloo C Con~sim

f lgur!' 7.2: 111&SlIe slrcngth mll'mCllon bct"een membmne Sires!> stale

In

resultants

the shell. first

Mi.scs criterion IS used as an tween membrane stress rcsul.

... (7.2)

Ints. recognising thai the genInnb 11,. I1nand 11\1J

gi .. e a close approxima tion 10 ~h ie'ed llround Ihe emire cir1 of failure. Uowcvcr. where (lcs a very conscrvutivc csti. 1"1CTIt of the complcte plns tic .are nOi satisfied). In par· dlstnbution. this method of Morem'er. there is no simple notions of plastic rcdistribu-

F

~tress

7.2.3

Estim a tio n based on linear clasti( s hell be nding theory calcula tions

Where a linear shell bending thto'ory analysis has been u'\Cd to detennine the stress state. fil'\t yield on the surface at the most highl y stressed point in the ~hcll according to \'on Miscs criterion \:ould be used as an estimate of the full plnslie rcsistnnee. but this lends to an e .. en lower estimate of the true plastic limit loud than the membrane stress e\aluatioll of Section 7.2.2. If the criterion of first yield on lhe surfllee is ndoptoo. design D:1Sl.-d on lincnr bending thcory becomes more conser-uli, c than design based on membrune theory. which S/..'CIII!> inapproprintc. Howc\er. thc bending actions thm are calculated u!>ing linl'ar !>hell bending tln:ory are sometimes necessary to suppon the static equilibrium of the stnleturc (Rollcr. 1985). Consequently. these moments cannot be ignored en· tirely (it is difficul t to define geneml crileriu tlml identify when these moments are imponant to slmic equilibrium and when nOI). Tlu:sc difficulties all arise when an allcmpt is made to use u linear elastic anlilysis to idcntify the plastic reference resistance. Funher discussion and advicc may be found in Section 8.2.3. In tenns of the definitions of EN 1993-1·6. Ihe stl\.'SS s tatc in the shellthnt is required to cquilibrnh! with the louds (irrespective of \~ hether clastic or plastic stresses are involved) is tcnned the prill/an· :flre.U Sf(ll(? whilst the stresses arising from compatibility considerntions are tenned S(-'COlldt,,)' Slrf!.ue.~. J-1owe\-er. although this conceptual difference is very imponam in the pR.'SCm discussion. it i~ not possible to idcmify cU!>ily Yo hieh strcliS/..'S belong in which class \\ hcn lhey nre cnlcullm.-d using a linear elastic numericalnnalysis (LA). In nn allempt to rcsoh e the unfonunate scriou~ con'iCrvatism of the abo\-e assessment of the linut statc. EN 1993·1..() uses Yo hat is commonly temlCd the Ilyus hin yicld critcrion for combined bending and membrane SI.rcSSCS to provide a measure of increase in the enlculmed surface stresses. This criterion first combines the hcnding moments and membrane forces in cach direction to produce an internction that is sufe for uniaxial bending. and then combines them into a biaxial von Miscs mlcrnction. It is still e\'aluali.o'd in tcnns ofstn.'SSl.'S. but the outcomc represents a closer approueh to full plasticity of the section than can be obtained using surface strcsSC!> (Fig. 7.3):

... (7.3)

BlId:/i"K n/Slul SI,,>/'s - ElIrtJl)t!{II/ De.~igll RecommclldtlliollS

7.3

in which: UO ·

r.rO

!!JJl.. ±, m \0 . I

r

r '4

'"

design

!!JJ.. ±-,InO I

Rules of

... (7.4)

r 4

7.3.1

". !l..!!L

,

... (1.5)

This criterion now pennits the full phu;tie moment of the shell to be dc\elopi.."!. J.M.F.G" Docrieh. C, and ROller. J.M. (:W05) "Aeeurale detennilllltion of the plastic collapse IOOld!> of !lhells when using fintte clement analyses", Proc .. Founh IntenllltiOllll1 Conference on Ad,ane(.'S in Steel Suuctures. IC ASS'OS, Shanghai. July 200S, pp. 1789-179-1. Kalnins. A. and Updike, D.P. (1998) "Plasticlty and changing geometry in pressure ,"cssel dc ..ign", PVP-Vo1. 360, Pressure Vessel and Piping Codes and Standurds· 1998, American Socicty of Me· chanical Enginecrs. pp 91·99. Massonnet, CE. and 53\ e. M.A. (1972) Plaslic Aml/I'Si.f am/III(' DesiX" of Pltlt~.'i SlII'II.\· tllIIl Disks, Nonh·'Iolland. ROllcr. J.M . (198S) " Analysis :lIld Design of Ringbcllllls". in Dc!oign ofSteclllin!o for the Stornge of Bulk Solids. edited by J.M . Rolter. Un;\er..ity ofS),dncy, pp 16-1-183. Rotler, J.M. (2002a) "Shell Buckling and Collapse Analysis for Structuml J)c.,jgn: The Ne\\ Fnllncwork of the European Stllndard". in Nc\\ Approaches to Structuml Mechanics, Shells and

92

Biological Structures, 3SS-37R. Rotter, J.M. (20021,) .'Aj temational Conference 42. Rotter, J.M. (200S) Analysis". m Shell Tuylor and Franci!>.

Rideslor lite plastic lim;t slt/I£' utld plastic- refere"ce lood assessmellt

ad. but this is only a referencc 'mcd in EN 1993·1·6 as Direct In.~. Where numcrical analYliois lJralcly and \pecial proccdun.~ ct al.. 2(05). II ..hould also be :rainmg in pans of the ~ hell, so the usc of a mesh that h, nde· 'SC\crc Mraming invol .. ed in a

)(b

for Pla.o.ticity", Computers

~ufClie buckling rcsii>l ..hell is sct out in S perfonn because mun must be related to to the fully nonlinear d .. trueturcs. and for Iili lolandardiscd shell hu Recommendation\, The GMNIA allalys methodology in any represents II con~idc not be taken'" ithout through the usc of SI 8.1.6

Sumnllu)

0

Summansmg th ...' in these RI..'ContTocOOations. II in\ oh cs the follo\\ ing loteplo. illustrotcd previously in Fig. 1.1. and hcre repealed as Fig. 8.1:

In Ihe follo\\TTlg. MNA LilA fom131 in thc ri},lhl column. executed and the fo, thcbucklin" 5"cn~

Geometry

EIII.'>Iic criticlIl resistance Rcr

Plastic reference fC'iiT,m

,,,·.JI/I

jo

,Y·M'!-) OJ

iJ,WII ;JIll

~q Jr.ILlII" ,([[U:lI\,{I[d

IUll!iill RU1~11 i')tj~

[~III,ll lll"I[lm~

100

p"~n:l

atjl JO Jnol\U\pq

UI Ja[[IlJ II U!I!\qo

""',"U;)ih;) JI;')III pun 'Iqnop JO S;)'iU:I III

r

no'l \Il111

,)111 ji) ,1110 I l"IJ,l1

')(1

fJI

P,),J/I

(lJ POOl uOIlu:unJ'q)

(I

ndJI,I· ,JIIIUqlll,111/

JlIIIlIY ,1q 1\·///11 (f·R '''3)

,),I!I.).-'.//iJ

II"')//,'j!fl ;}III 111!.11 1I;}1/1'

'''II II! III!tHl,l'll III/II 1',1111/1 ,J of (9) cunnot be

WilerI' Ihe dlt)iidercd.

(II) If lIpL'Cific valucs of a, ...

'I, ~ tIIlll ArlO " ('W/lIU! Iw "~lOhli51/('d Il"i!II UI)I)mpriflfl..' CfJll{it/"rlCt', 1I IllOrollgll GMVlA ;'1I'''Sliglllioll ()f Ihl! proh/{'''' dlOlild Iw IlI/dent/A('11 h{'e 8,3 heloll'/. Where gr,'{/I I/I/{'ertail/~I' t'Xlll.1 am/IIII' mHC(II//(' is 1)(lrliclI/llr'" (·rilicul. I/Il' fiflll/llrbiler mml Iw all IIpproprime I('.fl programme (~ee 8.3.3,6 heloll'),

If a GMV/A flll(I~nis afld /{'w~ ('lIImol ht.' IlI/den"J.efl. wid douh, r('m"II/,\' tm ,11(' approprillll' m/tle~ of '''''.fe p(lrame!"I'.\·, 11,;1' mle ~il'e,\' dt"lI/llIm/II('.\' 11m/ are det'med 10 he WI(e{(1r mool h"lIIm ~IIell hllckling (y.llldilioll.(. I, is b(l~cc! Oil ,''(' plaw,i"'!' (/.HIIIII/"iofl ,lwll/()f"ell huckling case cafl be imal!inec! to he more impl'lf('('I;'''',\emilll'l! WI /l1/.\liffened circillar C.:I'lilld"r u"d"r lI11ifimll (u;ol c()mpre.Hiflll (ll'hich bd"n't'l sllllilClr(1' 10 WI eXlenwlfl' pn''ullri,w'll '~P"{"·I..,). Ollt! 111lI1 (l1)lllring Ilf btlckling IHII'(II/Ief('l'S

(I:!) The .. hlluld b..: obtained

R,

.lUI

I

Rp/

\~heT'l!:

is th~ pla~tic

R,

(11) I he tk-.lgn

bo.! ubtnintxl fmm:

\1

here:

rill

I~ thc panial budding ",,"d~

K.2.2.J (I)

It !>oould '" "nl

F fJ S

r,u ~ RJFI:J

' '(111

H.2.3

r,dendl'ti eel

a" /Jr, '1: hi!> parlit'lIlal' "rue tlgllillSI II,,, bad.grolmd sel 0111 III Ihe ('xlelldetl romRlet/lul)',

106

The second of thc'>C t bound estlmatc uf the Ihen the outcome ror !>tn."l>!> pattern ineh~\; identical), but it enn I because II finite Tone mechanism. Where procl.-dure e...aluUles I~ consequently leads to

Httlt,~ for

---I /Ie hi.~/u',' (or a. .. hili Ihe ,~ " WI' /1.... ell/maled lI'ilh flholll wrioll\/l' adn'I"'t'~)' 't' n,.'prillllln' fix:uI ,/multl III/it· illll''''''/t',"lim, rn/ll('lilm 11I1"iI'.,~1' nlllfmll II/t' daHi/" 1II1t} Ilhid, (11/)' hal I I 11111/01' ! rdalil'(, .l'it."";,,.,,(,,H i.1' wlI/II illlh(' pla.l /i,· I"f'Mitm). I Irong indication tllllt the !>Inleture is likely to be ,ery scnsi th e to gcomclric impcrft'Ctions, with a corn..-spondingly 10\\ valuc of a".. for these modt'S. Since a low value of a leads to a con... idcrable change in the clastiC impcrfL'Ct shell buckling resistancc. II I§ \ Ital that these modes arc identificd and used al> P.1r1 ofthc MNA LBA de~ign proccdurc. It i... recommended here that. '" such eircUll1lo1ll1lCCS. euch cigelllnode should be lreated as a l>Cpar.llc dc::-ign casc and taken through thc dcsign buckling re~i"tnnce process (Fig. H.l) using approprime ,alues of the par.lmetcrs a,. fl,. 'I. and Ad/ for each mode. according to its location nnd stress state. This methodology will remo\\! much of Ihe unccrlainty that has been expm...cd in somc of thc eommcnts conceming Ihe approprilllC choice or a.... P,. 'I, lind A\o :lIld should lead to reliably s,1 1c designs.

Discretely stiffcned !>hclls presenl many addilionul difficultics. Commonly. these struCIIII'CS huve \cry nex iblc shell walls with much stitTer elemcnt:.. ns stiffeners. When an LBA analysis is pcrfomled. the first 100 or so modc::- may all be local modt'S in the shcll wall and do not Impact on the global strength of the strucmre significantly. In stich situations. the LBA outcome will greatly tlndcrpredict the intcndt-d buckling stI'Cngth of the structure. and some special measun.'s may be needed in analysis to find an appropriate LOA modd that captures the glob:11 buckling mode including stiffener defomlations. FUrlhcr form.1lisation of procedures to address thL'SC issues is an aClive l'CSC'.trch urea. but thc abO\ e appears 10 providc a safe proccs.o; on the basis of current knowledgc. It il> hoped that the mcthodology will be refincd in thc ncxt fcw years. with more pn:else inform:ttion available on the proximity of cigenmodcs and formalised mellstlrcs to lIddress these ~ues. H.2,3.4 C .. rc uith the interp retation orthc MNA .. na l}'sis

The MNA analysis is USL-d to obtain a fomlal plastic fC.'iistancc for the shell. For simple geometries und load CIUCS. closed ronn unulyticlli solutions can be found, and many ofthcsc nre documented in Massonnet and Sa\ e (1972) lind Annex B of EN 1993- 1-6 (2007). 1I0wc\'er. for II'K)l>t cases. a finite clcment nnalysis will be n:(IUtred. Where a finile clement analysis is used to obtain this resistance. the analysis must be malcrially nonlinear and may in\'ohe high local ).IrainS at some locations. Mnny programs. c .. cn well known commercial program!> of high quulity. ha .. e diniculty in reaching It full plastic IIlt'ChanislIl. so the plastic resistance may be difficult 10 define pn..-ciscly. In design calculations. this is not of great imporlance as any load ltttained before colillpse may be udopted as a conSCI"\'lIti\e estimate of R,,,,

110

StudIes by Uo!-.t l·' a/, found b)' u~ing a pl(lt deh.'nninatilln". Here. It Ihe load factor llrl de"i!:!n R" al:!i.IUht IN" 11'), \\ lead to \ cl) prcr.:i~ result~ of computation, givc'l "ngle pmhctcd I 8.2.3.5

dctenninL'ti by u"ng. an conSCl"\ati\e prO' idcd to d..:\ I~e a gencral conl>Cnati\c fllr '>lIme l>\ructun.... under cia'tic The ","1n1e is Inn: In cffects ure sufficiently \\UY'" of \Cpardtmg oot LA anaIYl>i~.

an LA IInaly~" th.11 is strictly safe if applied system is bending is be safe for shells in stnlcturc~ in which guaranteed to be 'tOfc oftcn cause

",,,,,i'''""l

In cOl1loidcring the usc resistance, it i... important separatc conditions at These are: n) Ihl 000111"."",." ~ b) the coliaploC occur; plastic strain... cun c) the collaJhC occur.> circle: and d) These are ,"d,,,dluailly di~ Whcre ntCmbr.lne yield yields exactly thc same

Rilles for the hlldlillM limit _Hilt/! II.Ue,ulllc"t II.fillg global IIlII1l('ri('ul ufwl\,s;j

Uing parameters \\ hcn u~ing ) :u.ldrt."l>" IhLo;,c concems. the

~tmcture. the procedure is eigcnmodc. I-unhcr, where comprL-ssion. thc scn each other. this i~ a ,trong .ctric imperfections, \\ ith a of a k...tL. 10 a consIderable I thctIfe

nonly, these structures 11lI\-e When nn LOA analysis is I wall and do not impact 011 : LUA outcomc will greatly ~ special TTK.'IIsurcs may be the global buckling mode

rc!oCllreh llrea. but the above It is hoped that the rformation 11\ uilablc on the

~ge.

i hell. For simplc gcometries ofthcsc are docurncntL-d in e\er. for most cuses. u finite

blysis mUst be materially progrnms, e,en \\-cll known I plastic mcchunhm. so thc ,lations. this is not or greal onsc.... ali\e estimate or Rpl.

but \\here thIs methodolob'Y ts used for research 'Iudles, It 1 unhelpful in design: designers using simple hand calculations and engineering j udgment would obtain a correct and 501fe cstimate. whilstlhosc ut-ing LA would find thai the referencc resiSlance nppears to be vcry low. Thus. LA lo\\cr bound estimates of the plastic reference resistance should be used with some caution for complex stres!. statcs.

'"

Whcre plastic collapse occurs (c) undcr a combinalion of bendlllg and stretching on a complete circumferentiul cirele. the lo"er bound LA estimate is signi fica ntly but nOI so drnm3ticully conservative. A \ery ~imp l e cxample is Ihe ring loadL-d cylinder (Massonnct and Suvc. 1972). whcre the membrane stress criterion of Eq. 8. 1 produces 3n estimntcd plustic reference l\."Sistunce Ihal is j ust 80010 of the true vul ue. The estimated l\.'Sistance tends to fall lower as Ihe stress state becomes more complicated. but the LA cstim.1te is not so very conservative for uxisymmetric load cascs. Where plustic collapse occurs (d) by n combination of bending and stretching on a limited 70ne of the shcll. which docs nol extend far around the circumference or along the cylinder length. the lo"er bound LA estimUle can easily become dmmatically conS(.TVlllivc. Unfonunlllcly Ihcre arc \ery fewlligebrnic solutions for problems of Ihis kind. nnd Ihe d e~igller must always I'Csort to finite clement anulysis to obtain a good L'S timate of the plastic resistance. Where such situations are crilical 10 the design (relathely Ihick shells widl highly localised stress development). it is recommended that a full MNA analysis is USL-rl. 8.2.3.6 Uuekling and plastic collapse in differentlocatiolls in the structure COlleem is sometimes exprL'SSL-d thnt if the plastic collapse (or the highesl stress condition) and the buckling mode occur in dilTerent parts of lhe structure. then perhaps the above methodology might not be secure. It is certainly true Ihut thcre will be lillie interoction between plasticity and stability elTects ifthcse two phcnomena are widcly separatcd. But this concern may be addressed by considering the 7..one in thc structure in which buckling is found to occur. If the true plastic colilipsc strength of this local region \\ere found. it would pro\C 10 be higher than thc vlllue thnt is takcn for the structure us n wholc. because the lowest plastic collapse load is associnted with a mode elsewhere. So if thc true clastic-plustic interaction associated with the buckl ing mode wcre used. it would Icad to higher eltlslic-plastic buckling lo.1ds than will be obtaincd by assuming un intcmction betwccn two phenomena thnt do nol interact. Thus the calculation procedure is conservative in Ihese cases. The only real shon coming of this methodology is Ihnl the LBA analysis finds the lowesl elastic critical resiSlnnce of the Slrueturc. nnd this may be locllted in a struclurel clement Ihat is nol very imperfection sensitivc. If there is a higher elnstic critical resistance in 3 dilTerent structural element

11 2

",,,.,r.""i,,.

8.2.3.7

In Rulc 8.2.2.2 (II) II is . parameter.. a.... P- '1 . the commentary on this simple concept. Th~ the shcll stnbility i cylinder for "hich thc assumptions eonccrn method of assessing Ihe adopted. then the two I (a)

Geometric "."h~"~ panicular. special

'~~":~:;;~

Thc>cmildly only buckling. This commcnt on thiS

(b)

'."'P"o!

The a.l[ially scnsiti\ ity "hl.'Il is therefore on the yield in a thicker value of p (i.e. i firsl nlTccts the the \ alue of p for. load lind the true One exnmple of where global global bending).

due to the ""'"'"'""~ the values of the

8.2.3.8 As noted in Chapter 2.

buckling. The m,c. ~~k~::~ the criticnl clastic analysis is always the

Rilles for the bllckling limit slate usses:"III('III IIsing globallllllllerical ana(I·.fi.f

There is then no differencc calculation. he shell (b) so that indel1nite ~ analysis agam dclivers the ones. A good e.~amplc of this at. 2(05). 1I0wc\·cr. here the In LA analYSis and any of the the LA anlllysis predicts thc "aluc for a typical stnlctu re e is 5 to 6 times too smllll. mely 10" and is unhelpful in ~ment "ould obUlJn a correct I resistance appcllrs to be very oce should be Il\.t-d \\ ith some

stretching on a complete nl)' but not so dranuuically 1assonnet and Sm e. 1972). plU!.lic reference resistance fall lo"cr as the stress state l"\"3ti\e for axisymmetric load

tching on a limited 70ne of long the cylinder length, the I\e. Unfonunatel y there lire mu!tt always resort to finite Whcre !ouch situMions are H stress de ... clopmcnt). it is

:r

Irurture hcst stress condition) and the ~e abme methodology might 'tween plasticity and stability

rueture in "hich buckling is "cre found. it would prove :. because the lowest plastic JC clastic-plastic interaction ~Iastic·plastic buckling loads 11:1 that do not internet. Thus 1

ysis finds the lowest clastic Jrul clement that is not very l different structural clement

that is highly imperfcction sensiti... c. thcn thc eharactcristic resistance may be found to be higher for the entire structure than is valid. For this reason. it is important that care is taken with thc adoption of It si ngle clastic critical resistance for the structure where the structuml clemcnt concerned is not as imperfcction sensitive as is possible in othcr clements ill the structure. 8.Z.3.7 Clue when making simple choices for the parameters 8.2.2.2 (1\»

a.•. fl••.

11.. and )..... (Ru le

In Rule 8.2.2.2 ( II ) it is indicated that. where it is difficult to determine appropriate values for the p..1r3meters «.... {J.... II.... and Au" ,IIo the values for an axilllly compressed cylinder may be IIdoptcd. In the commentary on this rule. it is noted thutthere arc two implicit assumptions which lie behind Ihis simple conccpt. nlCSC two w.sumptions depend on the simililrity bel\\ccn the basic mechanics of the shell stability situation under considcration and thut of the unifonnly axilllly compressed cylinder for which the buckling parameters lire taken as an approximation. These two key assumptions concern the role of geometric nonlinearity in the prebuckling behaviour. and the method of assessing the plastic reference resistance. If the axially loaded cylinder values are to be adopt/..-d. then the two following aspects should be cllrefully eonsidcl\.'d. (a)

Geometric nonlinearity should not have u great impact on the stability behaviour. In particular. special attention is required with regard to the possibility of snap-through buckling. These effects cannot be covered by the val ue of a. becausc the axially compressed cylinder is only mildly affccted (IS-f.) by geometric nonlinearity and is not susceptible to snap-through buckling. This mUllcr is addressed in E 1993· 1·6 in the Note in 8.2.2.2(9). A further comment on this mailer is given in S.2.3.g.

(b)

The axially compressed eylinder has becn adopted for this purpose because its imperfection sensitivity when susceptible to clastic buckling (when thin) is btfC3t. Thc focus of this choice is therefore on the par:lmeter a. However. hccuusc the prcbuckling stresses are unifonn. first yield in II thicker structure may be vcry close to the plastic col1l1psc lo.1d. giving a rather low value of P (i.e. lillie separation of the plastic collupsc load from the load at which yielding first affects the behaviour significantly). Thus, consideration should be given to inerensing the value of P for situations in which the LA lower bound estimate of the plastic reference load and the true MNA value of the plastic reference lo.1d are widely separated. One example of this situation is found in the long cylinder under wind loads (chimneys). where global bending dominlltcs (giving a large separation of first yield and fully plastic global bending), coupled with an interaction with circumferential compression and bendi ng due to the unsymmctrical pressure distribution. For such situations. it is recomlllended that the values of the parameters a.... II.... and Ao~, O be taken for the axially compressed cylinder. but that the value of ~ should be iocreasl.-d from the current Pr (- 0,6) to p.,. =: 0.7 to ensure that the earlier yielding in sueh structures is accounted for properly.

8.Z.3.8 Situations thllt require a GNA IInal)"sis

As noted in Chapter 2, a shell may be subject 10 either bifurcation buckling or snap-through buckling. The methodology of the MNA/lBA procedure covers only bifureation buckling. where the critical elastic buckling strength may be found using a linear elastic eigenvalue anillysis. This analysis is IIlways the basis for the reference clastic critical resistance.

11 3

!lo\\c\cr. for condition~ whcre ~nap-through buckling may OCCllr. thl! IInl!ar cla~tlc clgcmuluc ilO unable to detcct the buckling c,ent. Instead the progrc~:-.i\C chungc of g...'Ollll!lry under the applied lo.1d:o mu~1 be followed 10 dele mIme the true budding !>Irenglh of the pcrfloct Mructure. and thi~ require)o II (iNA or more sophi .. l1emcd analysis. l\c,cnhdess. thIs Inle perfcci shell buckling strength i!t not u:.t'ii II!> thc reference critical rcloi~t:lIlee. but I~ e,ulualcd I\!oo u lnocL.-do\\n efTcct due to geometric nonlllle~lrI\) from the clastic bifurcation reloi,\lInce. :1Ilalysi~

§ § § §

~

-

nlC cla.... ~ic e~umplc of a .. hell buckling case where snap.though bucklmg controls is Ihe shallo\\ spherical cap. !lowc\cr. cylindrical !>hclls or panels subjeci to locttl Ill.. d~ tiT \\llh loeul boundury conditions or SI1f1cners may also be sU.!oceptiblc to such ~nup'lhough buckling. Wlh.:n.: Ihi~ is the caloc. it i~ Illo..t Illlporwnt Ihut the efTect of geometric nonlinearily i!t mdudcd in Ihe clustic imperfccllon reductIOn r.1ctor a and adopted into the design procc!;S. For Ihi ... type of nllaly~i .... thc reader i... rcfcrred to SL'tllon IU on global nonlinear analysis.

t .. ~

t .. I

Gi\cn its imponallcc. il is unfununute thut a eOlllprehen ... i\ e Ii ... t of con(lIlion ... thnt lead to snap· through buckling cannot be 'Hillen at Ihe present time. 11.2.3.9

Sh('l1~

oh 11'1 ing radius (rones lind

doubl~

01500

cuned shel!!)

One final aspect conccnling the conscrvuti,c "'Iimatc of a; by mcall'i of I'_N 1993-1-6 Rule 8.2.2.2(11) should be mentioned here. Where thc shell radiu ... \arie .... it i.. unclear \\hal \alue of radiU',> ... hould be u'iCd \\hen thc characlerislic imperfection amplitude i... defincd for a cylinder in tcnns of ib radiu~ (I·.N 1993·1--6 Anncx 0 Fxprcssion 0 .15 and lq. 10.15 of these Rccommcndallom). Thi .. choice influences the value of a •. and thU!. the com-sponding \uluc of 0" if thc lVIiully compre.'>:.t-o cylinder lrealmem is used. One cxample of Ihis diffieullY is the case of a conical shell "here Ihe critical buckling mode e)(tends o'er a range ofnldl1. "hil"t another is Ihe case of doubly cuned shel ls \\hcre there arc two radii of curvnture. For !luch case~ it is perhnp ... morc uscfulto formulate thc expression for Q, in tenns of the relati"e shell slcndeniL'SS i. III plnce oflhe rtI. In plnee of Ihe exprc~sion 0 .15 in Annex D. it is proposed Ihal thc dimensionless slendenless be used to definc the ehnmcleriSlie imperfcction Llw. If it is rccogniSl.->d that lIlost of the data uioed 10 define thL'SC imperfcclions relale 10 steel with/. 250M?a. Ihe expression D.15 CEq. 10.15 in these RL'tollllllcndutionio) lIlay be rcplIlCL'ii by ;;0

6". -( ~)"

... (K. IO)

Fij!urc 8.5: Sleel (hlmn~')

~

Technical dat,l MUICrilli: StroClUml ..,,:d S 1 1-.: 110< ::m~

f.t

Fabriclltion quality: ("I:h'" n ~igTl

aetl0n

Prcloading force of one ten"i Rudi:11 strip lo'ltI of one tcnsi

An LA analysis of the cyli cqUl\,lIlcn\ .. t(l'S'" field sho'" :malysi .. (LBA) yields the r R.... 1.64. The de-;ig.n chL'l:1.

in "hich the \ aluLos of Q for different fabriclllion qualities remain uochangL-d. max 8.2.4

CT.n".J

I-: ulllplt·s of use of the MN rV LBA buckling design procedure a",-

8.2.4.1

114

0.65; jJ.

11:

fill

[0.65 (1~.60I['"

Steel chillln(') \\Hh encireling prl'lolid from aeri:ll attacbml'nlll'nsion hl'lts

Mobile phone acrials arc 10 be allnchct1 10 all existing steel chimney by means of four nm sleel tensioning belts (Fig. 8.5). In order 10 e)(clude slipping of the acrinls do\\n the chimncy. il is intcnded to prelo.'1d Ihe lensionlng belts by means of high strength Ml6 bolts. It shall be pro\cd that under the chosen prclO!

FI):urt' 8.5: S!("-cl chimnl')' "ith tem.ion mg bells

cil;cnmodc

be cOn'L'!>ponding \alue of a. ..

TL'Chnical data

!hi ... difficulty is Ihe case ofa f radii, ""hil.,1 [Illolher is Ihe For such casc, il i, perhaps : shell slendemess I., III pluce

Mnlcrinl: Slruclur.lI steel S 235 E = 2 10GPa f,A - 235 MPa Fabrication qUlllilY: Class B Pc:-ign aclion

dimcn~ionlcss

slendemcss be Ihal mosl of Ihe data used 10 sion D.1 5(l.;q.lo. I 5 in Ilu:sc

... (8. 10)

Prc\ooding forcc of one tensioning bell:

f~•. ,

50 kN

F•.d 1.5·50 75 kN , Radinl Sirip load of onc lensioning belt: P~.J 75/(0.75' 0.14) - 7 14 kN, m·

An LA analysis of the cylindrical shell under Ihe rour design strip loads yields Ihe membrune equivalent stress field shown in Fig. 8.00) with a maximum of83.5 MPa. A follO\\ il1g eigcnvaluc analysis (lBA) yields Ihc first eigclllnode shown in Fig. 8.6b) wilh an eigenvalue load faclor of R..,. 1.64. The dLosign check now reads as follows: that the MN for a case of this kind. Note also that RGIL\U

RII/es lor ,lie bllCklillg /im;'~llIte {b.W!\Wllet!l IU';/Ig g/oballlllmeric{l/ llll{/~I'SjS

1I compression Ilnd thc lo.... cst Sb). very similar to the basic Ie to choose fld as the overall

eket supports

•

Boundary conditions and a~sumpti{Jns~ Out-of-round defonnations lire constrained Ul the top and the bollom of the cylinder to simulate a stiffening element (i.e. roof and hopper). The bmd:eHo-columll connection is assumed to be stiff only allowing radial displacement of the brockct. Thc bmckct. which is locatcd at mid-height of the cylinder. is modelled as a rigid element. Calculation f\.-sults: From thc fini te clemcnt calcul3tions in which I:""

R",

2.238

Rf~ =

WIIS

IIpplied to thc struc ture

3.917

For fabrication toler.mce ci:ISS A.

Q 40. from which the imperfection amplitudc is 11l1"w' 0.612

For a unifonnly compresSl.-d cylinder without intcnllli pressure. H

Q)l"" :---:~o,:".6-=-2--:T"144 = 1+1.91(.111".. I)'

~• . p

0.62 t4-1

0.319

1+1.91{0.612) ·

ao. .. a, = 0.319 The assumed \'alm:s of the othcr global p.1r3mcler.; for this locally axially compressed cylindcr nre as follows:

fk.,

'I<

fJ,. = 0.60 " "" I

..1.....,,-0.20

Applying the calculation procedure

A." • JFRpl I FRrr - JRpl l R,. l,,. ..p =-

fJ

[Qo1,l(I - Pm' )

(2.23813.917)"

[0.315/( I-O.60)J°o5

(8.2)

0.756

(8.6)

0.MM7

Since -'\" .. /1 > 1,,1' > 1,,1•.0 or 0.887 > 0.756 > 0.20. the shell fa lls into the elastic-plustic region. The relcvant clastic-plastic interaction is then

23.19 N mm~

"'O,.•

'-P",[ Auv.p-Avl'.o A"..- A.". J' -

1_ 0.6[°·755 - 0.2]1.0 0.887 - 0.2

R" "" Z(II' HpJ l r", = 0.516·2.238 / 1.1

0.516

(8.4)

1.05 > 1.00 OK

The corresponding GMN IA calculation of this structure was undcnakcn using the rules of SL'Ction 8.7 of EN 1993- 1-6. The required imperfection al11plimde for GMN IA analysis is 1l\\·O.rqll 0.980. The impcrfeclion foml of the first lincar eigenmodc was adopted. with the peak displacemcnt ill\\urdly oricnted. The result was RGM,\"H - 1.78 so Rd """ Hc...I/\/f ' rll 1.78 / 1.1 1.62. This shows that the MNAILBA calculation procedure is significantly conscrv:lli\'c (1.62 » for 3 case of this ki nd. Note also that HG.\f~·',j 1.78 < 2.238 Hpland RG.U,\I4 ... 1.78 < 3.917 = R..,.

1.05)

11 7

RI/I

8.204.3 Ctlnicill rOtlf of II silo ~upportinJ! a

eentrall~

loeuled nIler n

0 2000

filter

35m

l

r,.t.

~

-2.'1

boundary

~ I cond;tlons of the

,f ~

«

FE l..-Jh_

_,,&odel

~~'td~!it -9

•

Flgurt M.9: br.t eigemnodc

T !""£hnical data Mutcriul: Structuml aluminium A1Mg £ 63 GPa f" KOMPa Fabriclllion qUlllity: C la!>s I) Dc5ign aCllon!ies to the e GMNIA analysis with the Juce a safe udJustmcm of the cd to cn~ure thut the rcsuil of n 'C not been omitted nnd that

alyscs. since il is not olwuys !)SI critical.

",('tlradolog)" is ha.~f...'li Oil ulmiolls. Ihe prillCiple 0/ does not app(I' lmd UC:liOlIS (ur '00(/ cases. IlOt possibll.'. nll!rl.'/ort' Ihe Ffes 0/ complele comhilll.'d lSeJ. phenomella are physically f.prcss·iI'lIlue p NOTE: The National Annex m:ly chootrehou ld be incorporutcd to cmer thc effects of imperfections that cannot be avoided in pructicc. including: a) geomctric imperfcction ... !>uch as: de\iallons from the nominal geomelric shape of the middlc surface (pre-dcfomlatlOns. oulof-roundne..s); - irrcguhlritic"lII :tnd ncar", cld!> (minor ccccntricitics, !>hrinkagc depressions. rolling curvature errors): - dcviation!> from nominal thickness: luck of e\'CnnCliS of ~upport,\;. b) TIllllcrial ifl1pcrft,.'ctions. such as: l'C!>iduul !>tresses cuu..cd by rolling. pressing. welding. )otruighlening 1.'11.',: inhomogcncitie~ und unisolropies,

it doe,f Iml IK'rmii afly plaMic:ih' to dew!!op at all (lIdllter throllgh lhic:klfess 1I0r spa/i(//~)' 011 Ihe shell sllf:fill"e),

It ;s (Il!mys Il('C('SS(II)' 10 ll'tlm 111m bijurcafioll e;gem'alue checb are required, .fitlce Ihe cOII.\'e{llIeIlC(·.~ o//ai/ul'l! w delet'l II "ijIlIT'lIt;ml tire I'ery' ,\eriOI/,\'.

]1,;s

;S

Ihe firsl ami mOSI gelleml .\·Wlemefll impery('("iOlI,\ ,ha' ,'wuld be l'OIIsidercd lI.\ IJUrt 0/ the de.\'igll, Mml,' a/ tl,(' ;tt'ms Oil IM\ l;sl IUII'(' 1I0t beell e.\'tell.fil,(,~I' researched, lIlId mme mll)' pili)' a rery' mulll mil!, The Iisl is gil'CII 10 ell.\lIre Ihlll 'he desigller cOIlsider.. all IHJ.Hihilitit·s I\},ert d(-'Iernlillillg 1111(1t qllC'SliollS tltc' 1I/1(I1y.ris ,rllol/Id midr('ss.

tlOOII' Ih"

(12) 1 he pattern of the iOlpcrfl'Ctinn~ ~hnuld

NOTE: hlr1hcr possible neglllil'e innucnccs on the impcrfcct chlstie-plll.o.tie buckling resistance R(;"'lA' )ouch alo ground 5Cttlements or nexibilities of conru.'Ctions or supports. are not classed as imperfcctions in thc ~n'iC of these pro\ iloions, (10) itnpcrfl'Clion!> lohould be allowed for in the

GMNIA nnalYlois by mcluding appropriate additIOnal quanulics in the anlllytical modcl for the numcrical computation,

EN 1993-1-6 I'c.'ql/irt'.\ 'hat imperfecliolls arc exp/icit~I' 1110&..'11('(/ 'IIImC'riclIlly ami ,,01 ,fimp~I' Irealed lI.f small perll/rhotimn to ,ItI.' pt.'r/(!('/ gt~(}lIIetn'. The real(}II i.\ 11t,1/ IH.'rwrho'iml '''('Or\' IIIUI' wlmelime.\ gil'e ,figllificWII errors at ;mperfi'('tioll amplillu/C'S Iltal appcar ill COIIstnlcted .flnlCtllre.f. II may m' 1I00ed that mltr!r lIpproac:ll(,.f 10 tlt(' imlJf!ry('l'lioll \(,II.\itil'ill' flf ,/tell,' IIlII'e heell proPfllUi. I" olle upprouc:ll, .\fK'('ifk pe,.tll,.lHllioll·hlic-plu"lic R(,\"IA of the :-.hd1. If pattern cannot be rea.\Onnblc doubt. the out for a sufficlcnt imperfcction pattern... Ilnd value of R(,\{\l,d .. hould be (13) Tho c;""H",>I'-"r.~ u:.\.'d unlelo~ a ditlercnt

bcjustified.

'OTE: The eigcnmodc critical buckling ~kk ' critical bucklin~ rc'I,Wm:c analysllo of the pcrrl..'Cl 1

Rules for till! bllCklillg /imit slllie {IllCSSmem /lsillg global II"merical OIllllpi.r

holllld Ire{llme/ltS ftJr a/l shells (e.g. Croll ami E/liIULf. 1985). AIIII'ese aPIJrtJlJChe.f slill m.'I!d to be relall.'d 10 measurable lo/enmccs, ,fo af'(' IIOt yel applictlhle ill design. For tltis reaSon, lite neulral wordillg "{lppropriole mldilimllli qllalllilie!J'" has beell dtosell ill EN 1993· 1·6.

IiI lin.\' pltl.\liC"it\' 10 den"op 01 Ilgh lhid:tle'ss /lor SfHlliull.' ml I. to U'Cln! Iltal hiplr(:tlliml h lire' rl'qlliri!d, ,rilln' lite fllilure 10 th'I('C'1 (I hijilrcatioll

'Hat:\,

IIlId mO,fl gl'/Il'nll SUltt'IIIC'1I/ '1t.'rji.'('liul/!J 11t"1 \'hollld !H.' In of III,' tks(e.II, \I(IIn' of tltc' \! huw.' 1/01 !H.'c'n C,I'lemil'e/I' 10m.' Rlm- pltll, tI \'('/)' ,s1//all ;s gin;1I 10 t'lI.ntre Iha/ the' it'n all po.5.fihiliti,''\ II-I/e", I qllntiOl,.f Ih.' lIflcl~\"!;i.f .\I"mld

(II) The imperfcctions should geneOilly be illlroduecd by means of equivalcnt geomctric imperfcctions in thc fonn or initial shlll)C dcviations perpendicular to the middle surfacc or the perfect shell, unless a bcller tcchnique is used. The middle surruec of the gCOIllI:tricully imperrcct !then should be obtained by superposition or the equivalent geometric imperfections on the perfect shell geometry.

Ti,e domilllllll ftm" of imlX!Ifi'Climl Iltal is knoll'n 10 Ital'c a dclele,,;olls 'ffcct 01/ sltell slrc"glh is a {Iel'ill/ioll oflhe slwpe from i/~ ideal forlll , I;'urlher, more research IIlL\' bel'li I",denakell 0" IIIi.f form of illlperfi"'lioll IIt(III {lilY otltcr. both ill rel"tioll to lite !Jltell gt.'olllclric fOl""'s, Ihe 100ui cases alld tlte holll/dm}' co"diliolls. Geometric impeifectiOlu of llti.f form are l!terefore Cllrrelll(,' b)' for lite be.fl choice for lise lIS represelllalilY! imJJ4!rfi'Clioll tyIX'. HOlI'el'Cr, tlte desigller/allalysl sltolild 1101 forgt'l IIt"1 Ihese shape del'illliol/S are "royj\'(lfCIll" impeifecliollS. {Illd Ilml lite\' nllISI al.w COIY!r lite effect'i of all other '-'7N!.f of gcometric illllX!rfi'Clion (e.g. mCIIlbr{me lad of fil, bolllllltl'J' errors, fad: o/foodillg aligllml'"1) and all types of material imlX!ifL'Cli()l1S (e.g. re.\'idllal wreues). as indicated ill lite liM ill (9), The IImllre of l'lillimiem imlX!ifL'l.'fiOll.1 i.\ disclls,l'cd il/mOl'e dewil ill 8.3,3,3 10 8.3.3,5 ..

/lin'.f Ihal illlpt'rfi'dimu arc: I/IIR1t'rit'a/~r IIrre/,wl si"'IJh' p.'rtllrlotJliOlI\ 10 Ihe paft-oct Tl'U)()1I i.1 Ih"l pertllrhalio/l 'ml'~ girl' _figlli(it'tllll e:mJrs al 'Plilllt!..,'j Illal appt!ur 11/ In'f. approtlChes 10 lite: ili\'j,\, o(llrd/l hm'e hel'" Ollc.' tlpprudc-II. VH.·ci(ic jilalgorilhRl!J m¥.' lI.fe:cI 01/ Ihe: rcJdIlI·elllllllaru/,.'IIRIIt /0.1,1 1985; DillJ.ler mill Knoke, rht'fl han' propo.lI'd 1(}II'c'r ,/ldl Ofha

( 12) The pancm of the equivalent geometric imperfL'Ctions should be chosen in such a rorlll Ilmt it hns the most unravoumblc effcct on thc imperfect clastic-plastic buckling rcsistance R(,""IlA or the shell. If the most unravoumble pallcm cannot be readily idelllifk-d beyond reasonable doubt. the analysis should be carried out for a sufficielll number or different imperrection panems, and the worst case (Io"cst value of R('~L'lA) should be idl.:ntified.

a IIItllfer (If prillciple, Ihe forlll of lite geometric imlX!ifecliolls .fltould l"fI'e lite mOSI IIII!m'OlIrtlble l1J"ect Oil lite hllCklillg r£'sisttlllce of lite .ftrllclllrC'. Tlte cltil1 rea,I'OI' \l'h)' {III Iltese forntf IIeed to he cO/I.~ider!!{1 is illllN!rfi't:lioll!o; ojlIIa"y clifferelll sill/IX'S alld form!J' al"e foulld ill real shell cOllsln/clioll, alld a reliallcc on (III as.fllmed f0ntl of illlpeifecliOll is lII/SlIfe IIIIII.'SS il i.f a l'e/)' tmftn'OlIrtlble form_ Ne\·erll,eleu. lite strici reqlliremelll giI'C.'I/ ill lhi!>' principle is .fOfielled ill laler dlll/sf!S.

(13) The eigenmode-affine pallcm !thould be used unless a diffcrent unfa\'ournble pallenl can be justified.

Tile mode Ihal is fOlmd ill all LBA tlllairsis is ofiell 11,(' mosl mt/m'Ourahle, til snmll amplillldc.f, 10 lite elastic bllCklillg re.fi.~ItIllCC oj ,fltelk TI,i.f dttl/se Ihere(ore ,flllte.f tl\ill/ple llep l!tal call he etn'i~)' takell 10 uscerwill whelher lite stru('{lIral cOII{tgllrtllion i.f likely 10 he I'ery' imperfection sellsitiw.'.

NOTE: TIle cigcmnode affine pallem is the critical budding mode associated With the clastic critical buckhng resistance RN' based on an LUA nnalysis of the perfcct shell ,

A,\'

tL~Sllmed

,It",

110II'L'l'('r, lhis simple impeifel'tioll fon" i.\ IIIIICit less reliably de/eleriO//.~ IIt{lIl il i,1 for hellll/, COIIlIllIl alld pfme ,flrllCfllr!!J. Ti,er!! are lIIallY sltell fmcklillg exall/ples where litis lillem'

Bl/d,lillg 0151(-'('1 Sltell.1 - E"'"fl/Je.-'illl D"_ligll R{'C(III1111elldc,liom

I'/gl'II1111H/e 1.1' 1/01 Ihe IIIO~I IItlfill'OlII·lIh/l'. ('.\/x'(:i(ll/\' IIhere lite Imdlillg /IIfNle if /tN:III (SOftg "I tIl, !O(}5) or 1I11I.'re Ihe prehm'4/illg hl'hol'iollr is high~l· lIunlill('tll" (Greill('r tlml /J('r1er, 1995: f)fH.'rk/, ('( a/., 10(5) or "hen' plc,.lIicil.\' IIUlI' olIO ul[e('/ IIIl' fmU'Oflll' (St:hm'idl'r. ](}04). II i~ IllI.'njh/"(' illlpo/"/tI1If Ihal Ihe ill/IIIY.VI doc:. 1101 place too /llIIcli re/i(I//('t' Oil calc"lalioll\' Iltllt ".Il' IM~ mud" alolle. F'lI"lht'/·l1w/·e, corefltl t'fl//I'lderalifm :,'lIoltld ht, WI'l'lI Iu hij.!/lI.'/" I//IHie.\ fillllld ill (1/1 LHA (l/Ia(I·~i;, .5illce Ihell.' ,\'olllelilll(,\ are /l/O/"(' c/"iticul IlwlI III(' 101\"('51 t'igclI/l/ot/e (set· (Ilso 8.3.3 ..5 ami R.3.3.6).

(14) The pallem of thc equivalcnt geomctric

impcrfl."ctlOns should. if prncllcablc. reflect the conMnlctional dctlliling and the boundary condition ~ in un unflllournbic manncr.

(15) Not"'ithManding (13) And (14), patterns

may be eJ(clud(..-d from the il1lc!otigation if they can be eliminated as unrealistic bccllUSC of the method of fabrication , Illanulncture or crection.

The IlIIper/eeliall fanll.f Ilwl ore /l/a.~1 criliclIllO lite dc_\igll lire Iho.\"e 1i/tIi l\'ill lwife ill prtll'licl', so lhis ~lal(''''elll i.\ ('/"itic(11I1' i/l//Nlrlflltl_ II /l/oy he diffkllil to delel"l"itle II'hal 1)1X'l' of imper/t'Clioll lite fllhricllliall ami COl/slmClioll IJ/"()ceue.~ lI'ill elltlSe, hili )I'here Ihif i.f pauihie, Ihe,\e illI/Jt-'/fccliO/I.\5/lfmld he tKiupled illlo Ille model.

'''1'

abu\'e gt'lle/"(Il ,H(I/(,lIIell/f, il il FolIOIrillg permiltl'd I{J igllo/"(-' IIIIIW,!i-l"lic forms o( imperfection ely.'" if IIIc~I' IKllcmially "m·l' 1I \'e/':t' lkleleriolls effi.'(·1 WI hlldr,lillg re,fi.f/(IIIC('. Ollt' eXlilI/ple of ufoI'''' IIltll COIlld he idemified he/"e is lite eige//l//(xle a.Bo!'illled lI'ilh high ~hl'llr sm'ues (IOl".fioll), 1I'/IIdl il /1II!ik£'lI' W be Icell (II a re,\'II11 o//(lII";('(lli(1I/ p'·(N·(-'HC'f.

( 16) Modification of thc adoph:d mode of geomctric imperfcctions to includc realistic ~t nlc tur.tl dct!!ils (such as ax isymmctric wcld dcprcM,ions) ~ h ou ld be cxplon.-d,

The jim" dcfined ill (14) i.f he/"e c!(II"ijit-d ill' I"f.jerellce tolpel"ijk ((1/"//1.1' ,hal (II"(' A·llml'l/ In he u-ide(l· jiJllml ill pra{'lit'lIl C'O/Ml"llclimt, One itle/llijied /ol"m 'WI lIIelltimlt-'d here il Ihe dimple WIISl'd hy iIllIN.'Ifi·{·1 Jiltillg 0/ phlll"~ NO II'.: I he Nat ional Anncx may definc additional (HoI.\! el aI, 1999), 11111 rhi5 i.f mort' requirements for thc 1I!(l1/ t'(II/~it!.'ruliOIl

hiMher m(xl.',1 /iI/lilt! ill UII tlw,W! ,fOlllt'limt',f urt' 11101'(' 1111\oIA'Rc.W'~A) will be much larger than the clastic imperfection

Rcwmmcndc

d \alue of V o '

0,010

factor a. and no c1ol>C comparison enn be

0.016

expected. However, where the resistance is controlled by buckling phenomena thut tire

0,025

'J geometric ilflpe'fi'Clioll (lIId /I'raflcf' meusllre are required alell, the imperfection ,f/WI/lt! the light of the metlSlIremell' rI/