ELASTIC AND PLASTIC BUCKLING OF SPHERICAL SHELLS UNDER VARIOUS LOADING CONDITIONS SHAHIN NAYYERI AMIRI

Transcription

1 ELASTIC AND PLASTIC BUCKLING OF SPHERICAL SHELLS UNDER VARIOUS LOADING CONDITIONS by SHAHIN NAYYERI AMIRI B.Sc., Univesiy o Tbiz, M.Sc., Univesiy o Tbiz, M.Phil., Univesiy o Tbiz, 6 M.Sc., Knss Se Univesiy, AN ABSTRACT OF A DISSERTATION sbmied in pil lillmen o he eqiemens o he degee DOCTOR OF PHILOSOPHY Depmen o Civil Engineeing College o Engineeing KANSAS STATE UNIVERSITY Mnhn, Knss

2 Absc [Spheicl shells e widely sed in eospce, mechnicl, mine, nd ohe indsil pplicions. Accodingly, he cce deeminion o hei behvio becomes moe nd moe impon. One o he mos impon poblems in spheicl shell behvio is he deeminion o bckling lods eihe expeimenlly o heoeiclly. Theeoe, in his sdy some elsic nd plsic bckling poblems ssocied wih spheicl shells e invesiged. The is p o his esech sdy pesens he nlyicl, nmeicl, nd expeimenl esls o modeely hick nd hin hemispheicl mel shells ino he plsic bckling nge illsing he imponce o geomey chnges on he bckling lod. The hemispheicl shell is igidly sppoed ond he bse cicmeence gins hoizonl nslion nd he lod is veiclly pplied by igid cylindicl boss (Loding co) he pex. Kinemics sges o iniil bckling nd sbseqen popgion o plsic deomion o igid-peecly plsic shell models e omled on he bsis o Dcke- Shield's limied inecion yield condiion. The eec o he dis o he boss sed o pply he loding, on he iniil nd sbseqen collpse lod is sdied. In he nmeicl model, he meil is ssmed o be isoopic nd line elsic peecly plsic wiho sin hdening obeying he Tesc o Von Mises yield cieion. Finlly, he esls o he nlyicl solion e comped nd veiied wih he nmeicl esls sing ABAQUS sowe nd expeimenl indings. Good geemen is obseved beween he lod-delecion cves obined sing hee dieen ndmenl ppoches. In he second p, he Sohwell s nondescive mehod o colmns is nlyiclly exended o spheicl shells sbjeced o niom exenl pesse cing dilly. Sbseqenly inie elemen simlion nd expeimenl wok shown h he heoy is pplicble o spheicl shells wih n biy xi-symmeicl loding oo. The esls showed h he echniqe povides sel esime o he elsic bckling lod povided ce is ken in inepeing he esls. The selness o he mehod lies in is geneliy, simpliciy nd in he c h, i is non-descive. Moeove, i does no mke ny ssmpion egding he nmbe o bckling wves o he exc loclizion o bckling.]

3 ELASTIC AND PLASTIC BUCKLING OF SPHERICAL SHELLS UNDER VARIOUS LOADING CONDITIONS by SHAHIN NAYYERI AMIRI B.Sc., Univesiy o Tbiz, M.Sc., Univesiy o Tbiz, M.Phil., Univesiy o Tbiz, 6 M.Sc., Knss Se Univesiy, A DISSERTATION sbmied in pil lillmen o he eqiemens o he degee DOCTOR OF PHILOSOPHY Depmen o Civil Engineeing College o Engineeing KANSAS STATE UNIVERSITY Mnhn, Knss Appoved by: Mjo Poesso D. Hyde A Rsheed

4 Copyigh SHAHIN NAYYERI AMIRI

5 Absc [Spheicl shells e widely sed in eospce, mechnicl, mine, nd ohe indsil pplicions. Accodingly, he cce deeminion o hei behvio becomes moe nd moe impon. One o he mos impon poblems in spheicl shell behvio is he deeminion o bckling lods eihe expeimenlly o heoeiclly. Theeoe, in his sdy some elsic nd plsic bckling poblems ssocied wih spheicl shells e invesiged. The is p o his esech sdy pesens he nlyicl, nmeicl, nd expeimenl esls o modeely hick nd hin hemispheicl mel shells ino he plsic bckling nge illsing he imponce o geomey chnges on he bckling lod. The hemispheicl shell is igidly sppoed ond he bse cicmeence gins hoizonl nslion nd he lod is veiclly pplied by igid cylindicl boss (Loding co) he pex. Kinemics sges o iniil bckling nd sbseqen popgion o plsic deomion o igid-peecly plsic shell models e omled on he bsis o Dcke- Shield's limied inecion yield condiion. The eec o he dis o he boss sed o pply he loding, on he iniil nd sbseqen collpse lod is sdied. In he nmeicl model, he meil is ssmed o be isoopic nd line elsic peecly plsic wiho sin hdening obeying he Tesc o Von Mises yield cieion. Finlly, he esls o he nlyicl solion e comped nd veiied wih he nmeicl esls sing ABAQUS sowe nd expeimenl indings. Good geemen is obseved beween he lod-delecion cves obined sing hee dieen ndmenl ppoches. In he second p, he Sohwell s nondescive mehod o colmns is nlyiclly exended o spheicl shells sbjeced o niom exenl pesse cing dilly. Sbseqenly inie elemen simlion nd expeimenl wok shown h he heoy is pplicble o spheicl shells wih n biy xi-symmeicl loding oo. The esls showed h he echniqe povides sel esime o he elsic bckling lod povided ce is ken in inepeing he esls. The selness o he mehod lies in is geneliy, simpliciy nd in he c h, i is non-descive. Moeove, i does no mke ny ssmpion egding he nmbe o bckling wves o he exc loclizion o bckling.]

6 Tble o Conens Lis o Tbles... ix Lis o Figes... x Acknowledgemens... xv Dedicion... xvi Pece... xvii A Review o Liee... xxi I. Hisoicl Bckgond o he Spheicl Shell Bckling... xxi II. A Bie Hisoy o Yield Line Theoy... xxxii III. Hisoicl Bckgond o he Sohwell Mehod...xxxiii CHAPTER - Plsic Bckling o Hemispheicl Shell Sbjeced o Concened Lod he Apex.... Inodcion nd Ppose o his Chpe.... Peliminy Consideions.... Anlyicl Fomlion Kinemics Assmpions The Iniil Collpse Lod nd Revesl o Cve Popgion o he Dimple Solion o he Complee Poblem Nonline Finie Elemen Anlysis (FEA) Elemens nd Modeling Nmeicl Resls Expeimenl Pogm Pmees nd es sep Expeimenl Resls: Resls nd Discssion Conclsions... 5 Noion sed in his chpe CHAPTER - Nondescive Mehod o Pedic he Bckling Lod in Elsic Spheicl Shells vi

7 . Inodcion nd Ppose o his chpe Sohwell Mehod in Colmns Ageemen o Tes Resls in Colmns An Exension o he Sohwell Mehod o colmns in Fme sce Fomlion o he Govening Eqions Membe Fomlions nd Solions Sohwell Plo Cse Sdy Discssion Resls nd Conclsion The Sohwell Mehod Applied o Shells Deomion o n Elemen o Shell o Revolion Eqions o Eqilibim o Spheicl shell Eqions o Eqilibim o he Cse o Bckled Sce o he Shell Bckling o Uniomly Compessed Spheicl Shells Sohwell Pocede Applied o Shells Nonline Finie Elemen Anlysis (FEA) Expeimenl Pogm Resls nd Discssion Expeimenl wok indings Nmeicl Sdy esls Fo niom dil pesse cse wih hinge sppo: Fo niom downwd pesse cse wih hinge sppo: Fo niom dil pesse cse wih olle sppo : Fo Ring lod cse wih hinge sppo: Conclsion Noion h sed in his chpe Reeences Appendix A - Collpse lod o cicl ple Solion o Tesc Ple... 6 Appendix B - Axisymmeiclly Loded Cicl Ples... 6 vii

8 Appendix C - Eec o Axil Foce on he Siness o he Fme Membe viii

9 Lis o Tbles Tble. : Nos.,,, b, 4, 4b, 5 nd 6. Mild seel: Modls o Elsiciy 7 kg cm Tble. : T. Von Kmn s ess Tble. : Robeson s S No:5. Mild seel: Eecive lengh-.5 inches Tble. 4: Chceisics o Wood s Fme Tble. 5: Resls o Membe 8 o Wood s Fme sbjeced o he given loding pen Pkips, inches ix

10 Lis o Figes Fige. : Smple conscion pocede o he expeimenl sdy... Fige. : Geomey nd pos bckling o hemispheicl shells sbjeced o concened lod... Fige. : Iniil bckling nde concened lod... 4 Fige. 4: Pos bckling deomion iniil collpse lod P... 5 Fige. 5: Plsic bckling deomion exends owd nde n incesing lod P... 6 Fige. 6: n-sided egl polygon ple cying single concened lod is cene... 8 Fige. 7: Poile o deomion ding iniil bckling nd pos bckling behvio... Fige. 8: Eqilibim o n xisymmeic shell elemen... Fige. 9: Deomed p o hemispheicl shell shpe beoe he secondy bicion poin9 Fige. : Eec o xil oce on plsic momen cpciy... 9 Fige. : Deomion pen o he hemispheicl shell sing 8 node xisymmeic ecngl shell elemen.... Fige. : Dieen cs o he deomed modeely hick shell R /... 4 Fige. : Deomion pen o he modeely hick shell R / sing six node ingl shell elemen... 5 Fige. 4: Bckling iniiion o hin shell R / sing six node ingl shell elemens Fige. 5: Sbseqen deomion o hin shell R / showing he secondy bicion phenomenon Fige. 6: Dieen cs o he deomed hin shell R /... 8 Fige. 7: Deomion pen o he hin shell R / showing he secondy bicion phenomenon Fige. 8: Dieen hemispheicl shell smples wee mde o expeimenl sdy... Fige. 9: Thee dieen boss size sed o loding ( b.5,.5,5. mm )... Fige. : Gooved bse ple s sppo o wo sizes o hemispheicl shells... x

11 Fige. : Riehle Univesl esing mchine o displcemen conol mesmens... Fige. : Sess sin digm o coppe lloy... 4 Fige. : Sess sin digm o Bonze Fige. 4: Sess sin digm o Sinless Seel... 5 Fige. 5: Deomion o he modeely hick shell (Coppe lloy R / 75)... 6 Fige. 6: Iniil bckling nd pos bckling o he hin shell (Sinless Seel R / 66 ).. 6 Fige. 7: Degeneion o xisymmeic deomion o he hin shell (Sinless Seel R / 66 ) Fige. 8: Secondy bicion deomion o he hin shell (sinless seel R / 66 ). 8 Fige. 9: Iniil bckling nd xisymmeic pos bckling o he hin shell (Bonze R / 5)... 9 Fige. : Degeneion o he xisymmeic deomion o he hin shell (Bonze R / 5)... 4 Fige. : Secondy bicion deomion o he hin shell (Bonze R / 5)... 4 Fige. : Dieen sges o he ingl wih secondy bicion (Bonze R / 5)... 4 Fige. : Finl deomion o he hin shell (Bonze R / 5 )... 4 Fige. 4: Dimensionless Lod delecion cve o coppe lloy shell ( R / 75 ) Fige. 5: Compison o iniil collpse esponse o coppe lloy shell ( R / 75 ) Fige. 6: Dimensionless iniil collpse lod vs boss size o coppe lloy shell ( R / 75 ) Fige. 7: Dimensionless iniil collpse lod vs iniil collpse ngle o coppe lloy ( R / 75 ) Fige. 8: Iniil collpse ngle vs boss size o coppe lloy ( R / 75 ) Fige. 9: Dimensionless Lod vs knckle meidionl ngle in d o coppe lloy shell ( R / 75 ) Fige. 4: Dimensionless Lod delecion cve o Bonze shell ( R / 5) Fige. 4: Compison o iniil collpse esponse o Bonze shell ( R / 5) xi

12 Fige. 4: Dimensionless iniil collpse lod vs boss size o Bonze shell ( R / 5) Fige. 4: Dimensionless Lod delecion cve o Sinless Seel shell ( R / 66 ) Fige. 44: Compison o iniil collpse esponse o Sinless Seel shell ( R / 66 )... 5 Fige. 45: Dimensionless iniil collpse lod vs boss size o Sinless Seel shell ( R / 66 )... 5 Fige. 46: Nmeicl bicion poin o Bonze, Sinless Seel, nd Coppe lloy vess ( R / ) Fige. : Colmn nde compession oce Fige. : Lod Delecion cve in Sohwell Mehod... 6 Fige. : Noion o membe nd sce... 7 Fige. 4: Wood s me... 8 Fige. 5: Loding o me, Sowell plo o membe Fige. 6: Elemen ken om shell by wo pis o djcen plns noml o he middle sce Fige. 7: Spheicl shell elemen nd coesponding oces Fige. 8: Meidin o spheicl shell beoe nd e bckling... 9 Fige. 9: Deomion pen o hemispheicl shell wih hinge sppo nde dilly niom pesse... 7 Fige. : Sbseqen deomion o hemispheicl shell wih hinge sppo nde dilly niom pesse... 8 Fige. : Deomion o hemispheicl shell wih hinge sppo nde mximm dilly niom pesse... 9 Fige. : Dieen cs o he deomed hemispheicl shell wih hinge sppo nde dilly niom pesse... Fige. : Sbseqen deomions in he cs o he deomed hemispheicl shell wih hinge sppo nde dilly niom pesse... Fige. 4: Bckling iniiion o he hemispheicl shell wih olle sppo nde dilly niom pesse... xii

13 Fige. 5: Sbseqen deomion o he hemispheicl shell wih olle sppo nde dilly niom pesse... Fige. 6: Second mode o he deomion in hemispheicl shell wih olle sppo nde dilly niom pesse... 4 Fige. 7: Bckling iniiion o he hemispheicl shell wih hinge sppo nde ing lod in R... 5 Fige. 8: Bckling o he hemispheicl shell wih hinge sppo nde ing lod R.. 6 Fige. 9: Sbseqen deomion o he hemispheicl shell wih hinge sppo nde ing lod R... 7 Fige. : Lge deomion o he hemispheicl shell wih hinge sppo nde ing lod disibed R... 8 Fige. : Bckling iniiion o he hemispheicl shell wih hinge sppo nde ing lod in R... 9 Fige. : Bckling o he hemispheicl shell wih hinge sppo nde ing lod R.. Fige. : Lge deomion o he hemispheicl shell wih hinge sppo nde ing lod disibed R... Fige. 4: Bckling iniiions o he hemispheicl shell wih hinge sppo nde gviy loding... Fige. 5: Sbseqen deomion o he hemispheicl shell wih hinge sppo nde gviy loding... Fige. 6: Hemispheicl shells smples mde o polyehylene... 5 Fige. 7: A es mde o R75 mm shell sing scion pesse nd hee displcemen gges vios poins Fige. 8: Deomion mesemen wih hee gges dieen locions in hemispheicl shells nde niom scion pesse (R 75mm)... 7 Fige. 9: Tess mde o R5 mm shells wih scion pesse nd hee displcemen gges dieen locions hemispheicl shells nde niom scion pesse (R 5 mm).. 8 xiii

14 Fige. : Deomion mesemen wih hee gges in dieen locions hemispheicl shells nde niom scion pesse (R 5 mm)... 9 Fige. : Iniil bckling o hemispheicl shells nde niom scion pesse (R 5 mm).... Fige. : Iniil bckling o hemispheicl shells nde niom scion pesse (R 5 mm).... Fige. : Iniil bckling o hemispheicl shells nde niom scion pesse (R 75 mm).... Fige. 4: Sevel ess mde on dieen smples sing scion pesse wih hee nd ive gges dieen locions.... Fige. 5: Plo o Fige. 6: Plo o Fige. 7: Plo o Fige. 8: Plo o Fige. 9 : Plo o Fige. 4: Plo o w gins w ( R 5mm,. 5mm p )... 5 w gins w ( R 5mm,. 5mm p )... 6 w gins w ( R 5mm,. 5mm p )... 7 w gins w ( R 75mm,. 5mm p )... 8 w gins w ( R 75mm,. 5mm p )... 9 w gins w ( R 5mm,. 5mm p )... 4 Fige. 4: Plo o p gins w ( R 75mm,. 5mm )... 4 Fige. 4: Plo o Fige. 4: Plo o w gins w ( R 5mm,. 5mm p )... 4 w gins w ( R 5mm,. 5mm p )... 4 Fige. 44: Plo o w gins w ( R 5mm,. 5mm p ) Fige. 45: Compison o Sohwell expeimenl pedicion o heoeicl bckling pesses xiv

15 Acknowledgemens Fis nd mos, ll pises be o soce o ll goodness who is he Lod o he nivese, he mos gcios, nd he mos mecil. I wold like o expess my deepes gide o my dviso, Po. Hyde Rsheed, o he immesble mon o sppo, gidnce nd vlble dvice hogho he esech, nd lso o his osnding peomnce in eching Advnced Scl Anlysis I, Advnced Scl Anlysis II, nd Theoy o Scl Sbiliy coses. His conined sppo led me o he elizion o his wok. I wold lso like o exend my ppeciion o my commiee membes: D. Asd Esmeily, D. Hni Melhem, nd D. Jones Byon o hei dvice ding my esech. I exend my sincee hnks o M. Seid Klmi o helping me o dw he iges nd o his iendship. I m lso indebed o he enie s o he Khk Kvn es Lbooy, which hs peomed he expeimens h e eled o his sdy wih pience, ce nd pesevence. Specil sincee hnks e de o M. Amili Mhoi he mnge o his lb, o his coopeive gidnce, cel spevision nd vey helpl sggesions Finlly, vey specil hnks e o ll he cly membes o he Civil Engineeing Depmen Knss Se Univesiy, o hei conibions o my edcion he gde level. xv

16 Dedicion This disseion is dediced o my mohe, who hs lwys given me he encogemen o complee ll sks h I ndeke nd coninosly sppos me wheneve I ce ny diicly in my lie s well s o he love nd sciice. To my sise, Ysmn, who hs lwys inspied nd helped me. xvi

17 Pece A shell cn be deined s body h is bonded by wo closely spced pllel cved sces. A shell is ideniied by is hee ees: is eeence sce h is he locs o poins which e eqidisn om he bonding sces, is hickness, nd is edges. O hese, he eeence sce is he mos signiicn becse i deines he shpe o he shell whee is behvio is govened by he behvio o is eeence sce. The hickness o poin o shell is he lengh o he noml bonded by he bonding sces h poin. Edges o he shell e designed by ppopie vles o he coodines h e esblished on he eeence sce. Shells my hve no edges ll, in which cse hey e eeed o s closed o complee shells. A spheicl shell is genelizion o n nnls o hee dimensions. A spheicl shell is heeoe he egion beween wo concenic sphees o dieing dii. A shell is clled hin i he io o is hickness o is minimm pinciple dis o cve is smll comped o niy. A shell is sid o be shllow i he io o is mximm ise o he bse dimee is smll. The nlysis o shells o evolion consideing nonlineiies is o imponce in vios engineeing es. When nlyzing shell sce sbjeced o given loding one cold mke se o he genel eqions o he hee dimensionl heoy o elsiciy o come p wih he se sess ny given poin. Howeve, hese eqions e qie compliced nd in only ew idelized cses cn solion be chieved. Fo his eson, hee dimensionl inciden is ppoximed by mking se o wo dimensionl heoy o elsiciy. The ollowing ssmpions e he bsis o he clssicl line shell heoy.. Shell hickness is smll xvii

18 . The displcemens nd oions e smll. The nomls o he shell sce beoe loding emin noml e loding 4. The nsvese noml sess is negligible The mos common shell heoies e bsed on line elsiciy conceps. Line shell heoies deqely pedic sesses nd deomions o shells exhibiing smll elsic deomions, h is, deomions o which i is ssmed h he eqilibim eqion condiions o deomed elemens e he sme s i hey wee no deomed nd Hook s lw pplies. The nonline heoy o elsiciy oms he bsis o he inie delecion nd sbiliy heoies o shells. Lge delecion heoies e oen eqied when deling wih shllow shells, highly elsic membnes nd bckling poblems. The nonline shell eqions e considebly moe diicl o solve nd o his eson e moe limied in se. Shells ply n impon p in ll bnches o engineeing pplicions, especilly in eospce, ncle, mine nd peochemicl indsies. The sophisiced se o shells incomponens e being mde, sch s missiles, spce vehicles, sbmines, ncle eco vessels, nd einey eqipmen is vey common. As he shells e sbjeced o vios loding condiions sch s exenl pesse, seismic nd/o heml lods, compessive membne oces e developed which my cse he shells o il de o bckling o compessive insbiliy. Among shell sces, he spheicl shell is sed eqenly in he om o spheicl cp o hemisphee nd ecenly, he poblem o he bckling o spheicl shells hs eceived consideble enion. Accodingly, in he pesen sdy, eise o wo independen ps elsic nd plsic bckling o spheicl shells nde vios loding condiions e invesiged. xviii

19 Objecives nd Resech Mehodology Spheicl shell sces e widely sed in sevel bnches o engineeing. The clss o shells coveed hee in e hin, nd modeely hick so ile by bckling is oen he conolling design cieion. I is heeoe essenil h he bckling behvio o hese shells is popely ndesood nd hen sible mhemicl models cn be esblished. The objecives o his sdy e sed below: The is chpe o his sdy pesens he nlyicl, nmeicl, nd expeimenl esls o modeely hick nd hin hemispheicl mel shells ino he plsic bckling nge illsing he imponce o geomey chnges on he bckling lod. The hemispheicl shell is igidly sppoed ond he bse cicmeence gins veicl hoizonl nslion nd he lod is veiclly pplied by igid cylindicl boss c he pex. Kinemic sges o iniil bckling nd sbseqen popgion o plsic deomion o igid-peecly plsic shells e omled on he bsis o Dcke- Shield's limied inecion yield condiion. The eec o he dis o he boss, sed o pply he loding, on he iniil nd sbseqen collpse lod is sdied. In he nmeicl model, he meil is ssmed o be isoopic nd line elsic peecly plsic wiho sin hdening obeying he Tesc o Von Mises yield cieion. Boh xisymmmeic nd D models e implemened in he nmeicl wok o veiy he bsence o non-symmeic deomion modes in he cse o modeely hick shells. In he end, he esls o he nlyicl solion e comped nd veiied wih he nmeicl esls sing ABAQUS sowe nd expeimenl indings. Good geemen is obseved beween he lod-delecion cves obined sing hee dieen ppoches. xix

20 In he second chpe, Sohwell s nondescive mehod o colmns is exended o spheicl shells sbjeced o niom exenl pesse cing dilly. Sbseqenly by mens o inie elemen simlion nd expeimenl wok, i is shown h he heoy is pplicble o spheicl shells wih n biy xi-symmeicl loding. Fo his echniqe ny mesble deomion my be sed. The esls showed h he echniqe povides sel esime o he ciicl lod povided ce is ken in inepeing he esls. The selness o he mehod lies in is geneliy, simpliciy nd in he c h, i is non-descive. Moeove, i does no need ny ssmpion egding he nmbe o bckling wves o he exc locliy o bckling. xx

21 A Review o Liee I. Hisoicl Bckgond o he Spheicl Shell Bckling The is poblems o insbiliy, concening lel bckling o compessed membes wee solved bo yes go by L. Ele. A h ime he pinciple scl meils wee wood nd sone. The elively low sengh o hese meils necessied so scl membes o which he qesion o elsic sbiliy is no o pimy imponce. Ths Ele s heoeicl solion, developed o slende bs, emined o long ime wiho pplicion. Only wih he beginning o exensive conscion o seel ilwy bidges did he qesion o bckling o compession membes become o pcicl imponce. A he beginning o he wenieh ceny, he conscion o hin einoced concee shell concee oos ws widesped in Eope. This oo is o he ype whee cylindicl shell wih spn beween. nd 5. m is bil beween ch bems h give he shpe o he oo. These ches hve ie bem o esis hss nd hee is heeoe only veicl ecion on he pies. Aches e plced he boom side o he shell. A his peiod concee ws consideed o be n elsic nd line meil h obeyed Hooke's lw nd he ches wee heeoe nlyzed in hese ems. In Gemny, Wle Beseld nd Megle, enginees Dyckeho nd Widmnn, bil he is spheicl dome o concee in 9. In ode o bild he dome, hey poposed inslling spheicl ne o seel bs nd Megle sggesed pojecing concee gins omwok. The spheicl shpe o he dome llowed he se o he sme pieces o omwok gin nd gin. The dome ws nlyzed like coninos sce. xxi

22 The conscion o he dome Jen ciy ws mde possible by Po. Spngenbeg's epo. Conscion begn in he wine o The bs close o he edge sed o bckle nd some sbilizion bs wee needed. In his conscion, Beseld nlyzed he bending momen nd deomion. In he is dome (Jen, 6. m spn), no only wee he in plne ension nd compession in he pln o he dome ken ino ccon, b bending momens nd deomion wee lso sdied. The heoy o he igid o dome oion ws pblished by Föppl, Dng nd Zwng. Second ode dieenil eqions wee needed o solve he poblem. Beseld ond n ppoch which yielded solion, in which he Zoelly oml ws sed o nlyze he poblem o bckling, which gives sey co o. Beseld sked D. Geckele o ndeke some expeimens. He did mny ess nd ond h in he lods close o he Zoelly oml bckling s. In he mn o 9 Tooj begn sevel pojecs wih shell sces. The is pojec he ndeook ws he oo o Algecis Mke. This ws dome o 46. m spn, sppoed by 8 pies. The shell consised o spheicl concee conscion. The shell ws bil sing wooden omwok on scold. Wih his mehod hee ws no poblem wih bs bckling s hd hppened o Beseld wih he conscion o his is dome in Jen. In 94 Flügge poposed vle o he ciicl bckling lod o spheicl shells. Howeve he expession ws given o ll sphee. Von Kmn nd Tsien (99) showed h he se o sbiliy o some sces, slly shell like sces, is wek. In ohe wods, smll disbnce migh cse hem o snp ino bdly deomed conigion. They lso emped o explin he discepncy beween he clssicl nd excemenl bckling pesses o clmped shllow spheicl shells xxii

23 nde niom pesse. Ae he sdies o Von Kmn nd Tsien (99), he bckling poblem o spheicl shell hs been exmined boh heoeiclly nd expeimenlly by mny invesigos nde vios ypes o loding. Tsien (94) showed h smll disbnce in es wold cse he shell o jmp o new conigion wih lge displcemens s soon s he bckling lod ws exceeded. Kpln nd Fng (954) nd Simons (955) sdied he bckling behvio o spheicl cps om pesse delecion cve. Thei nlysis ws bsed on inegion o nonline inie delecion eqions. Kpln nd Fng (954) mde some expeimens o vey shllow clmped spheicl cps nde niom pesse. They comped hese esls wih he ones obined by pebion solion o he govening nonline eqions nd obseved h he geemen ws siscoy. Bckling o clmped shllow spheicl shells nde exenl pesse hs been sdied exensively boh expeimenlly nd heoeiclly. In 954, Kpln nd Fng peomed n nlyicl nd expeimenl invesigion o clmped shllow spheicl shells. Thson (96) obined nmeicl solion o he nonline eqions o clmped shllow spheicl shells nde exenl pesse nd pesened he esls in he pos bckling nge no peviosly comped. Then he comped he ppe bckling nd lowe pos bckling pesses wih he expeimenl d o Kpln nd Fng (954). Hng (964) woked on he poblem o clmped shllow spheicl shells o symmeic nd nsymmeic bckling s well. Hng comped his nmeicl inding wih he expeimenl esls. Fmili nd Ache (965) invesiged he bckling behvio o shllow shells by sing he nonline eqions, consideing he symmeic deomions he beginning o he xxiii

24 bckling o be inie. The nonline eigenvle poblem ws solved nmeiclly. Thei esls wee in geemen wih hose o Hng (965). Thson nd Penning (966) condced n exensive expeimenl nd nlyicl invesigion o he bckling o clmped shells wih xisymmeic impeecions. They bsiclly comped he pesse sin nd pesse delecion esls obined boh expeimenlly nd heoeiclly. They ond o h he eec o xisymmeic impeecions is no lge enogh o give good geemen beween heoy nd expeimens o vey hin shells. Hchinson (967) sdied he iniil pos bckling behvio o shllow secion o spheicl shell sbjeced o exenl pesse. He ond o h impeecion in he shell geomey hve he sme sevee eec on he bckling senghs o spheicl shells s demonsed o xilly compessed cylindicl shells. Bdinsky (969) nd Weinischke (97) lso deemined he xisymmeic bckling pesses o shllow spheicl shells nmeiclly. Thee is good geemen mong ll he esls obined. Fich (968) sdied he elsic bckling nd iniil pos bckling behvio o clmped shllow spheicl shells nde concened loding. He deemined h bicion ino n symmeic pen will occ beoe xisymmeic snp- bckling nless he io o he shell ise o he hickness lies wihin now nge coesponding o modeely hick shells. Fich (97) lso invesiged he elsic bckling nd iniil pos bckling behvio o clmped shllow spheicl shells nde xisymmeic lod. He ond o h s he e o he loded egion incese, he bckling behvio chnges om symmeic bicion o xisymmeic snp-hogh, nd hen bck o symmeic bicion. xxiv

25 Sicklin nd Minez (969) sdied nonline nlysis o shells o evolion by he mix displcemen mehod. The nonline sin enegy expession ws evled sing line ncions o ll displcemens. Five dieen pocedes wee exmined o solving he eqions o eqilibim, wih Hobol s mehod o be he mos sible. Solions wee pesened o he symmeic nd symmeic bckling o shllow cps nde sep pesse lodings nd wide viey o ohe poblems inclding some highly nonline ones. The diicly o epeed solions o lge nmbe o eqions hs been cicmvened by plcing he nonline ems on he igh hnd side o he eqions o eqilibim nd eing hem s ddiionl lods. The solions o he govening eqions wee obined by ieions nd ond o yield cce esls o some pcicl poblems. Fo highly nonline poblems, he eqions wee solved by he Newon-Rphson pocede, wih he copling beween hmonics being ignoed when he nonline ems wee eed s psedo lods nd ken o he igh hnd side o he eqions. Hng (969) sdied he behvio o xisymmeic dynmic snp-hogh o elsic clmped shllow spheicl shells nde implsive nd sep loding wih ininie dion. I ws obseved h he dynmic snp-hogh bckling ws no possible nde implsive lods b i ws chieved nde sep loding condiions. The esls obined o sic niom pesse nd dynmic loding omed benchmk o mny invesigos in he veiicion o hei esls. Axisymmeic nd dynmic bckling o spheicl cps de o cenlly disibed pesse ws sdied by Sephens nd Flon (969). Sndes xisymmeic nonline elsic shell heoy ws ppoximed by inie dieence eqions inclding he Hobol bckwd dieence omlion in ime. The eqions wee lineized sing n ieive Newon-Rphson pocede. Axisymmeic bckling lods wee given o spheicl cp sbjeced o consn xxv

26 sic pesse o sep plse o ininie dion disibed xisymmeiclly ove poion ove he cene o he shell. The inlence o he size o he loded e nd o momen nd inplne bondy condiions on boh sic nd dynmic bckling ws sdied, s well s vios bckling ciei o deine dynmic bckling wee sed. Gossmn e l. (969) invesiged he xisymmeic vibions o spheicl cps wih vios edge condiions by cying o consisen seqence o ppoximions wih espec o spce nd ime. Nmeicl esls wee obined o boh ee nd oced oscillions involving inie delecions. The eec o cve ws exmined wih picl emphsis on he nsiion om l ple o cved shell. In sch nsiion, he nonlineiy o he hdening ype gdlly evesed ino one o soening. Tillmn (97) pesened he esls o heoeicl nd expeimenl invesigion ino elsic bckling o clmped shllow spheicl shells nde niom pesse, ocsing minly on low vles o he geomeic pmee, o which he symmeicl nd is wo symmeicl deomions e vlid. Ache (98) sdied he behvio o shllow spheicl shells sbjeced o dynmic lods o sicien mgnide o esl in inie nonline xisymmeic deomions. Mgee s eqions o he smll inie delecions o shllow shells wih he inclsion o inei ems wee ken s he govening eqions. Resls o he qsi siclly loded shell beoe nd e snp hogh nd snp bck wee sdied nd comped wih known esls. The dynmic esponse o he shell o ecngl plse loding nd bckling lods wee obined. Dynmic bckling o ohoopic shllow spheicl shells by Gnphi nd Vdn (98) nd xisymmeic sic nd dynmic bckling o ohoopic shllow spheicl cp wih cicl hole by Dmi (98) wee invesiged. xxvi

27 Geomeic nonline D dynmic nlysis o shells bsed on ol Lgngin omlion nd he diec ime inegion o he eqion o moion ws deived by Woes (98). Dmi e l. (984) invesiged xisymmeic bckling o ohoopic shllow spheicl cp wih cicl hole. Anlysis hs been cied o o niomly disibed lod nd ing lod he hole. Zheng nd Zho (989) developed semi-nlyicl compe mehod o solve se o geomeiclly nonline eqions o ples nd shells. By his mehod, nlyicl solions sch s exc expnsion in seies, pebions nd ieions o he eqions cn be obined. Hsio nd Chen (989) sed degeneed isopmeic shell elemen o he nonline nlysis o shell sces. Six ypes o oion vibles nd oion segies wee employed o descibe he oion o he shell noml. In picl, inie oion mehod ws poposed nd esed. Boh he oion viions beween wo sccessive incemens nd he oion coecions beween wo sccessive ieions wee sed s he incemenl oion (oion vibles) o pde he oienion o he shell noml. Chn nd Chng (989) sed highe ode inie elemens o he geomeiclly nonline nlysis o shllow shells. Bsed on K. Mgee s shell heoy, mily o highe ode inie elemens ws developed. A sep ieion Newon-Rphson scheme ws doped in solving he inl sysem o nonline eqions. Bhimddi nd Moss (989) developed she deomble inie elemen o he nlysis o genel shells o evolion. Xie, Chen nd Ho (99) sdied he nonline xisymmeic behvio o nced shllow spheicl shells nde nsvese loding. Lod-delecion elion wee obined xxvii

28 hogh ieion nd nmeicl inegion. Shells sbjeced o niom pesse nd combined niom pesse nd concened ing loding wee invesiged. Elle (99) deived inie elemen pocedes o he sbiliy nlysis o nonline peiodiclly excied shell sces. Sing om geomeiclly nonline shell heoy nd pplying Ljpnow s is mehod s well s Floqe s heoy, nmeicl sbiliy cieion ws dedced. Lo e l. (99) invesiged he inlence o pe bckling deomions nd sesses on he bckling o he spheicl shell. They obined om Von Kmn's lge delecion eqion o he ple nd by ssming h ple hs n iniil delecion in he om o spheicl cp, he eqilibim eqions o spheicl cp sbjeced o hydosic pesse wee wien. Chng (99) developed non-line she-deomion heoy o he xisymmeic deomions o shllow spheicl cp compising lmined cved-ohoopic lyes. He expessed he govening eqions in ems o he nsvese displcemen, sess ncion nd oion. Nmeicl esls on he bckling nd pos-bckling behvio o spheicl cps nde niomly-disibed lods wee pesened o vios bondy condiions, cp ises, bse dis-o-hickness ios, nmbes o lyes nd meil popeies. Delpk nd peshkm (99) developed viionl ppoch o he geomeiclly nonline nlysis o symmeiclly loded shells evolion. The omlion ws bsed on king he second viion o he ol poenil enegy eqion. The nlysis commenced by king he is nd second viion o he ol poenil enegy o he elsic sysem by ensing h lod incemens wee pplied ininiely slowly. Ae seping he lod nd he siness ems nd coizing he nodl vibles, disinc demcion in he conibion o line nd xxviii

29 second ode ems ws obseved which povided cle mehodology in clcling nonline nd geomeic mices h led o he geneion o he ngen mix. A lge deomion elsic plsic dynmic nlysis o sqe ple nd spheicl shell sbjeced o shock loding ws sdied by Ling, Lio nd M (99). A nsien dynmic inie elemen mehod ws poposed o shock loding dynmic nlysis. An incemenl pded Lgngin inie elemen pocede ws dived. A 6-node isopmeic shell elemen ws chosen o he sdy o he sqe ple nd 8-node wo dimensionl xisymmeic elemen o he spheicl shell. Gonclves (99) invesiged he xisymmeic bckling behvio o clmped spheicl shells nde niom pesse. He exmined he bckling chceisics o he spheicl shells sing lly nonline Glekin solion pocede, clssicl bicion nlysis nd edced siness bicion nlysis. Polssopolos (99) pesened new nlyicl mehod o he deeminion o he sengh o sces sbjeced o bicion bckling eced by smll scl impeecions. Choic dynmic nlysis o viscoelsic shllow spheicl shells ws peomed by Kesmen (99). The nonline dynmic bckling sengh o clmped spheicl cps nde niom sep loding ws invesiged by Lee, Lie nd Lio (99). The geomeic coodines wee pded evey ime sep. Ths, lineized inie elemen incemenl eqions bsed on he pinciple o vil wok cold be deived. A hee dimenionl shell elemen wih biy geomey ws sed in he inie elemen omlion. xxix

30 Tendp e l. (995) sdied he bckling behvio o impeec spheicl shells sbjeced o dieen loding condiions. They nlyzed he bicion nd iniil pos-bckling behvio o highly impeecion-sensiive lge spheicl.shells, sch s cgo nks o ship nspoion o liqeied nl gs nd lge spheicl coninmen shells o ncle powe plns. Zhng (999) sdied he osionl bckling o spheicl shells nde cicmeenil she lods. He sed Glekin viionl mehod, o sdying he genel sbiliy o he hinged spheicl shells wih he cicmeenil she lods. Uchiym e l. () sdied nonline bckling o elsic impeec shllow spheicl shells by mixed inie elemens. They sed nine-node-shell elemen nd mixed omlion o sess esln vecos hen hey comped inie elemen esls wih iy-wo expeimens on he elsic bckling o clmped hin-wlled shllow spheicl shells nde exenl pesse. Güniz () exmined he bckling sengh o clmped nd hinged spheicl cps nde niom pesse wih cicmeenil weld depession by sing he inie elemen mehod. The esls obined show signiicn decese in he bckling sengh de o hese impeecions depending on he locion o he weld. Dmi e l. (5) pesened xisymmeic bckling nlysis o modeely hick lmined shllow nnl spheicl cp nde nsvese lod. In hei sdy, bckling ws consideed nde niomly disibed nsvese lod, pplied siclly. Annl spheicl cps hve been nlyzed o clmped nd simple sppos wih movble nd immovble in-plne edge condiions nd ypicl nmeicl lods nd hve been comped wih he clssicl lminion heoy. xxx

31 Jones e l. (7) invesiged he poblem o hin spheicl linely-elsic shell, peecly bonded o n ininie linely-elsic medim. A consn xisymmeic sess ield is pplied ininiy in he elsic medim, nd he displcemen nd sess ields in he shell nd elsic medim e evled by mens o hmonic poenil ncions. Nie e l. (9) deived n sympoic solion o nonline bckling o ohoopic shell on elsic ondion. They peomed n exensive pmeic sdy o deomion nd bckling o sch sces. The oegoing liee eview is by no mens exhsive. Howeve, he eeences cied nd sveyed cove some o he impon sdies h hve been conibed in his e. xxxi

32 II. A Bie Hisoy o Yield Line Theoy As ely s 9, he Rssin, A. Ingeslev pesened ppe o he insiion o Scl Enginees in London on he collpse modes o ecngl slbs. Le on yield Line heoy s i is known ody ws pioneeed in he 94s by he Dnish enginee nd eseche KW Johnsen. Ahos sch s R. H. Wood, L. L. Jones, A. Swczk nd T. Jege, R. Pk, K. O. Kemp, C.T. Moley, M. Kwiecinski nd mny ohes, consolided nd exended Johnsen s oiginl wok so h now he vlidiy o he heoy is well esblished. In he 96s, 97s, nd 98s signiicn mon o heoeicl wok on he pplicion o yield line heoy ws cied o ond he wold nd ws widely epoed. To sppo his mehod, exensive esing ws ndeken o pove he vlidiy o he heoy. Excellen geemen ws obined beween he heoeicl nd expeimenl yield line pens nd he lime lods. The dieences beween he heoy nd ess wee smll nd minly on he consevive side. xxxii

33 III. Hisoicl Bckgond o he Sohwell Mehod Si Richd Vynne Sohwell (888 97) ws Biish mhemicin who specilized in pplied mechnics s n engineeing science cdemic. Richd Sohwell ws edced he Univesiy o Cmbidge, whee in 9 he chieved is clss degee esls in boh he mhemicl nd mechnicl science ipos. In 94, he becme Fellow o Tiniy College, Cmbidge, nd lece in Mechnicl Sciences. Sohwell ws in he Royl Nvl Ai Sevice ding Wold W I. Ae Wold W I, he ws hed o he Aeodynmics nd Sces Divisions he Royl Aic Esblishmen, Fnboogh. In 9, he moved o he Nionl Physicl Lbooy. He hen ened o Tiniy College in 95 s Fellow nd Mhemics Lece. Nex, in 99, he moved o Oxod Univesiy s Poesso o Engineeing Science nd Fellow o Bsenose College. Thee, he developed esech gop, inclding Demn Chisopheson, wih whom he woked on his elxion mehod. He becme membe o nmbe o UK govenmenl echnicl commiees, inclding he Ai Minisy, he ime when he R nd R iships wee being conceived. Sohwell ws eco Impeil College, London om 94 nil his eiemen in 948. He conined his esech Impeil College. He ws lso involved in he opening o new sden esidence, Selkik Hll. As scienis, in 9, Sohwell pesened his nlysis o he specil cse o pin ended s o consn lexl igidiy o EI. Sohwell mehod o deemining he minimm bckling lod is nondescive es o pined-end, iniilly impeecs ss. Sohwell showed h he lod delecion cve o sch membe is hypebolic in he neighbohood o he smlles ciicl lod, while he sympoe is hoizonl line, P Pc. By sible xxxiii

34 nsomion o vibles his hypebolic poion o lod delecion cve my be conveed ino sigh line o which he inveed slope is he minimm ciicl lod. xxxiv

35 CHAPTER - Plsic Bckling o Hemispheicl Shell Sbjeced o Concened Lod he Apex. Inodcion nd Ppose o his Chpe De o he incesing se o shell ype sces in spce vehicles, sbmines, bildings nd soge nks, inees in he sbiliy o shells hs ccodingly incesed by eseches nd pcicing enginees. Becse hemispheicl shell is ble o esis highe pe inenl pesse loding hn ny ohe geomeicl vessel wih he sme wll hickness nd dis, he hemispheicl shell is one o he impon scl elemens in engineeing pplicions. I is lso mjo componen o pesse vessel conscion. In pcice, mos pesse vessels e sbjeced o exenl loding de o hydosic pesse, o exenl impc in ddiion o inenl pesse. Conseqenly, hey shold be designed o esis he wos combinion o loding wiho ile. The lod nsmied by cylindicl igid co pplied he smmi o he sphee is consideed common exenl lod. Ths, i is impon o sdy is eec on he iniil bckling nd plsic bckling popgion o his ype o shells. This sdy pesens he nlyicl, nmeicl, nd expeimenl esls o modeely hick hemispheicl mel shells ino he plsic bckling nge illsing he imponce o geomey chnges on he bckling lod. The hemispheicl shell is igidly sppoed ond he bse cicmeence gins veicl nd hoizonl nslion nd he lod is veiclly pplied by igid cylindicl boss he pex. Kinemic sges o iniil bckling nd sbseqen popgion o plsic deomion o igid-peecly plsic shells e omled on he bsis o Dcke- Shield's limied inecion yield condiion. The eec o he dis o he boss, sed o pply he loding, on he iniil nd sbseqen collpse lod is sdied. In he nmeicl

36 model, he meil is ssmed o be isoopic nd line elsic peecly plsic wiho sin hdening obeying he Tesc o Von Mises yield cieion. Boh xisymmmeic nd D models e implemened in he nmeicl wok o veiy he bsence o non-symmeic deomion modes in he cse o modeely hick shells. In he end, he esls o he nlyicl solion e comped nd veiied wih he nmeicl esls sing ABAQUS sowe nd expeimenl indings. Good geemen is obseved beween he lod-delecion cves obined sing he hee dieen ppoches. The pepions o condc expeimenl veiicions e lso shown in Fig... Fige. : Smple conscion pocede o he expeimenl sdy

37 . Peliminy Consideions This sdy is ocsed on he ollowing physicl phenomenon. A hemispheicl shell is compessed by concened lod he smmi. A he lod below cein ciicl vle, clled he iniil bckling lod, he shell emins spheicl o nbckled b when he incesing pplied lod eches he ciicl iniil bckling vle, he shell snps ino non-spheicl bckled se which is chceized by ond dimple ond he pex o he hemispheicl shell. Theeoe, i cees deomion se which exends o popges ove he sce o he shell leving ndeemined he mplide o deomion vios levels o lod (Fig..). Fige. : Geomey nd pos bckling o hemispheicl shells sbjeced o concened lod

38 . Anlyicl Fomlion... Kinemics Assmpions The behvio o modeely hick mel hemispheicl shell nde concened lod he smmi my be nlyzed s ollows: ) The peecly-igid se clmining he inmen o he iniil collpse lod P. Fo concened lod cing on hemispheicl shell he iniil collpse kes plce only in vnishingly smll egion o he shell, Fig... The collpse lod P depends on he plsic momen M o he shell meil. I igid cylindicl boss is sed o loding pposes, he size o his boss inlences he egion o collpse nd hence he collpse lod P. Fige. : Iniil bckling nde concened lod 4

39 b) Deomion nde he collpse lod P. A he lod P, he shell snps o evese is cve nd conines o deom nde he sme lod, esling in he omion o dimple. The dimple is ken o be conicl in shpe nd he pex o he cone is he poin whee he lod is cing. This ssmpion is no vince wih he obseved behvio. The exen o he dimple depends gin on he plsic momen M o he shell meil nd on he dis o he loding boss o co. A secion o he shell hogh meidionl plne, immediely e he deomion nde he iniil collpse lod P, is shown in Fig..4. The oe ndeomed poion o he shell (o dis R nd consn hickness ) nd he conicl dimple e conneced by n nnl zone o which he cone is ngen, nd which shes common ngen wih he ndeomed p o he shell. Boh he conicl dimple, nd he nnl zone which looks in secion like knckle o dis ρ symmeicl bo he xis o evolion, e plsic. Fige. 4: Pos bckling deomion iniil collpse lod P 5

40 c) Popgion o he nnl zone. This is he hid sge o deomion. I kes plce only e he deomion nde he consn lod P is complee. The dimple exends owd wih n xisymmeic deomion nde n incesing lod P o ende gee poion o he shell plsic Fig..5. The deomion involves conicl shpe nd n nnl zone. Fige. 5: Plsic bckling deomion exends owd nde n incesing lod P d) Degeneion o he shpe o deomion. Ae he nnl zone (which is cicl in pln) hs popged o n exen depending o given meil on he R / io, he xisymmeic deomion descibed bove begins o chnge. The nnl zone becomes ingl nd hen polygonl in pln. A new mechnism which involves he olding o he shell meil bo he edges o pymid-like sce kes ove nd eplces he conicl p o he deomion. This phenomenon cold be ssocied wih some so o secondy insbiliy. This sge o deomion will no be ddessed in p o his sdy, becse i is nlikely o ke plce in modeely hick shells. 6

41 ... The Iniil Collpse Lod nd Revesl o Cve Shells e commonly sbjeced o nsvese lods, i.e. lods h c in he diecion pependicl o he sce o he shell. Sch shells my il loclly by so clled n mechnism, wih posiive yield lines diing om he poin lod. Conseqenly, sicienly high lod, he shells my expeience exensive plsic deomion loclly nd evenlly lose ll is scl ncion nd chnges is cve diecion his phenomenon known s locl plsic collpse. Unlike elsic nlysis, exc solions o he plsic collpse lod e no vilble in mos cses. Even o he idelized igid peecly plsic consiive elion, he collpse lod cn only be ppoximed ove nge o vles. The echniqe sed o deine he bondy o he collpse lod is known s limi lod nd he heoem ssocied wih i known s limi nlysis. Conside n n-sided egl polygon ple cying single concened lod is cene nd igidly sppoed long he n sides, Fig..6. I smll vil displcemen δ is imposed nde he lod, he exenl wok done is W e P δ nd he inenl wok exeed ding he ssigned vil displcemen is ond by smming he podcs o plsic momen M pe ni lengh o yield lines imes he plsic oion θ he especive yield lines, consisen wih he vil displcemen. I he esising momen M is consn long yield line o lengh i nd i oion θ is expeienced, he inenl wok is M θ o ech yield line. Becse W i i hee e n yield lines, he ol inenl wok isw i n M Ti i θ. The oion he plsic hinge cn be clcled in ems o he delecion hs, δ θ OH δ. In view o he c OK 7

42 h OK i n ndoh i i n, Fig..6 By eqing W nd i Ti W e one n obins P M ( co ) co i i. Becse in n n-sided egl polygon i π π i i nd ccodingly, n π π π P nm co nm n. I n ends o ininiy, n n π n-sided egl polygon conves o cicle nd P lim nm n n. Using LHopil once n P πm.ths, o cicl ple, he vle o he concened lod necessy o iniie collpse is given by P πm nd s i cn be seen, he collpse lod is independen o he size o he ple. This oml cn be poven sing nohe mehod oo (See ppendix A) Fige. 6: n-sided egl polygon ple cying single concened lod is cene Following he sme pocede, i cn be esily poven h he lod pplied hogh igid boss o dis b, o cicl ple o dis podces collpse lod level given by: P M π b (.) 8

43 The egion o hemisphee sbjeced o downwd concened lod he pex h iniilly collpses wih evesl o cve is qie smll nd cn be esily consideed o be vey shllow spheicl cp. I he boss size is ignoed, π M is known o be he exc collpse lod o ny cicl ple nd heeoe i cn seve s lowe bond on he iniil collpse lod o shllow spheicl cp. The iniil collpse lod o hemispheicl shell nde concened lod shold hs ppoch he vle π M becse he locl ne o he collpse my men h he collpse lod is less dependen on he shell cve. When he shell is loded by mens o inie igid boss o dis b, he collpse lod P hs vle which is obseved o be gee hn π M while i is dependen on he size. As menioned elie, his is lso e o ple loded wih boss, nd so he sme modiicion o he collpse lod oml eeed o bove cn be mde. The dieence is h while he l ple dis is known, he dimple plnne dis iniil collpse in he cse o hemispheicl shell is no edily vilble b hs o be clcled. The vle o his dimple dis o he shell is ond by eqing he iniil collpse lod P wih he lod pediced by he mechnism o dimple popgion he s o he hid sge o deomion, s shown below. The iniil collpse nd sbseqen deomion mechnism cn be seen in Fig..7. The shell iniilly collpses lod vle o P which is eql o o gee hn π M o n exen depending on he boss dis b. A poion o he shell shown s doed line is iniil posiion s p o hemisphee o dis R kes p he bckled posiion shown by he bold line, compising cone nd n nnl zone, Fig..7 The exen o he deomion is mesed by θ he meidionl ngle coesponding o he bondy o he 9

44 plsic egion. Ding his deomion, he lod P emins consn. I is beween hese iniil nd inl posiions h he ooidl nnl zone wih he knckle dis ρ nd he cone come ino being. I is only e his sge is complee h he hid sge o deomion wih dieen ype o mechnism kes ove. This compises he popgion o he dimple nd he owd movemen o he nnl zone. Fige. 7: Poile o deomion ding iniil bckling nd pos bckling behvio

45 In he idel cse o concened lod cing he shell pex, i is nl o expec h he second sep o deomion shold begin lmos immediely e he shell lod eches he vle π M. The ollowing geomeicl elions in which he boss will no ply signiicn p cn be deived o he iniil collpse sing he incompessibiliy condiion, nd ssming no dieence beween he hickness o he shell in he dimple egion beoe nd e collpse: The sce e o he spheicl cp which eveses cve is: Sce Ae iniil πr ( cosθ ). (.) This ms be eqed o he sm o he sce es o he cone nd he nnl zone, which e eql o: Sce Ae BckledSce e o nnl p Sce e on conicl p π ( ρ ) π ( R ρ ) R sinθ.ρθ (.) cosθ sin θ Eqing eqs. (.) nd (.) hen simpliying, he ollowing eqion is obined: ρ R sin θ cosθ cosθ ( ρ / R) θ sinθ ( ρ / R) (.4) 4 As θ is smll, cosθ θ / θ / 4, nd sinθ θ θ / 6. Neglecing he second ems on he igh hnd side o he cosine nd sine seies expnsions nd ignoing he oh powe o θ wold mke eqion (.4) ivilly sisied. Sbsiing hese vles ino eqion (.4) nd neglecing powes o θ highe hn he oh, he eqion edces o: 4 ρ ρ θ /6 R R (. 5 )

46 Fo ny non-zeo vle oθ, he solion gives ρ / R. 5 nd/ o.75. Fo he lge vle o ρ / R.75, R ρ becomes negive, which mens h he conicl p o he dimple cnno exis. Ths he elevn vle o ρ is R / 4. Alhoghθ hs smll vle, i cn be ssmed h ρ is eql o R / 4 hogho he sbseqen deomion o which θ is gee hnθ oghly nil θ d since ll ssmpions e sisied.... Popgion o he Dimple Ding he omion o he iniil dimple, he deomion is smll nd he bckling hppens nde consn lod P. As he delecion inceses, he eec o geomey chnge ss o become signiicn nd he lod inceses wih conining deomion. When he non-plsic meil sonding he deoming egion cnno sppo lod highe hn P, he plsic egion ms gow in size wih incesing lod. I is ssmed hee h he deoming sce minins geomeicl similiy ding he popgion o he dimple s evidenced by he he nmeicl esls. The deomion sge being ideniied by single pmeeθ, which is he ngl posiion o he sce he bondy o he plsic egion (Fig..7). I is ssmed h he dis o cve o he ooidl knckle emins consn while is cown moves wy om he xis o evolion by coninos oion nd nslion o he igid meil eneing ino he plsic egion. The middle sce o he deoming shell oms sce o evolion nd he se o sess is compleely speciied by he diec oces, esln momens nd nsvese she. I N φ nd N β denoe he meidionl nd cicmeenil oces pe ni lengh, M φ nd M β he co-

47 esponding bending momens, nd Q he nsvese she oce (Fig..8), he meidionl eqions o eqilibim o shell o evolion cn be wien s ( Nφ ) N β cosφ Q (.6) φ ( M φ ) M β cosφ Q (.7) φ whee is he disnce o he elemen om he xis o evolion nd is men meidionl dis o cve, which is eql o ρ he nnl zone. Fige. 8: Eqilibim o n xisymmeic shell elemen

48 In o ssmed deomion model, he nnl zone hs consn dis o cve ρ, I is esily vislized h N φ chnges sign om posiive (ension) o negive (compession) s we move om he inne o he oe p o he ooidl knckle. N φ he bondy o he cone nnl zone is ension nd he bondy o he nnl zone- ndeomed shell is compession o esis he downwd P sch h F y. In ohe wods, nde incesing lod he shell meil om he oe ndeomed egion is pshed ino he nnl zone nd meil om he nnl zone is plled ino he conicl dimple. Ths N φ my be ssmed o vnish he cown o he ooid. Consideing he oe p o he knckle deined by φ θ nd noing h N φ is compessive in his egion, i is eviden h N β shold lso be compessive. This is de o he c h compessive Nφ cses expnsion in he hoop diecion while he igid shell esins he knckle om expnsion hs indcing compessive sess: N σ, (.8) β whee σ is he (consn) yield sess o he meil nd is he cen hickness. The is eqion o eqilibim, eqion (.6), heeoe edces o φ ( N ) ρσ cosφ Q φ (.9) This eqion ms be spplemened wih he eqion o veicl eqilibim, nmely ( N sinφ Q φ) P π φ cos (.) whee P is he ol veicl downwd lod he conicl pex. Elimining Q beween (.9) nd (.), 4

49 φ P ( N secφ ) ρσ φ φ sec π (.) Assming h he hickness viion my be negleced in he eqilibim eqions nd noing h (he iniil hickness) φ θ, eqion (.) cn be ineged o obin P N ρσ φ cosφ sinφ, φ (.) π whee N φ is sed s bondy condiion. φ Consideing he momen eqion o eqilibim, i is obseved h he second em o eqion (.7) is o he ode o / imes h o he ls em, s poven by Dcke & Shield (959), nd hence he second em my be negleced when he egion o inees is no close o he xis o evolion. Eqion (.7) hen edces o φ ( M ) ρ Q φ (.) Fom eqs. (.) nd (.), Q P ρ σ φ sinφ cosφ (.4) π nd eq. (.) becomes ( M ) P φ ρ ρ σ φ sinφ cosφ (.5) φ π Since he cown o he ooidl knckle ses he mos sevee bending, i is nl o ssme h M φ he cown is eql o he yield momen M. I my lso be 5

50 ssmed h he bondy o he igid egion, M φ ins he vle o he yield momen. Neglecing gin he hickness viion, he bondy condiions cn be wien M φ M φ nd φ θ. Inegion o eq. (.5) beween he limis o nd θ gives: P M d d ρ σ φ sinφ φ ρ cosφ φ φ cosφρ σ ρ σ φ π ρp ρ σ ( sinφ φ cosφ ) sinφ C π ρp cosφdφ sinφ C π φ M M C R sinθm ρ sinθ φ φ M φ θ M M ρp R sinθm ρ σ M π ( sinθ θ cosθ ) sinθ RsinθM ρ sinθ This will hen led o: P M sinθ ρσ ( sinθ θ cosθ ) sinθ (.6) π Inseing he vles M σ R nd ρ, eqion (.6) edces o 4 4 P πm θ R / nθ (.7) This oml diecly eles he downwd veicl lod o he ngl posiion o he dimple denoed byθ θ. I is independen o he size o he boss, povided he dimee o he boss is smll in compison wih he dimee o he shell. Fo ly concened lod θ is vnishingly smll nd he solion edces o he well known esl P πm. 6

51 ..4. Solion o he Complee Poblem I he shell is loded by concened lod he pex, he oml (.7) povides he complee lod- deomion chceisic o he shell. Howeve, lly concened lod is only mhemicl convenience which cnno be elized in pcice. In c he iniil collpse lod is vey sensiive o he size o he boss nd smll boss cn considebly incese he collpse lod om he vle P πm. I is heeoe essenil o inclde he boss size in developing elisic heoy. The iniil ngle o he popging dimple θ is lso mese o he plsic egion coesponding o he iniil collpse. Assming h his smll poion o he shell behves like ple ding collpse even when he lod is pplied by inie boss in he om o igid pnch o smll bse dis b, eqion (.) cn be ewien s: P πm b (.8) Rθ (.7) gives Since he lod coesponding o he beginning o he dimple popgion is lso P, eq. P R θ πm nθ (.9) Since θ is smll, heeoe; θ θ 5... nθ θ θ θ 5 7

52 I he powe gee hn hee is negleced so, θ nθ θ θ θ nθ θ Ths, θ θ P πm R (.) θ Eqions (.8) nd (.) nish he ollowing eqion oθ : R θ θ, (.) b R which cn be solved nmeiclly o ech picl cse. An immedie conclsion om (.) is h θ is lwys gee hn b / R. I is esily seen om he geomey in Fig..7 h he pnch peneion coesponding o he posiion o he dimple given by θ is h R( cosθ ) ( R ρ)sinθ nθ (.) nd h R ( cosθ )( secθ ) (.) when θ θ. by sing he esl ρ R / 4. Eqs. (.7) nd (.) give he lod-delecion elionship pmeiclly hogh θ 8

53 When R p, his solion is ond o coec o lge mplides o delecion, howeve o R he deomion ollows he sme xisymmeic pen in he beginning. Then, he nnl zone degenees ino vios n-sided polygon deomion modes nd bicion in deomion pen is obseved. In ode o deemine he bicion poin ding deomion, ssme he deomed p behves simil o clmped cicl ple o dis c wih lge deomion beoe bicion poin (Fig..9). Fo hin cicl clmped ple he dil nd ngenil momens e (see Appendix B): Fige. 9: Deomed p o hemispheicl shell shpe beoe he secondy bicion poin Fige. : Eec o xil oce on plsic momen cpciy 9

54 M M P c 4π ( υ) ln P c υ 4π ( υ) ln (.4) (.5) In he bicion poin ρ R 4 c Rsinθ ρ sinθ c R sinθ 4 Assming h he bicion poin, M eches o he plsic momen c ρ sinθ R sinθ (s evidenced by he nmeicl nd expeimenl indings). By combining eqions (.7) nd (.5) nd inseing he vle θ P πm R / i nθ cn be wien: M θ πm R / R nθ sinθ M 4 υ plsic 4π R sinθ ( υ) ln Becse o he membne eec, shell plsic momen is no M σ nymoe nd i will be 4 lmos eql o M p M ˆ σ 4. In his eqion, ˆ Pυ nd conseqenly: π sinθσ M P θ M R / nθ υ [( υ) ln.5 ] Hence: σ θ σ R / ˆ θ 4 n M [( υ) ln. 5 υ] (.6) 4

55 In he ls sge o deomion, hemispheicl shell will be pnched nde he poin lod. Sbseqenly; he lime being cpciy is oghly eql o: P U π b σ (.7)

56 .4. Nonline Finie Elemen Anlysis (FEA).4. Elemens nd Modeling Finie elemen nlysis (FEA) is cpble o modeling elsic-peecly plsic meil behvio wih nonline geomey whee he nlysis is bsed on he iniil geomeicl shpe. This shold give limi lod nd lod-deomion esponse lmos eql o he igid-plsic limi lod nd ovell esponse since he elsic deomions e negligible comped o he plsic deomions. This mkes i possible o compe he esls sing he inie elemen mehod wih hose o he nlyicl solion. All FEA o his invesigion ws peomed sing he genel ppose pogm ABAQUS Vesion 6.7. The boss sed o lod he spheicl shell ws modeled s igid elemen. Boh eigh node xisymmeic ecngl nd six node ingl shell elemens wee sed o model he hemispheicl shell Figs..-.. In he pesen nmeicl nlyses, he hybid elemen ws chosen o ll inie elemen models in ode o void he poblem o mesh locking nd o ge coec elemen siness nd cce esls. The shell is pinned he bse nd he meil behvio e yielding is ssmed o obey he Tesc o Von Mises yield cieion.

57 Fige. : Deomion pen o he hemispheicl shell sing 8 node xisymmeic ecngl shell elemen.

58 Fige. : Dieen cs o he deomed modeely hick shell R /. 4

59 Fige. : Deomion pen o he modeely hick shell R / sing six node ingl shell elemen. 5

60 .4. Nmeicl Resls The nmeicl esls o modeely hick shells ( R / p ) wee idenicl in he cse o xisymmeic nd genel D models. These esls wee ond o compe well o he nlyicl nd expeimenl indings. On he ohe hnd bicion om xisymmeic esponse is obseved some poin e iniil collpse in hin shells wih ( R / ) nd his phenomenon is shown in Figs The level o bicion lod depends on he eec o geomeicl pmees o shell (wll hickness, dis R, R / io), he meil popeies s well s he size o he boss. Fige. 4: Bckling iniiion o hin shell R / sing six node ingl shell elemens. 6

61 Fige. 5: Sbseqen deomion o hin shell R / showing he secondy bicion phenomenon. 7

62 Fige. 6: Dieen cs o he deomed hin shell R / 8

63 Fige. 7: Deomion pen o he hin shell R / showing he secondy bicion phenomenon. 9

64 .5. Expeimenl Pogm.5. Pmees nd es sep Dieen smples wih vios R / ios wee designed, mnced, nd esed, Figs..8. These wee mde o o Bonze ( E GP nd F y MP ), sinless seel ( E GP nd F y 5MP ), nd coppe ( E 6GP nd F y MP ). The dis o he shells ws R 5mm nd R 75mm especively. The hickness o he shells wee. mm nd mm. These pmees yield R / io o 5 o Bonze, 66 o Sinless Seel, nd 75 o Coppe lloy. Ding esing, hese shells wee sbjeced o concened lods he pex by mens o igid l-bsed cicl ods o hee dieen boss sizes, nmely b.5,.5,. 5mm (Fig..9).

65 Fige. 8: Dieen hemispheicl shell smples wee mde o expeimenl sdy Fige. 9: Thee dieen boss size sed o loding ( b.5,.5,5. mm )

66 The hemispheicl shells wee igidly sppoed gins nslion ond he boom cicmeence by sing gooved bse ple s shown in Fig... Fige. : Gooved bse ple s sppo o wo sizes o hemispheicl shells The lod-delecion cves o he shells wee ecoded on Riehle Univesl esing mchine (Figs..). The meils wee seleced o mncing e ension copons wee esed o ense h hei meil behvio coesponds closely o igid-plsic. This ws he cse in ode o illy compe expeimenl nd heoeicl esls.

67 Fige. : Riehle Univesl esing mchine o displcemen conol mesmens The yield sess o ech meil ws ond om he lod-delecion digms o he mel copons since in oming he shell specimens only negligible mon o wok hdening ws involved. Figs..-.4 shows ypicl sess sin digms o he Coppe lloy, Bonze, nd Sinless Seel sed in expeimens.

68 Fige. : Sess sin digm o coppe lloy. Fige. : Sess sin digm o Bonze. 4

69 Fige. 4: Sess sin digm o Sinless Seel..5. Expeimenl Resls: I is eviden om Fig.5 h he deomion o he modeely hick coppe lloy shell R / 75 is xisymmeic hogho he dieen loding sges. On he ohe hnd, he deomion o he hin sinless seel shell R / 66 is xisymmeic pon iniil collpse nd ely sbseqen deomion Figs Howeve, Figs.7-.8 show he le bicion in deomion s n ineesing secondy phenomenon. This phenomenon kes plce in hin shells, which hve R /, s shown by he nmeicl esls (Fig.46). 5

70 Fige. 5: Deomion o he modeely hick shell (Coppe lloy R / 75) Fige. 6: Iniil bckling nd pos bckling o he hin shell (Sinless Seel R / 66 ). 6

71 Similly, he deomion o he hin bonze hell R / 5 is lso xisymmeic pon iniil collpse nd some sho sbseqen deomion Fige.9. Then, Fig. shows he degeneion o he secondy bicion deomion which is clely ideniied in Figs..-.. Fige. 7: Degeneion o xisymmeic deomion o he hin shell (Sinless Seel R / 66 ). 7

72 Fige. 8: Secondy bicion deomion o he hin shell (sinless seel R / 66 ) 8

73 Fige. 9: Iniil bckling nd xisymmeic pos bckling o he hin shell (Bonze R / 5) 9

74 Fige. : Degeneion o he xisymmeic deomion o he hin shell (Bonze R / 5). 4

75 Fige. : Secondy bicion deomion o he hin shell (Bonze R / 5) 4

76 Fige. : Dieen sges o he ingl wih secondy bicion (Bonze R / 5) 4

77 Fige. : Finl deomion o he hin shell (Bonze R / 5 ) 4

78 .6 Resls nd Discssion As i is seen in Fig..5 o elively hick shell R / p, shell deomion pen s ssmed in he nlyicl solion consiss o cone nd os which sped o xisymmeiclly om he pex wih incesing lod. Theeoe, he nlyicl solion, expeimenl indings nd nmeicl esls e expeced o mch o lge mplide o deomions, Fig.4. Howeve, when R /, he deomion ollows he sme xisymmeic pen in he beginning. Then, he nnl zone degenees ino vios n-sided polygon deomion modes nd bicion in deomion pen is obseved (Figs..6-.). In ode o ind bicion poin in hin wll hemispheicl shell ( R / ) o which he bending nd membne sesses e simlneosly impon nd ssming he boss pesse o be niomly disibed ove he egion o conc, new solion hs been deived sing eqilibim ppoch. Figs..4,.4, nd.4 give he lod delecion cve o he coppe lloy, Bonze, nd sinless seel shell especively. The nlyicl solion esls seem o be lwys slighly less hn he nmeicl solion becse he inie elemen solion gives lowe bond on he mximm displcemens o given se o oces nd gives n ppe bond on he mximm sesses o given se o displcemens. The secondy bicion esponse is shown o yield highe lods hn he xisymmeic esponse o he sme h vle Fig.4 nd.4, he R xisymmeic nlyicl solion my sill be sed o design o hin shells since i is on he consevive side. Figs..5,.4, nd.44 show he boss eec on he iniil collpse lod. De o ignoing he shell cve in he smll viciniy o he lod, he collpse lod is lwys 44

79 slighly on he se side. Howeve, becse o he smple impeecion in he expeimenl sdy, he expeimenl indings e on he lowe side Fige. 4: Dimensionless Lod delecion cve o coppe lloy shell ( R / 75 ) Fige. 5: Compison o iniil collpse esponse o coppe lloy shell ( R / 75 ) 45

80 Fige. 6: Dimensionless iniil collpse lod vs boss size o coppe lloy shell ( R / 75 ) Fige. 7: Dimensionless iniil collpse lod vs iniil collpse ngle o coppe lloy ( R / 75 ) 46

81 Fige. 8: Iniil collpse ngle vs boss size o coppe lloy ( R / 75 ) Fige. 9: Dimensionless Lod vs knckle meidionl ngle in d o coppe lloy shell ( R / 75 ). 47

82 Fige. 4: Dimensionless Lod delecion cve o Bonze shell ( R / 5) Fige. 4: Compison o iniil collpse esponse o Bonze shell ( R / 5) 48

83 Fige. 4: Dimensionless iniil collpse lod vs boss size o Bonze shell ( R / 5) Fige. 4: Dimensionless Lod delecion cve o Sinless Seel shell ( R / 66 ) 49

84 Fige. 44: Compison o iniil collpse esponse o Sinless Seel shell ( R / 66 ) Fige. 45: Dimensionless iniil collpse lod vs boss size o Sinless Seel shell ( R / 66 ) 5

85 (h/r) vs (R/) (h/r ) Bonze Sinless Seel Coppe Alloy (R/) Fige. 46: Nmeicl bicion poin o Bonze, Sinless Seel, nd Coppe lloy vess ( R / ). 5

86 .7 Conclsions In his esech he eec o he geomey chnge on he plsic bckling behvio o hemispheicl mel shells loded inwdly he pex hs been sdied heoeiclly, nmeiclly nd expeimenlly. An nlyicl expession hs been deived o plsic bckling o hemispheicl shell in om which is especilly convenien o pplicion de o is simpliciy. In ddiion, i shows h qie simple model o deomion eqilibim is sicien o mke pedicions concening posbckling behvio o modeely hick hemispheicl shell sbjeced o concened lod on op. The iniil bckling lod hs been shown o be highly depending on he dis o he loding boss. A omlion is sed o evle his eec bsed on he iniil collpse o simply-sppoed cicl ple nde concened igid boss. The esls o iniil bckling lod omlion de o ignoing he shell cve in smll viciniy o he loding co e lwys slighly on he se side. I is ond h he shell cying cpciy e iniil collpse inceses coninosly wih he delecion. The hemispheicl shell deomion cn be epesened by mhemicl model consising o cone nd os which sped o symmeiclly om he pex wih incesing lod. The whole egion o he shell oming he cone ms be plsic o mke his deomion possible. The nlyicl solion esls e shown o closely mch hose o nmeicl nd expeimenl vles. I is lso eviden h lge boss size coesponds o highe iniil collpse lod nd lge dimple size. This solion is igoosly pplicble o shells hving smlle vles o R / (no exceeding ppoximely ). This is becse he symmey o he popging nnl zone bo he veicl xis cnno be ssmed hogho he lod-delecion esponse o lge, R /. Fo R / gee hn, he oe p o he 5

87 nnl zone ns o be nsble e cein deomion level whee he plsic p o he shell degenees ino n n-sided polygon. Fo hin shell which R / is lge hn, he bicion poin is ppoximely ond sing nlyicl solion s well. On he ohe hnd, o vey hin shells lime lod cying cpciy is limied by pnch sengh o he shell meil. 5

88 Noion sed in his chpe Rdis o Cicl Ple c Rdis o Cicl Ple sed o model behvio bicion poin b Rdis o Rigid Boss Iniil Shell Thickness Cen Shell Thickness M Shell Plsic Momen/Uni Lengh in shell M φ Meidionl Momen/Uni Lengh in shell M Cicmeenil Momen/Uni Lengh in cicl ple β M Rdil Momen/Uni Lengh in cicl ple M Tngenil Momen/Uni Lengh in shell N ϕ Meidionl Foce/Uni Lengh in shell N Cicmeenil Foce/Uni Lengh in shell β Q Tnsvese She Foce /Uni Lengh in shell P Shell iniil Collpse Lod P Shell Cen Lod R Shell Rdis θ φ Knckle Meidionl Angle Tooid Angle ρ Knckle Rdis h Veicl delecion o he pex o he hemispheicl shell 54

89 CHAPTER - Nondescive Mehod o Pedic he Bckling Lod in Elsic Spheicl Shells. Inodcion nd Ppose o his chpe As he viey nd he qniy o shells incese, he deeminion o shell behvio becomes moe nd moe impon. One o he mos impon hings is o deemine he bckling lod o shells eihe expeimenlly o heoeiclly. The ciicl lod o n xilly loded elsic sce is he lod which sigh (ndeomed) nd he deomed om o sce e boh possible. Theeoe, smll incemen o his ciicl lod cses sdden deomion clled bckling. In n iniilly sigh membe, i he weigh o membe is negleced nd no ecceniciy exiss, nil he bckling lod is ined hee is no nsvese deomion heoeiclly. B his deiniion o ciicl lod holds e only in heoeicl sense. Becse, in eliy, de o he mncl impeecions, vey smll ecceniciy o non homogeniy, which e nvoidble o mos cses om he is poin o pplicion o he lod, nsvese deomions begins. When he ciicl lod hs been ined hee will be excessive deomions. Theeoe, he heoeicl minimm bckling lod is no elible one, since he cl ciicl lod is less hn h. Theoeiclly speking, hee e n ininie nmbe o ciicl lods, b in pcice only he smlles one is necessy, since, his lod o slighly below h lod bckling shold be expeced. Accodingly, he ge is o deemine his minimm bckling lod eihe heoeiclly o expeimenlly. 55

90 The moden design echniqe goes ino he model invesigion, especilly, o compliced sces s shells. Since in mos cses, he e behvio o he shell is no known o vey diicl o know, he bes hing is o mke some ssmpions nd hen o veiy hese ssmpions by mens o model ess. Accodingly, his chpe is invesiging possibiliy o nondescive mehod o inding he ciicl bckling lod in spheicl shells. Fo his ppose, Sohwell s nondescive mehod o colmns is exended o spheicl shells sbjeced o niom exenl pesse cing dilly, nd hen by mens o expeimens, i is shown h he heoy is pplicble o spheicl shells wih n biy symmeicl loding. In ddiion, Sohwell s nondescive echniqe o colmns is exended o he med colmns. Theeoe, pocede is developed h he ciicl lods o colmns in mli-soy me cn be deemined by sing lel delecions obined hogh mix omlion.. Sohwell Mehod in Colmns I x is mesed long he line o he hs, nd y deines he nsvese delecion, which is vey smll eveywhee mening h he smll deomion heoy is pplicble, nd ssming P hs consn inensiy ove he spn lengh L, wiing he sm o he oces in hoizonl nd veicl diecions nd he momens bo n biy poin, (eqilibim eqions) he condiion o eqilibim o he ben conigion my be obined, Fig... Smming p he oces in veicl diecion nd eqing o zeo; 56

91 Fige. : Colmn nde compession oce ( V V ) q F y V x (.) heeoe, e king he limi: dv q (.) dx The sm o he momens, ( M M ) ( V V ) x q x P( y) M (.) king he limi, 57

92 dy dm V P (.4) dx dx b since d y EI dx M d M dx dv d y P dx dx d y q P dx ( EIy ) Py q (.5) whee EI is he lexl igidiy, P is he xil oce nd q is he lel lod. In he cse o zeo lel lod, i is possible o wie eqilibim eqion s: EI y Py (.6) clling, P, he genel solion eqion will be: EI ( x ) y Asin x (.7) whee A nd x e wo biy consns o inegion. The condiion h y ms vnish boh ends o he s will be elized i, sin l which is possible o, b in h cse, he s will emin sigh o in ohe wods, hee nπ will no be ny deomion. The he solions e whee n,,, l P P n π Sbsiion o his vle ino he eqion yields o by sbsiing EI EI l he vles o n, Pl EIπ,4,9,..., n (.8) Fo pcicl pposes, only he smlles vle o h ciicl lod is needed. Theeoe, sing he smlles vle o n, which is one, 58

93 P c π EI (.9) L Sppose now h he s is no qie sigh iniilly, nd le y be he iniil nsvese delecion o he s. Then he eqion o eqilibim o zeo lel lods will be; ( y y ) py y y y EI (.) Povided h y vnishes boh ends o he s. A genel solion my be obined by expessing boh y nd y in ems o Foie s seies. Theeoe; y n w n nπx sin L (.) nπx y wn sin L n (.) Dieeniing hem wih espec o x weice, y n n π nπx w sin n L L (.) n π nπx y w sin n n L L (.4) Sbsiing bck ino he dieenil eqion (.), w n wn (.5) L n π I P n is he π n EI n h ciicl lod, hen P n.theeoe, L w n wn P P n (.6) 59

94 As P ppoches o is is ciicl vle P, w will be vey lgely mgniied, while w will be ppoximely in he io o ove hee, w nine ove eigh nd so on. This esl explins why s he lod is incesed, evey s ppes o ben ino sine wve o one by, ohe hmonics e pesen, b hey e vey lile mgniied by he lod, whees he is hmonic soon becomes lge. Also in h cse, om he dieenil eqion i is possible o noice h he delecion begins om he is pplicion o he lod, since he dieenil eqion is no homogeneos nymoe de o he consideion o he iniil impeecions, which e lmos nvoidble in pcice. The delecion o he s is cene my be wien s; nπ L δ wn sin w w w5... (.7) L n o w δ w (.8) P P Ths povided h he bove menioned ssmpions hold e, he lod-delecion cve is ecngl hypebol hving he xis o P, nd he hoizonl line P Pc s sympoes. (See Fig..) 6

95 Fige. : Lod Delecion cve in Sohwell Mehod Now, i hypebol psses hogh he oigin nd hs sympoes o eqions. x y β (.9) The eqion o his hypebol will be: xy β x y (.) o dividing by y ; x β x (.) y clling x v y he eqion o he hypebol becomes; x βv (.) 6

96 dx which is sigh line. Theeoe, i x is ploed gins v, he inveed slop is he dv mese o he smlles ciicl lod (See Fig.b). I insed o hving n iniil cve, n iniil ecceniciy e hs ppeed since in he cse o single lel lod Q he deleced shpe my be deined s; QL nπc nπx y sin sin π EI n P L L n n Pn (.) whee c is he disnce beween he sppo nd he poin o pplicion o he lod. Mking c ininiely smll, he condiion o bending by cople is obin. Theeoe, sing he noion M Qc ; ML y π n sin EI n n P Pn nπx L (.4) is obined. Fo wo momens pplied boh ends, by speposiion; ( L x) ML nπx nπ y sin sin (.5) π EI n n n P L L Pn nd he delecion he midspn is: 4 ML δ... (.6) π EI P P P 9 P Since de o symmey, he even ems do no ppe. I he coples boh ends e csed de o he smll ecceniciy e o he pplied xil lods P, hen clling Pe M 6

97 4Pe δ... (.7) πp P P 9 P P As P eches P he io P P ininiy, he ohes ppoch o zeo heeoe, eches o niy. So, while he is em ppoching o 4Pe δ (.8) π P P P o in moe compc om, 4e δ (.9) π P P Fo moe genel cse o combinion o boh n ecceniciy nd cve: 4e δ w n (.) π P P which is nlogos o he oiginl eqion: w P P δ (.) The min dvnge o he mehod lies in is geneliy nd simpliciy. In ll odiny exmples o elsic insbiliy he eqion EI y y Py (.) govens he delecion s conolled by is iniil vle, povided h boh e smll. I ollows h he delecion is eled wih he pplied lod by n ppoxime eqion o he hypebolic om 6

98 w δ w (.) P P c. Ageemen o Tes Resls in Colmns As sed in he peceding secion, he elion beween mesed lod nd delecion will no be hypebolic i he delecions e so lge h he elsiciy o he meil is impied; moeove, when he delecions e lge he ppoximions will no hold e. On he ohe hnd, i boh he delecions nd lods e smll, he exc mesemens will no be possible. Ths, sme sce o he obsevionl poins ms be expeced. Moeove, in his nge w (he is em o he Foie seies) does no necessily domine he expession o delecion (Sohwell 9). The d eqied o siscoy es is eled vles o lod nd cenl delecion o colmns which hve been loded s xilly s possible. The ecoded obsevions o his ne e given by T. Von Kmn (99) in n ingl disseion pblished in 99. In his ppe, he expeimenl ss e clssiied in hee gops, descibed elively s slende, medim o hick. Slende ss hving io k gee hn 9 nd o medim ss he slendeness io nges beween 45 nd9, nd hick ss e hose o which he slendeness io is less hn 45. In he slende gop, Von Kmn esed eigh ss, nmbeed,,, b, 4, 4b, 5, nd 6. These hve been nlyzed by R. V. Sohwell (9) in Tble. nd Fig.. exhibiing he elion o x o v. In some insnces he iniil obsevions hve been ejeced in esiming he bes iing sigh lines; sch obsevions e disingished in he ble by seisks. The 64

99 emining obsevions hve been nlyzed by he mehod o les sqes by R.V. Sohwell, in ode h he bes iing lines migh be deemined wiho inodcion o pesonl jdgmen. Tble., gives he vles o slope s deemined om he bes iing lines. This ble shows h he geemen wih heoy in egd o he ciicl lod is in c vey close. The cl vle o he modls o elsiciy, mesed by Kmn is 7 kg cm As pplied o he ss in Kmn s medim nd hick gops, he mehod iled o he eson h pciclly evey obsevion eled o delecions, which cn be shown o hve involved in elsic ile o he meil. I, hs, ppes h he mehod hs given good esls in evey cse whee hese cold be expeced, b h only il cn show whehe in ny insnce sicien obsevions cn be ken o delecion which on he one hnd e lge enogh o give esonble ceiny o v, nd on he ohe hnd, e no so lge h meil is sill elsic. One o he ss esed by Po. Robeson (9) ws loded wih sch smll ecceniciy by R. V. Sohwell s o povide i es o he mehod. Tble. pesens he nlysis o his cse, nd eled vles. As i is seen om h ige, he ploed poins ll on wo disinc sigh lines; he is, coveing vles o he mesed delecion nging om 7 o 8 hosnds o n inch, indiced n iniil delecion o bo. inch nd ciicl lod o 4.5 ons, which is some en pecen in excess o he vle 96 kgs obined om Ele s heoeicl expession when. modls o elsiciy is given he vle. ons, which ws mesed by Po. Robeson. in The second es epesenion vles o he mesed delecion in excess o 8 hosndhs o n inch (o which.64 inch is he mplide o he is hmonic in he Foie seies o he 65

100 speciied ecceniciy o.5 inch) nd ciicl lod o bo.9 ons, which is less hn hl pecen in eo s comped wih he heoeicl ige. Tble. : Nos.,,, b, 4, 4b, 5 nd 6. Mild seel: Modls o Elsiciy 7 kg cm S No: P, end lod in kilogms x, mesed delecion (mm) 6 v x P S No: P, end lod in kilogms x, mesed delecion (mm) 6 v x P S No: P, end lod in kilogms x, mesed delecion (mm) 6 v x P

101 S No:b P, end lod in kilogms x, mesed delecion (mm) 6 v x P S No:4 x, mesed delecion (mm) 6 v x P P, end lod in kilogms S No:4b P, end lod in kilogms x, mesed delecion (mm) 6 v x P

102 S No:5 P, end lod in kilogms x, mesed delecion (mm) 6 v x P S No:6 P, end lod in kilogms x, mesed delecion (mm) 6 v x P

103 Tble. : T. Von Kmn s ess S No: A dedced om bes iing line in Fig. (mm) P, esimed om slop o bes iing line in Fig. P, s given by heoeicl oml kg b b Tble. : Robeson s S No:5. Mild seel: Eecive lengh-.5 inches. Dimee-.999 inches. Slendeness- 89. Ecceniciy-.5 inches. P, end lods in ons x, mesed delecion in hosndhs o n inch. v x P

104

105 .4 An Exension o he Sohwell Mehod o colmns in Fme sce This secion shows n ppoch wheeby he ciicl lods o colmns in mli-soy me cn be deemined by sing lel delecions obined hogh mix omlion. The delecion d o colmns in mli-soy me e obined by inclding he eec o pesence o xil lods in he membe nd sce siness mices. Eecs o iniil bending momens e lso inclded. The esls pove h Sohwell ploing echniqe sed o deemine he ciicl lod o single colmn is lso pplicble o med colmns..4. Fomlion o he Govening Eqions Fo single colmn, omlions cn be mde diecly o obin he ciicl lod. Simple omlions cn lso be exended o simple pol mes s done by Zweig (968). When he poblem ises o nlyze nd design ll bilding sbjeced o gviy nd lel lods, howeve, se o moden mix mehods becomes necessy. Becse o lge nmbe o degees o eedom involved in mli-soy bilding me, one is cononed wih coespondingly lge nmbe o eqions which cn be epeedly solved only by se o mix mehod. When bckling is involved hen eicien mix omlions cn be mde sing slope delecion eqions modiied by sbiliy ncions. 7

106 7 Fige. : Noion o membe nd sce Wih eeence o noion sysem shown in Fig.., membe siness eqions,,, nd 4 e omed sing eqions o eqilibim (Bleich (95)). The xil oce eqion is given by: ( ) b L EA (.4) The she oce eqion is given by ( ) b b L EI L EI L EI β β β (.5) The bending momen eqions e given by ( ) 6 4 b b L EI L EI L EI β β β (.6) ( ) 6 4 b b b L EI L EI L EI β β β (.7)

107 whee is he end oce, is he end displcemen, he sbscips i nd j ee o he picl end o he membe nd he diecion especively, β i epesens coesponding sbiliy ncions E, A, I nd L e modls o elsiciy, coss secionl e, momens o inei nd lengh o he membe especively. Sbiliy ncions β hve been widely pblicized (Gegoy (968)). i They my be concisely expessed s (see ppendix C) ( co ) S (.8) n sin C sin cos (.9) β S 4 (.4) β SC (.4) ( ) 6 β S C (.4) β 4 co β (.4) 5 β.β 4 whee P wih P being he xil lod on given scl membe nd P E PE being he Ele Lod o h picl membe. By se o ppopie nsomion mices, he membe siness eqions e nsomed o globl coodine sysem. Sce siness eqions e hen omed by sing he eqions o compibiliy. The seps leding o he compe pogm e smmized s ollows: 7

108 ) Fo given me dop nmbeing sysem o membes nd joins, omle he nmbe eqions, nsom hem o he sce coodine sysem, nd omle he sce siness mix by sing compibiliy eqions. b) Adop popoione mliplie o lod, λ, o given exenl loding pen. Then o ny vle o λ, omle n exenl oce veco. In he exenl oce veco, he ixed end momens o ech membe ms be modiied o he pesence o xil lod in h membe. This is done by deiving he ppopie sbiliy ncion coesponding o he picl lel loding on he membe. The sbiliy ncion coesponding o niomly disibed lod ove he complee lengh o membe is deived s. β This ncion is sed o modiy he ixed end momens in he oce veco de o he loding o his me. c) Fo ny vle λ o loding pen, iniil solion necessies ll β i be se o niy (i.e. xil oces ). By sing n inve sboine, solve he sysem o eqions o he nknown displcemens. Wih he displcemens nd sing eqion, veco o xil oces o membes is obined. Fo ech membe, β i vles e clcled nd eqions (.5), (.5) nd (.6) e modiied in ode o eomle he membe nd sce eqions. d) The ieion is conined nil elemens o oce vecos nd displcemen vecos o wo sccessive ieions e coespondingly close o one nohe. The conol poin in conining he ieions is he es vle o he deeminn o he siness mix. I he vles o he deeminn e posiive, hen ccoding o he pinciple o posiive deinieness o sbiliy, he ieion becomes possible. 74

109 e) Once he inl se o displcemens nd oces o vle o λ e obined, hen hey e soed he se. ) Seps c, d nd e e epeed o dieen vle o λ. Solions o he sce e compleed; end displcemens nd end oces hve been obined. Now, i is possible o he sdy he membes..4. Membe Fomlions nd Solions Fo sigh pismic membe, he genel eqion o eqilibim is given by L IV EI q dx le P (.44) whee q is he inensiy o he lel lod on he membe. Solion o eqion (.44) is q Acos x Bsin x Cx D L L P x (.45) In he membe xis sysem ( x, x x ), o ech membe, he bondy condiions, he inl end oces nd displcemens coesponding o ech vle o λ o loding pen (obined in sep e o he pocede) e: End End b Displcemens Foces Displcemens Foces i i b i b b b i b b b 75

110 Using hese, he coeiciens o eqions o eqion (.45) cn be deemined. Le π sin, cos, w nd L P P E hen, w w w L A B ql C b P D ql b P (.46) Knowing he igh hnd side o eqion (.46), i is possible o deemine he coeiciens nd se hem in eqion in eqion (.45) o clcle he lel displcemen ny poin x o he membe..4. Sohwell Plo Using he compe pogm, o given me nd loding pen, ll inomion egding he xil oces nd displcemens dieen poins o colmns e clcled nd soed. Fo evey colmn membe, ploing xil lod/lel delecion io gins xil lod o vios poins wihin membe lengh yields sigh lines conveging o one poin on he xil lod xis. This is he elsic ciicl lod, P o he colmn nde he given loding o he me. c The ciicl lod coesponds o h vle o he xil lod when he displcemens ppoch ininiy. A cve iing oine gely edces he mon o wok involved in deemining Sohwell Plo which leds o ciicl lod. 76

111 .4.4 Cse Sdy In ode o demonse he pocede, he me designed by Wood nd invesiged by Bowles nd Mechn (956) is doped. A loding pen is cceped nd he solion is cied o. Fig.4 gives he dimensions nd he coss secion poiles o membes o he me. Also, is illsed he nmbeing sysem sed in he compe pogm. Tble.4 gives he chceisics o he membes o his me ccoding o he nmbeing sysem o Fig.4. The loding pen sed is given in Fig..5. In ble.5 e given he esls o membe 8 o vios vles o λ o his loding pen o he me. The Sohwell Plos e illsed, nd s seen in Fig..5, he P i vs. P cves yield sigh lines h convege o he sme poin deining he ciicl lod, Pc which is eql o 8 kips. 77

112 Tble. 4: Chceisics o Wood s Fme Membes ( in) 4 L I ( in ) ( in ) A ( ksi) P E 78

113 Pkips, inches Tble. 5: Resls o Membe 8 o Wood s Fme sbjeced o he given loding pen λ P P P P Discssion Resls nd Conclsion In pesen dy pcice o design igidly conneced colmns o mli-soy me, one ses he AISC (Ameicn Insie o Seel Conscion) nomogphs (Mnl o Sell Conscion 8) o deemine he eecive colmn lengh co k. In eeing o he ssmpions behind he nomogphs, one inds h pimy bending momens inheen in ll mlisoy me e ignoed. In ohe wods, only xil lods e consideed. Hee, he eec o pimy bending momens ( de o lods pplied poins ohe hn he joins) e inclded in he omlions hogh he se o sbiliy ncions o modiy he 79

114 ixed end momens ( o niomly disibed lod, β is sed). Theeoe, he pocede leds o moe ionl evlion o he ciicl lod. Once he ciicl lod hs been deemined, he eecive lengh co k is given by k P E P c Fo membe 8 o he me in he exmple, he El lod PE is in Tble.4. Then k is ond o be eql o.. AISC nomogphs yield k eql o.4. The echniqe demonses h hee is n eec o he loding pen o me on he lod cying cpciy o colmn in h me. This c hs been long discssed, nd he Gemn Codes pilly ke ino ccon. Zweig (968) poved h lod cying cpciy chnges when wo dieen vles o exenl lods e pplied o pol me. In conclsion, by sing mix omlions, he Sohwell ploing echniqe is poved o be pplicble o deemine he ciicl lods o colmns in mlisoy mes nde given loding pens. 8

115 Fige. 4: Wood s me 8

116 Fige. 5: Loding o me, Sowell plo o membe 8 8

117 .5 The Sohwell Mehod Applied o Shells To pply he Sohwell mehod o pedicing he ciicl lod wiho disbing he model, niomly compessed spheicl shell is consideed. I spheicl shell is sbmied o niom exenl pesse, i my ein is spheicl om, nd ndego niom compession whose mgnide is in his cse; p σ (.47) So, o vles o pesse incesing om zeo, he shell will is deom in he niom mnne. This pocess pesiss nil he exenl pesse, p, eches cein ciicl vle, p c, clled he iniil bckling pesse. A his vle o pesse he shell no longe deoms in niom mnne b jmps o snps ino nohe non-djcen eqilibim conigion. The pesse o which he shell jmps is clled he inl bckling pesse. Ths, i he pesse inceses beyond cein limi, he spheicl om o eqilibim o he compessed shell my become nsble, nd bckling hen occs. In ode o clcle his ciicl pesse, he bckled sce is ssmed o be symmeicl wih espec o he dimee o he sphee. B beoe going ino he deil o he bckling poblem i is dvisble is o conside he bending heoy o shells. 8

118 .5. Deomion o n Elemen o Shell o Revolion In he Fig..6 le ABCD epesens n ininiely smll elemen ken om shell by wo pis o djcen plns noml o he middle sce o he shell nd conining is pinciple cve( Timeshenko e.l (96) nd Timeshenko e l. (959)). Tking he coodine xes x nd y ngen n biy poin o o he middle sce nd he xis z noml o he middle sce, he elemen my be deined. In bending heoy o shells, i is ssmed h, he line elemens, which e noml o he middle sce o he shell emin sigh nd become noml o he deomed middle sce o he shell. Ths he lw o viion o he displcemens hogh he hickness o he shell is line (Novozhilov (959)). Ding bending, he lel ces o he elemen ABCD hve oion nd displcemen; speposing nd is consideing oion only wih espec o hei lines o inesecion wih he middle sce. The ni elongions o hin lmin disnce z om he middle sce e; ε x z z x x x (.48) z ε y (.49) z y y y 84

119 Fige. 6: Elemen ken om shell by wo pis o djcen plns noml o he middle sce I in ddiion o oion, he lel sides o he elemen e displced pllel o hemselves, owing o seching l o he middle sce, he elongion o he lmin consideed bove, l l ε (.5) l b since; l z ds nd, l ( ) z ds ε x x sbsiing hem bck ino he eqion (.5) nd smming p wih ε x de o he oion only, ε z ε x z z x x ( ε ) x x (.5) 85

120 nd similly, ε ( ) z ε y (.5) z z ε y y y y Since he hickness o he shell lwys will be ssmed smll in compison wih he dis o cve, he qniies z nd x z my be negleced in compison wih niy. y Also neglecing he eec o elongions ε nd ε on he cve he expessions become, ε x ε z (.5) x x nd, ε y ε z (.54) y y Assming h, hee e no noml sesses beween lmins σ is obined. Then, om he well known omle; z ( ε υε ) E σ x x υ y ( ε υε ) E σ y y υ x (.55) (.56) Theeoe, sbsiing he vles o sin componens; 86

121 ( ε υε z( χ υχ ) E σ x x υ ( ε υε z( χ υχ ) E σ y y υ y x (.57) (.58) e obined. Since he hickness o he shell is vey smll, he lel sides o he elemen my be consideed s ecngles. Theeoe he coesponding noml oces cing o he cenoi o he side will be; N x N y σ dz x E y υ ( ε υε ) o dy (.59) E σ dz ( ε υε ) o dx (.6) υ nd he momens, M x zσ xdz D( χ x υχ y ), nd M y z ydz D( χ y υχ x ) σ in which E D υ z dz E ( ν ). Now knowing h he sheing sesses e lso cing on he lel sides o he elemen in ddiion o he noml sesses. I γ is he sheing sin in he middle sce nd χ xydx he oion o he edge BC elive o z xes, τ ( γ χ )G is obined. Also knowing h xy z xy Qx τ xzdz, y Q τ dz, N yx N xy τ xydz nd M xy M yx zτ xydz one my ind, yz M xy yx ( υ) χ xy M D,nd N xy N yx E γ ( υ) 87

122 Ths ssming h ding bending o shell o evolion he line elemens, noml o he middle sce he esln oces N, N nd x y xy N nd he momens, M x, M y, nd M xy my be expessed in ems o six qniies; hee componens o sinε, ε, nd γ o he middle sce o he shell nd he hee qniies χ, χ, nd x y χ xy epesening he chnges o cve nd he wis o he middle sce o he shell..5. Eqions o Eqilibim o Spheicl shell De o he symmeicl deomion, one o he displcemen componens vnishes, nd he ohes e only he ncions o ngleθ. Theeoe; v w ( θ ) ( θ ) 88

123 Fige. 7: Spheicl shell elemen nd coesponding oces In he cse o symmeicl deomion, hee e only hee eqions o be consideed s he pojecions o oces on he x, nd z xes nd momens o oces wih espec he y xes. Theeoe e simpliicion, he hee eqions o eqilibim become; dn dθ ( N N ) co θ Q x x y x dqx Qx co θ N x N y p dθ (.6) (.6) 89

124 dm x dθ ( M M ) co θ Q x y x (.6).5. Eqions o Eqilibim o he Cse o Bckled Sce o he Shell In wiing he eqions o eqilibim o he cse o bckled sce o shell, which is ssmed symmeicl wih espec o ny dimee o he shell, he smll chnges o he ngles beween he ces o ny elemen sch s ABCD de o he deomion hs o be consideed. Since hee is symmey o deomion, he oion will only be wih espec o y xes. Fo he ce OC, his deomion is; dw (.64) dθ Ths he ngle beween he ces OC nd AB e deomion becomes; d dw dθ dθ (.65) dθ dθ n ngle, The ces AO nd BC owing o symmey o deomion, oe in hei own plnes by dw dθ (.66) 9

125 Sch oion in he plne o he ce BC hs componens wih espec o he x nd dw dw z xis eql o; cosθdψ nd, sinθdψ especively. Ths e dθ dθ deomion, he diecion o he ce BC wih espec o he ce AD my be obined by he oion o he ce AO wih espec o he x nd z xis hogh he dw dw ngles, sinθdψ cosθdψ nd, cosθdψ sinθdψ especively. dθ dθ Using he bove deived ngles insed o he iniil ones; d θ, sinθdψ, nd cosθdψ, he eqions o eqilibim o he elemen OABC become; dn x dθ dw d w w ( N N ) θ Q N Q x y co x y x dθ dθ dqx d w d dw Q co. co x θ N x N y p N x N y θ dθ dθ dθ dθ (.67) (.68) dm dθ x dw ( M M ) co θ Q M x y x y dθ (.69) In his cse; d w ε dθ (.7) w ε coθ (.7) d w d χ x dθ dθ (.7) dw χ y coθ dθ (.7) 9

126 The dieenil eqion o eqilibim developed hve bove e bsed on Love s genel heoy o smll deomions o hin shells which neglecs sesses noml o he middle Sce o he shell nd ssmes h he plnes noml o he ndeomed middle sce emin noml o he deomed middle sce. Fige. 8: Meidin o spheicl shell beoe nd e bckling 9

127 .5.4 Bckling o Uniomly Compessed Spheicl Shells I spheicl shell is sbmied o niom exenl pesse, hee will be niom compession whose mgnide is; σ p Le, v nd w epesen he componens o smll displcemens ding bckling om he compessed spheicl om, hen p nd hey become, N x nd N y die lile om he niom compessive oce p N x N x p N y N y (.74) (.75) whee N x nd N y e he esln oces de o smll displcemens, v nd w. Also, consideing he smll chnge o pesse on n elemen o he sce, de o he ε seching o he sce, p becomes p( ε ). Theeoe sbsiing eqions (.74) nd.75) bck ino he dieenil eqions o eqilibim (.67), (.68), nd (.69) nd simpliying nd neglecing he smll ems, sch s he podcs o N x, N y nd Qx wih he deivions o, v nd w ; dn x dθ dw ( N N ) co θ Q.5p x y x dθ (.76) dqx Q dθ x d w d d w co co.5.5 co dw θ Nx Ny p θ p p θ (.77) dθ dθ dθ dθ 9

128 dm x dθ ( M M ) co θ Q x y x (.78) Fom he Eqion (.78): Q x dm x dθ ( M M ) x y coθ Sbsiing Qx ino he Eqions (.76) nd (.77); N x E ν d d θ w coθ w υ (.79) N y E d ( ) coθ w υ w ν d θ (.8) D d d w dw M x υ coθ (.8) dθ dθ dθ D dw d d w M y coθ υ (.8) dθ dθ dθ Now inodcing wo dimensionless pmees, nd φ which e deined s; D( υ ) p( υ ) nd, φ E momens in ems o nd w one obins; nd sing he elsic lw o expess he oces nd E 94

129 d d dw d w d w ( ) coθ ( υ co θ ) ( υ) coθ ( υ co θ ) dθ dw φ dθ dθ dθ dθ dθ d d d d ( υ) coθ w coθ ( υ cog θ) coθ ( υ co θ) dθ d w coθ dθ d w dw d dw d w ( υ co θ) coθ ( υ co θ) ] φ coθ 4w co dθ dθ dθ dθ dθ dθ 4 d w 4 dθ dθ dθ dw dθ (.8) (.84) These wo eqions my be simpliied by neglecing in compison wih niy in he is em, since he shell is hin, nd heeoe io is vey smll. Also, de lgely o ngl displcemen χ we mke good se o his siion by inodcing n xiliy vible U sch h dψ. Ths, he expessions in he bckes become idenicl. Then sing he symbol dθ H o he opeion; H ( ) (...) d(...) d dθ coθ dθ (...). Eqion (.8) my be wien s ollows, d dθ [ H ( ψ ) H ( w) ( υ)( ψ w) ( υ) w φ( ψ w) ] The oh em, conining he co, my be negleced in compison wih he hid. Ineging his eqion wih espec o θ nd ssming he consn o inegion is eql o zeo; ( ) H ( w) ( υ)( ψ w) φ( ψ w) H ψ (.85) 95

130 And similly o Eqion (.84) ; ( ψ w) ( υ) H( ψ) ( υ) H ( w) ( υ)( ψ w) φ[ H( ψ) H( w) ( ψ w) ] HH (.86) Now, ny egl ncion o cosθ in he inevl cosθ my be expnded in seies o Legende ncions; P P... P ( cosθ ) ( cosθ ) cosθ ( cosθ ).5( cosθ ) ( n ) 5... Pn ( cosθ ) cosnθ n n! which sisy he dieenil eqion, n n cos ( n ) θ n( n ) ( n )( n ) cos ( n 4) θ... d P n dθ dpn coθ n n dθ ( n ) P (.87) Ths, peoming he opeion H; one obins, H ( P n ) λnpn HH (.88) ( P n ) λn Pn (.89) λ In which n( n ) n Assming genel expessions o U nd w o ny symmeicl bckling o spheicl shell, n ψ A n P n (.9) w n B n P n (.9) 96

131 sbsiing hem bck, n n { A [ ( υ) φ] B [ λ ( υ) φ] } P n λ (.9) n n n { A [ λ ( υ)( λ ) φ( λ ) ] B [ λ ( υ) λ ( υ) φ( λ ) ]} P n n n n n n n n n n (.9) The Legende ncions om complee se o ncions. Theeoe, he wo seies cn no vnish ideniclly in θ nless ech coeicien vnishes; Ths, o ech vle o n, he ollowing wo homogeneos eqions e obined. n [ λ ( υ) φ] B [ λ ( υ) φ] A (.9) A n n n n [ ( υ)( λ ) φ( λ ) ] B λ ( υ) λ ( ν ) φ( λ ) n [ ] n n n n n n λ (.94) A n nd Bckling o he shells become possible i hese eqions o some vle o n, yield o B n solion dieen hn zeo, which mens ivil solion o in ohe wods eqies hving zeo deeminn o he sysem o eqions. Ths, [ ] φλ λ ( υ ) ( υ ) λ λ λ λ ( υ) [ ] n n n n n n (.95) A solion o which λ. Th coesponds o vle o n eql o niy. Sbsiing his vle o λn, one obins, n A B which coesponding o he displcemens, 97

132 dψ dθ w A cosθ A sinθ This is displcemen o he sphee s igid body displcemen bckling mode long he xis o symmey. This ms, o cose, be exclded when we wish o invesige he elsic insbiliy o he shell. Now o λ ohe hn zeo; φ n [ ] ( ν ) λ λ ( ν ) n n (.96) λ n ( υ ) dφ which yields o is minimm, o o e simpliicion; dλ n ν λn ( υ ) λn (.97) Ths, ν λn ( υ ) λn (.98) nd ( υ ) ν φ 6 (.99) min B since ( υ ) p φ (.) E 98

133 φ yields he is ciicl lod p, nd min c p c E φ E υ υ ( min ) ( ) υ υ (.) o neglecing he second em in he penhesis; p c E ( ν ) (.) o, σ c E ( ) v In he bove deivion coninos viion o λn hs been ssmed b λn is deined so h n is n inege. Hence, o ge moe cce vle o he ciicl lod, wo djcen ineges s obined om he eqion n( n ) λ n shold be sbsied in he eqion o φ nd he vle o λn which gives he smlle vle o φn shold be sed in clcling ciicl sesses. B his moe cce clclion o he ciicl lod will die lile om h given by he bove oml, since he vle o λn is so lge (Timoshenko e l.96). Alhogh in he deivion symmeicl bckling o shells ws consideed, moe genel invesigion shows h owing o symmey o he niomly compessed spheicl shell wih espec o ny dimee, he oml lwys cn be sed o clcling he ciicl sess. 99

134 .5.5 Sohwell Pocede Applied o Shells The Sohwell pocede ws is pplied o colmns by Sohwell in 94. In his p o his chpe, n emp is mde o show h Sohwell pocede is lso pplicble o niomly compessed spheicl shells. In he deivion o he oml, s i ws done o he clssicl heoy o bckling shells ( See pevios p) i is ssmed h he displcemens nd w my be expessed s, ψ d dθ n A n P n w s beoe. n B n P n whee P n is he Legende ncions o he odes n nd An nd Bn e he el consns Fhemoe, s explined in he poceeding secion, he mncl impeecions, which e nvoidble, e consideed nd i is ssmed h hey my be expessed s; ψ n A n P n w n B n P n Moeove, o he ske o simpliciy, i is ssmed h he mncl impeecions o ψ is eql o zeo. Ths, i is ied only wih he diecion w. When he compessive lod q is pplied o shell, ech poin o he middle sce ndegoes elsic displcemens nd w, nd is noml disnce om he eeence sphee is hen becomes w w. I is ssmed o cose, h w is o he ode o n elsic deomion,

135 nd hen he elemen o he shell looks like he deomed elemens, which e sed o esblish he dieenil eqions o he bckling poblem. Agin going hogh he sme pocede one inds h he ems o hose eqions belong in wo gops. (See poceeding secion) In hose ems which conin he coφ, he qniies nd w descibe he dieence in shpe beween he deomed elemen nd n elemen o e sphee. In hese ems w ms now be eplced by w w. On he ohe hnd, ll ems which do no hve he co φ, cn be ced bck o ems o he elsic lw, nd epesen he sess eslns cing on he shell elemen. Beoe he pplicion o he lod, he shell hs been ee o sess nd he sess eslns depend only elsic displcemens nd w. Conseqenly, in ll hese ems w is js w nd nohing else. Ths one ives he ollowing se o dieenil eqions: ( w) H ( w) ( υ)( ψ w) φ( ψ w w ) H ψ (.) HH ( ψ w) ( υ) H( ψ ) ( υ) H( w) ( υ)( ψ w) φ[[ H( ψ ) H( w w ) H( w w ) ( ψ w w ) ] (.4) In which H denoes he sme opeo s beoe; H ( ) d( ) d co g dθ dθ ( ) Agin ollowing he sme pocede h is sed o he clssicl bckling heoy o spheicl shells (see he poceeding secion) one obins he ollowing se o lgebic eqions:

136 n ( λ υ φ) B ( λ υ φ) B φ A (.5) n n n n A n B φ n ( λ n λ υλ υ φλ φ ) B ( λ n λ υλ υ φλ φ ) ( λ ) n n n n n n n n (.6) Ths he poblem is edced o solving his se o eqions. Elimining bove se o eqions, An om he [ ( ) λn ( φ φ) λn ( υ υ φυ φ φ υ υ υφ φ υφ) λ ] B B φ ( ) λ ( υ φ ) λ n n n [ ] n n (.7) Theeoe he coeicien Bn becomes; B n (.8) Bn φλn[ ( ) λn ( υ φ )] ( ) ( φ φ ) λ υ( φ ) [ λ n n φ υ ] λ n Ae cnceling λn, nd neglecing he smll qniies s, φ nd hei podcs in compison wih niy; B B n (.9) λ n n φ[ λn ( υ φ) ] ( φ φ ) λ υ n

137 Coming bck o he deiniion o he displcemen w one my wie he eqion, n w B n P n o wiing i in deil, w B P B P BP... Sbsiing he vles o he Legende Polynomils in hei plces; 9 w B.5B B cos 64 B B 8 θ θ [ B...] cos... Also ccoding o he deiniion o; ( ) n n λ n which is minimm o n heeoe i hs he sme vles o n eqls o mins one nd zeo. Since n ms be n inege, i is chosen s zeo, which yields, λ nd coesponds o B which is ncion o λn nd ges smlle when λn becomes gee. Ths, i is possible o neglec ll he ems nd simply wie w B since he ems which conins cosθ,cosθ... e mch moe smlle so, bckling is slly expeced he plces whee θ is lge. Accodingly, i is possible o wie; w 4 B φ[ ( υ φ) ] ( φ φ ) υ

138 o enging he ems, w φ B φ[ υ φ ] ( φ) υ Neglecing he smll ems s nd φ in compison wih niy, ( υ ) w B υ φ q( υ ) Now wiing φ in deil, φ o Em Em p in which m is he io o he υ φ hickness o he dis o he sphee. The clssicl ciicl lod o spheicl shell s ond beoe is; p c Em ( υ ) o 4E m 4 Em p c ( υ ) heeoe, ( ) c υ Em p Eqing he wo elions; q φ p Em c Ths, φ q c Em p Sbsiing bck ( υ ) B n w ( υ ) p 4Em q c o peoming he coss mliplicion, 4

139 ( υ ) w p ( ) w c B υ n 4Em q Which is he eqion o sigh line i one xis is ken s w nd he ohe one s Ths he invese slope o his line gives he ciicl lod wih mins sign. Theeoe obining he slop o his line expeimenlly, w. p p c 4Em ( υ )S whee S denoes he slope. Ths, he Sohwell pocede is pplicble o niomly compessed spheicl shells. 5

140 .6. Nonline Finie Elemen Anlysis (FEA) Finie elemen nlysis (FEA) is cpble o ind he ciicl lod ssocie wih elsic bckling behvio. The is sep in elsic bckling nlysis is o ind he ciicl lod, which shold be eled o he lowes eigenvle. All FEA o his invesigion ws peomed sing he genel ppose pogm ABAQUS Vesion 6.7. ABAQUS is highly sophisiced, genel ppose inie elemen pogm, designed pimily o model he behvio o solids nd sces nde exenlly pplied loding. Eigh-node shell elemen ws sed o model hemispheicl shells. This elemen is genel ppose qdic shell elemen. The meil o he shells is ssmed s homogeneos, isoopic, imcompessible nd elsic. In ode o check o he cccy inble by his mehod, nmbe o spheicl shells wih dieen kinds o bondy condiion nd loding wee solved (Figs.9-.5). 6

141 Fige. 9: Deomion pen o hemispheicl shell wih hinge sppo nde dilly niom pesse 7

142 Fige. : Sbseqen deomion o hemispheicl shell wih hinge sppo nde dilly niom pesse 8

143 Fige. : Deomion o hemispheicl shell wih hinge sppo nde mximm dilly niom pesse 9

144 Fige. : Dieen cs o he deomed hemispheicl shell wih hinge sppo nde dilly niom pesse

145 Fige. : Sbseqen deomions in he cs o he deomed hemispheicl shell wih hinge sppo nde dilly niom pesse

146 Fige. 4: Bckling iniiion o he hemispheicl shell wih olle sppo nde dilly niom pesse

147 Fige. 5: Sbseqen deomion o he hemispheicl shell wih olle sppo nde dilly niom pesse

148 Fige. 6: Second mode o he deomion in hemispheicl shell wih olle sppo nde dilly niom pesse 4

149 Fige. 7: Bckling iniiion o he hemispheicl shell wih hinge sppo nde ing lod in R 5

150 Fige. 8: Bckling o he hemispheicl shell wih hinge sppo nde ing lod R 6

151 Fige. 9: Sbseqen deomion o he hemispheicl shell wih hinge sppo nde ing lod R 7

152 Fige. : Lge deomion o he hemispheicl shell wih hinge sppo nde ing lod disibed R 8

153 Fige. : Bckling iniiion o he hemispheicl shell wih hinge sppo nde ing lod in R 9

154 Fige. : Bckling o he hemispheicl shell wih hinge sppo nde ing lod R

155 Fige. : Lge deomion o he hemispheicl shell wih hinge sppo nde ing lod disibed R

156 Fige. 4: Bckling iniiions o he hemispheicl shell wih hinge sppo nde gviy loding

157 Fige. 5: Sbseqen deomion o he hemispheicl shell wih hinge sppo nde gviy loding

158 .7. Expeimenl Pogm In his p, n emp is mde o ind he ciicl lod o hemispheicl shells pinned he bse nd sbjec o niom pesse in pely expeimenl wy. I is inended o show h he omlion which hs been deived in his sdy give coec esls o shells o evolion nde vios xisymmeic loding condiions. A ol o six hin wlled polyehylene hemispheicl shells wee consced nd esed nde niom scion pesse. The bse dimees o hese shells wee 5 cm nd cm nd hei wll hickness wee.5 cm yielding R ios o 5 nd especively. I is eviden h he conscion o hese shells hogh mchining wold hve been diicl nd o he ollowing esons, he shells wee mde o solid polyehylene plsic which posses good ensile, lexl, nd impc senghs nd is lexl modls is popoionl o he siness o he meil. Is ceep esisnce is excellen nd is sbsnilly speio o mos plsics. Is mechnicl popeies e s ollows: Flexl modls: 65 MP Poisson s io:.4 Densiy: 5 kg/m Poisson s io:.4 A complee mily o hemispheicl shells is shown in Fig.6. The mncing o hese shells ws cied o wih he id o mchined mle nd emle molds mde om cs lminm lloy. The lminm lloy molds wee mchined wih consideble pecision nd hen he spheicl shells wee cs by pddling echniqe. Ech shell ws inspeced by poliscope o ense h no i bbbles wee pped in he shell wll. 4

159 Fige. 6: Hemispheicl shells smples mde o polyehylene 5

160 Fige. 7: A es mde o R75 mm shell sing scion pesse nd hee displcemen gges vios poins. 6

161 Fige. 8: Deomion mesemen wih hee gges dieen locions in hemispheicl shells nde niom scion pesse (R 75mm). 7

162 Fige. 9: Tess mde o R5 mm shells wih scion pesse nd hee displcemen gges dieen locions hemispheicl shells nde niom scion pesse (R 5 mm). 8

163 Fige. : Deomion mesemen wih hee gges in dieen locions hemispheicl shells nde niom scion pesse (R 5 mm). 9

164 Fige. : Iniil bckling o hemispheicl shells nde niom scion pesse (R 5 mm).

165 Fige. : Iniil bckling o hemispheicl shells nde niom scion pesse (R 5 mm).

166 Fige. : Iniil bckling o hemispheicl shells nde niom scion pesse (R 75 mm).

167 Fige. 4: Sevel ess mde on dieen smples sing scion pesse wih hee nd ive gges dieen locions.

168 .8. Resls nd Discssion In his sdy, he Sohwell pedicions e comped wih expeimenlly, nd nmeiclly obined vles. The esls o he invesigion e smmized in Figs.5 o.45 which he pedicions o he Sohwell mehod e comped wih mesed bckling lods in expeimenl sdy nd nmeicl simlions. Sce in he expeimenlly obined bckling pesses is pobbly de o viions h exised occed in he specimens becse o he c h ech one ws cs sepely. The mnces did, howeve, ke consideble ce ding he mncing pocess, nd especilly wih he mix, nd o his eson he sce is vey smll. Ths, i is likely h he min soce o eo comped o heoy is becse o mesemen eding eos, nd impeecions in meil popeies. Howeve, he geemen beween mesed bckling lod nd Sohwell pedicion is emkble. Mosly, he Sohwell mehod ended o yield bckling lods which e slighly highe hn hose mesed nd he dispiy o bckling lod is somewh diicl o deec. Neveheless, he pediced lods e elible o be slighly highe (p o bo 7%) hn he cl lod enconeed. Theeoe, esonble degee o cion is ecommended o be execised.sce in he nmeiclly obined bckling pesses o xisymmeicl bckling cses e vey smll nd i is mos likely de o he ssmpions o he inie elemen solion. Fo he cse o bckling o spheicl shell wih olle sppo nde niom pesse, he bckled shpe is no xisymmeic nymoe (Fig.5). So, once d e colleced om he pinciple xes locions, he nswe is ccepble nd hee is only % eo ohewise, he deviion om he coec vles is consideble. Biely, he mehod povides vlble echniqe o esiming he bckling lod o spheicl shells wiho hving o condc descive es. The esls obined hve logicl cccy nd he mehod does no se om he ny mjo isses. Any bondy condiion 4

169 he edge my be ken ino ccon nd s long s he loding is xisymmeic, his pocede cn be sed wih esonble cccy..8. Expeimenl wok indings w/p-w Expeimenl w/p (mm /N) y -.x w (mm) Fige. 5: Plo o S. w gins w ( R 5mm,. 5mm p ) 4Em p c pc. 88MP S ( υ ) Fom es p. 78MP Eo o his cse:.8%.78 5

170 w/p-w.5..5 Expeimenl Nmeicl w/p (mm /N) y -.5x w (mm) Fige. 6: Plo o w gins w ( R 5mm,. 5mm p ) S.5 4Em p c pc. 87MP S ( υ ) Fom es p. 78MP Eo o his cse:.5%.78 6

171 w/p-w Expeimenl.6 w/p (mm /N) y -.79x w (mm) Fige. 7: Plo o w gins w ( R 5mm,. 5mm p ) S.79 4Em p c pc. 95MP S ( υ ) Fom es p. 8MP.95.8 Eo o his cse: 6.8%.8 7

172 w/p-w Expeimenl w/p (mm /N) y -.885x w (mm) Fige. 8: Plo o w gins w ( R 75mm,. 5mm p ) S.885 4Em p c pc. 4MP S ( υ ) Fom es p. 5MP.4.5 Eo o his cse: 4.8%.5 8

173 w/p-w Expeimenl w/p (mm /N) y -.8x w (mm) Fige. 9 : Plo o w gins w ( R 75mm,. 5mm p ) S.8 4Em p c pc. 48MP S ( υ ) Fom es p. 5MP.48.5 Eo o his cse: 6.5%.5 9

174 .8. Nmeicl Sdy esls.8... Fo niom dil pesse cse wih hinge sppo: w/p-w w/p (mm /N) y -.457x w (mm) Fige. 4: Plo o S.457 w gins w ( R 5mm,. 5mm p ) 4Em p c pc. 84MP S ( υ ) Fom es p. 8MP.84.8 Eo o his cse:.7%.8 4

175 w/p-w w/p (mm /N) y -.4x w (mm) Fige. 4: Plo o p gins w ( R 75mm,. 5mm ) S. 4Em p c pc. 9MP S Fom es ( υ ) p. 7 MP.9.7 Eo o his cse: 5.4%.7 4

176 .8... Fo niom downwd pesse cse wih hinge sppo: w/p-w w/p (mm /N) y -.97x w (mm) Fige. 4: Plo o S.97 w gins w ( R 5mm,. 5mm p ) Bckling pesse sing Sohwell mehod Bckling pesse p. MP 4Em p c pc. MP S ( υ ).. Eo o his cse: 4.5%. 4

177 .8... Fo niom dil pesse cse wih olle sppo : w/p-w 4 8 w/p (mm /N) y -5.4x w (mm) Fige. 4: Plo o S 5.4 w gins w ( R 5mm,. 5mm p ) Bckling pesse sing Sohwell mehod Bckling pesse p. 44 MP 4Em p c pc. 9MP S ( υ ).44.9 Eo o his cse:.%.9 4

178 Fo Ring lod cse wih hinge sppo: w/p-w..8 w/p (mm /N).6.4. y x w (mm) Fige. 44: Plo o w gins w ( R 5mm,. 5mm p ) S Bckling pesse sing Sohwell mehod 4Em p c pc.4mp q.57n / mm S ( υ ) Bckling pesse p. 5MP.57.5 Eo o his cse: 4%.5 44

179 Fige. 45: Compison o Sohwell expeimenl pedicion o heoeicl bckling pesses. 45