Lecture 5 Buckling and Ultimate Strength of Plates

Transcription

1 Topics in Ship Structurl Design (Hull Buckling nd Ultimte Strength Lecture 5 Buckling nd Ultimte Strength of Pltes Reference : Ship Structurl Design Ch.1 NAOE Jng, Beom Seon

2 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Fcts out MSC Npoli One of the orld s lrgest continer ships hen uilt (1991 Built to BV Clss nd chnged to DNV 00 Lst renel surve ried out in 004 in Singpore Built 1991 Length over ll m Bredth m Drught m Gross tonnge 53,409 GRT Cpcit 4419 TEU Slide

3 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Accident Jnur 007 MSC Npoli Ship left Anterp 17 Jnur 007 heding for Sines in Portugl 18 Jnur - ter ingress in engine room reported All 6 e memers sfel rescued Ship eched in Lme B ner Brnscome, UK on 19 Jnur 007 Slide 3

4 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Accident Jnur 007 MSC Npoli The vessels s split into to in Jul 007 Forrd prt s toed to Belfst for reccling

5 DNV Rule for Clssifiction of Ships Prt 3 Chpter 1, Section 13 B00 Plte pnel in uni-il compression Unstiffened Plte (Plting eteen stiffeners Elstic nd Inelstic Buckling Post-Buckling nd Ultimte strength Clssifiction Rule Idel elstic uckling stress el t 0.9kE 1000 s (N/mm For plting ith longitudinl stiffeners (in direction of compression stress: k=4 Johnson-Ostenfeld plsticit correction formul

6 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Wide Column The plte is cting more s ide column thn s plte. The product EI is replced the plte fleurl rigidit D. P D D E t t 1(1 v The thickness / length rtio pls the sme role s the slenderness rtio for columns. The idth pls no prt, no support long the unloded edge It is inefficient to use Simple support long the loded edges With no support long the unloded edges Buckling of ide column 6

7 Lrge-Deflection Plte Theor von Krmn Smll-Deflection Plte Theor Lrge-Deflection Plte Theor 9. Comined Bending nd Memrne Stresses-Elstic Rnge N N N p D 4 1 N N N p D D p

8 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Simpl Supported Plte From lrge-deflection plte theor 4 N t, p N N 0 D Since the edges re simpl supported, the deflected shpe cn e epressed in the form: C mn m n m n mn m n sin sin hich stisfies oth the oundr conditions nd the generl ihrmonic eqution. Plte is ssumed to e free to move inrd under the ction of the in-plne compression. The strin energ of deformtion is due to ending onl D U = m n U DCmn 8 m n t (1 v dd Buckled shpe of long plte

9 Reference Strin Energ Densit for plne stress (σ z =0 dzε z dz d dε dε dz d du 1 Lod pplied on ddz re 1 1 ( Elongtion in -direction ( ddz( d 1 dddz du ( ddz( d 1 ( du V 1 1 dddz ( ddz( dddz d d d thickness t

10 Strin Energ for plne stress (σ z =0 In Chpter 9 (Lecture 03, Plte ending (Derivtion of Plte Bending Eqution, the folloings re derived Reference ( 1 v z v E Gz = + u z u z v z ( dzdd du U t t 0 0 / / ( 1 ( 1 v z v E ( z ( z

11 Strin Energ Densit for plne stress (σ z =0 Reference dd v v D dd t v v v E v v Et dzdd Gz z v v E z v v E du U t t / / (1 (1 (1 ( (1 4 ( (1 ( (1 1 (1 v E G dd v D U 0 0 (1

12 Reference Work done for plne stress For unit-idth strip in Section 9. d W d N dd d 1 t 1 dd 1 d d (1 1 for smll

13 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Simpl Supported Plte Likeise, the ork done the in-plne compressive stress is 4 t t W W Cmnm 0 0 dd 8 m n Becuse of W=U, nd hence, m n DCmn m n t m C c1 c... cn min( c / d1, c / d,..., cn / dn m( c1 / d1, c / d,..., c d d... d 1 n n 1 n The minimum vlue of σ is given tking onl one term, s C mn, D m n ( tm here m nd n indicte the numer of hlf-ves in ech direction in the uckled shpe. When n=1, σ gives the smllest vlue. Hence the plte ill uckle into onl one hlf-ve trnsversel. m n mn / d ( D m 1 t m 13

14 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Simpl Supported Plte A uckling coefficient k is generll used. It depends on the tpe of oundr support. D m ( k k t m For design pplictions, in hich the plte thickness is to e determined, it is usull ritten like this: 3 t k Et ( KE K D 1(1 v 1(1- Q: Which iticl stress ill e higher?, hich stiffener rrngement is etter ginst in-plne compression?

15 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Simpl Supported Plte For long simpl supported pltes it is usull ssumed tht k=4. ( D k t k m m Assuming v=0.3 Homeork #1 Plot this curve D ( 4 t t ( 3.6E el Clssifiction Rule t 0.9kE (N/mm 1000 s k=4, s= (m 15 Buckled shpe of long plte

16 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Simpl Supported Plte /=1, m=1 /=, m= /=3, m=3 16

17 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Simpl Supported Plte For ide plte, in hich the spect rtio(/ is less thn 1.0, m=1 D ( 1 t For generl "ide plte, in terms of ecuse < ( D k t k k k For design purposes it m e ritten s: ( t KE K 1 1(1 v D 3 Et 1(1 v For v=0.30 K

18 1.1 Elstic Pltes Sujected to Uniil Compression Buckling of Simpl Supported Plte Longitudinl stiffeners: >>(=s, k=4 ( 4 D st Trnsverse stiffeners: <<, =s, =B, m=1 D ( 1 s t s B <<1 Longitudinll stiffened plting hve the gret dvntge over trnsversel stiffened plting in ship structures, nd the former is used herever possile. 18

19 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Reproducing the event in computer model Direct ve lod clcultions Liner strength nlsis Non-liner strength nlsis Lod nd strength comprisons Simultion of ck propgtion Slide 19

20 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Most severe ve for engine room re Hull forces: Sher force nd moment Aft ship out of ter Wve est round midship Verticl g force

21 Filure of MSC Npoli Continer ship Structurl rrngement in Engine room zone 1

22 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Not sufficient uckling cpcit The uckling cpcit might not hve een checked sufficientl hen the ship s uilt Potentill insufficient uckling strength in the engine room ulkhed

23 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Four stges of progressive collpse Outer shell

24 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Four stges of progressive collpse Inner structure

25 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Four stges of progressive collpse Inner structure

26 Filure of MSC Npoli Continer ship DNV Report Olv Nortun Alterntive correcting ctions The likelihood of reoccurrence is ver lo: Dmge sttistics re ver good Little likelihood of such hrsh se stte The ship s strength s elo the strength of similr ships Me not ll ships checked in this re Hoever the consequences re mjor Inese uckling strength Minor modifictions smll mount of steel to e dded Aft of the engine room ulkhed Cn e done hile in service

27 1. Other Boundr Conditions Solutions for Some Principl Cses Q: Which edge is more effective to inplne uckling? Loded edge or unloded edge? ( t KE A, B fied A : Unloded edge B: Loded edge B simpl supported A fied When unloded edge (A is replced simpl supported, the iticl uckling stress drops more thn hen loded edge (B is simpl supported. A & B simpl supported B fied A simpl supported 7 Buckling coefficient k in the design formul for flt pltes in uniil compress

28 1. Other Boundr Conditions Solutions for Some Principl Cses A 1, A clmped A 1, A pinned ( Loded edges clmped Loded edges Simpl supported A 1 pinned A clmped D k t In generl, 800mm, 3300mm, / 3~4 Unloded edge : clmped Loded edge : clmped Unloded edge : clmped Loded edge : simpl supported Unloded edge : pinned nd clmped Loded edge : clmped Unloded edge : pinned nd clmped Loded edge : simpl supported Unloded edge : simpl supported Loded edge : clmped Unloded edge : simpl supported Loded edge : simpl supported A 1 free A pinned A 1 A free A 1 free A clmped Buckling stress coefficient k for flt pltes in uniil compression 8 Unloded edge : free & clmped Loded edge : clmped Unloded edge : free & clmped Loded edge : simpl supported Unloded edge : free & pinned Loded edge : clmped Unloded edge : free & pinned Loded edge : simpl supported Unloded edge : free Loded edge : simpl supported

29 1. Other Boundr Conditions Solutions for Some Principl Cses ( D k t ( t KE A 1, A clmped Loded edges clmped Loded edges Simpl supported A 1 pinned A clmped A, B fied B simpl supported A fied A 1, A pinned A 1 free A clmped A & B simpl supported B fied A simpl supported A 1 free A pinned A 1 A free Buckling stress coefficient k for flt pltes in uniil compression 9 Buckling coefficient k in the design formul for flt pltes in uniil compress

30 1. Other Boundr Conditions Clmped Edges For in-plne lods, s in the cse of lterl lods, it is not possile to otin finite epressions for the solution of clmped pltes. Numericl solutions Fen, Muletsch, nd Lev. Buckling coefficient k for clmped pltes under uniil compression 30

31 1. Other Boundr Conditions Unloded Edges Rottionll Restrined Lundquist nd Stoell hve investigted the cse in hich the support long the unloded edges is intermedite eteen simpl supported nd clmped. Loded edges clmped The degree of rottionl restrint is specified in terms of coefficient of restrint, defined s C D Loded edges Simpl supported C : rottionl stiffness of the supporting structure long the unloded edge Buckling coefficient k for pltes ith loded edges simpl supported nd longitudinl edges OPen rottionll INterctive restrined Structurl L 31

32 1. Other Boundr Conditions Loded Edges Rottionll Restrined The importnt oundr conditions re those long the longer edges of the plte. Thus, for short ide pltes the edge restrint long the loded edges ecomes significnt. Similr to end conditions in column, using n effective length e: D e ( 1 et Using coefficient of restrint ζ : C D for clmped ends e = 1/ for one end simpl supported nd the other clmped e = C : rottionl stiffness of the supporting structure long the unloded edge The solution to this cse is otined from k1 k K1 tn K tn ( K1 K 0 in hich K1 nd K re relted to the uckling coefficient k. K1, ( k k 4 3

33 1. Other Boundr Conditions Loded Edges Rottionll Restrined Buckling coefficient k for ide pltes in compression elsticll restrined on the loded edges < k ( / k 33

34 1. Other Boundr Conditions Loded Edges Rottionll Restrined The corresponding coefficient in the "design" version of the ide plte formul t ( KE In ship structures the rottionl restrint is usull provided flnge-nd-e tpe of trnsverse stiffeners. In this cse ζ is given pproimtel 7 t Id J 3.6 d : depth of the e I : second moment of re of the stiffener out the midthickness of the e J : Sint-Vennt s torsion constnt for the stiffener 34 Buckling coefficient k for pltes ith loded edges simpl supported nd longitudinl OPen INterctive edges rottionll Structurl restrined L

35 1.3 Biil Compression All Edges Simpl Supported is prllel to σ nd to σ. Aspect rtio α=/. Appling the energ method ields the folloing epression for the iticl comintion: W 4 t 8 m 4 t Cmn m n Cmn n 8 m m m m U 4 4 m n DC mn 4 8 D m m 8 m m C mn m n U=W m D m n = n t If e denote the squre plte iticl stress nd nondimensionl form t (,1 3.6E m 1 m n = n ( ( 4,1,1 35

36 1.3 Biil Compression All Edges Simpl Supported σ α=0.5,m=1,n= t (,1 3.6E σ α=1,m=1 σ α=1,m=n=1 α=3,m=3 σ σ α=3,m=3 σ α=0.5,m=1 When σ σ, α=1 σ α=3,m=1 m=1 m= m=3 ( t ( E t 3.6E Buckling stresses of iill loded simpl supported 36pltes

37 1.3 Biil Compression All Edges Simpl Supported Plte under iil lod / = 3 / = 5 37

38 DNV Rule for Clssifiction of Ships Prt 3 Chpter 1, Section 13 B400 Plte pnel in i-il compression For plte pnels suject to i-il compression the interction eteen the longitudinl nd trnsverse uckling strength rtios is given c K c c c n 1 Homeork # Plot DNV i-il inte rction curve nd compre ith the previous interction curve (Fig. 1.8

39 1.3 Biil Compression All Edges Clmped For pltes sujected to pproimtel equl compressive stresses (σ σthe interction formul is t 3 ( 1.0E 3 When α=1, t ( 9.6E For squre pltes(α=1, iticl comintions re given for prticulr vlues of σ / σ, including cses in hich σ is tensile. When σ = σ t t ( ( E 10.15E 39

40 1.4 Other Tpes of In-plne Lods Pure Sher In ship structures the plting is commonl sujected to lrge sher lods. The shering lod cn cuse uckling since it gives rise to in-plne compressive stress. For the cse of pure sher, in-plne compressive stress is equl to the sher stress nd cts t 45 to the sher is. N In sher uckling, the coefficients re denoted s ks nd Ks. p N 0, N k s D t t t KE s 4 t D For simpl supported pltes k For clmped pltes s =5.35+4( / Buckling of n infinitel long, simpl supported plte ks = ( / 40

41 1.4 Other Tpes of In-plne Lods Pure Sher For simpl supported pltes k s D t ks =5.35+4( / 1(1 v k s E t 0.90k s E t B300 Plte pnel in sher (DNV Rule for Clssifiction of Ships Prt 3 Chpter 1, Section 13 el t s 0.9ktE (N/mm, kt s l

42 1.4 Other Tpes of In-plne Lods Pure Sher k s nd K s re given for vrious tpes of oundr conditions. Becuse of the smmetr of the pure sher loding, the choice of nd is independent of the lod. Buckling coefficient of flt pltes in sher Buckling coefficient of flt pltes OPen in sher INterctive (Design Structurl formul L

43 1.4 Other Tpes of In-plne Lods Biil Compression nd Sher For long pltes ks is given pproimtel : All edges simpl supported: k s All edges clmped: k s here 1/ 1/ 1/ 1/ = e e e e 1/ 1/ 1/ 1/ = e 3 e 3 e e D e t 43

44 1.4 Other Tpes of In-plne Lods In-plne Bending σ denotes the lrgest or edge vlue of the pplied stress. ( k D t Some pproimte formuls to clculte the vlues of k simpl supported edges: for for clmped edges: for one unloded edge clmped; the others simpl supported for unloded edges clmped; loded edges simpl supported for / / 3 / / 3 / 1 / 1/ / 0.4 k ( / 8.6( / k 3.9 k k k

45 1.4 Other Tpes of In-plne Lods In-plne Bending The figure illustrtes the cse in hich the ending is unsmmetric. For simpl supported edges the vlue of k is given pproimtel k 5 +4 ( simpl supported edges onl 3 45

46 1.4 Other Tpes of In-plne Lods Comined In-plne Lods: Interction Formuls Uniil compression nd in-plne ending (σ : iticl vlues of il loding (σ : iticl vlues of nd in-plne end ( ( Uniil lod(compressive or tensile nd sher For convenience e dopt the smol R to denote iticl lod rtio. In the present cse the strength rtios re Rc Rs ( The interction formul is s R R 1 1 c Rc Rs In-plne ending nd sher R R 1 ( >1/ ( Rs 46

47 1.4 Other Tpes of In-plne Lods Comined In-plne Lods: Interction Formuls Biil compression, in-plne ending, nd sher The to compression strength rtios re R ( R B performing series of four-vrile curve-fitting solutions, 0.65(1 0.6 / R 4 R ( R 1 (1 R ( Rs R B500 Plte pnel in i-il compression nd sher (DNV Rule for Clssifiction of Ships Prt 3 Chpter 1, Section 13 q c K c c q c q n 1 q 1 47

48 1.4 Other Tpes of In-plne Lods Comined In-plne Lods: Interction Formuls Interction curves for iil compression, in-plne ending, nd sher drn for α= 48

49 1.4 Other Tpes of In-plne Lods Comined In-plne Lods: Interction Formuls Homeork #3 Plot DNV i-il interction curve like the right figure nd compre ith the folloing curve for R =0 0.65(1 0.6 / R 4 R ( R 1 (1 R Rs R B500 Plte pnel in i-il compression nd sher (DNV Rule for Clssifiction of Ships Prt 3 Chpter 1, Section 13 n K cq c cq cq 1 q 1 49

50 1.6 Ultimte Strength of Pltes Pltes Without Residul Stress Uniill loded, simpl supported squre plte, ith sides free to pull in. some tpicl initil distortion in the form of hlf ve in ech direction. Plte slenderness t Y E The reltionship eteen the pplied lod (σ nd the il shortening Plte strength ithout elding (σ r =0 50

51 1.6 Ultimte Strength of Pltes Pltes Without Residul Stress Slender plte (β>.4 Buckling stress is ell elo ield stress nd elo the curve of collpse stress. After uckling (σ greter proportion of the lod is tken the region of plting ner the sides Non-uniform compressive stress distriution Deflected shpe of the uckled portion overll stiffness of the plte (dσ /dε is reduced. The center region ecomes more pronounced nd the mimum stress t the sides ineses. When the mimum stress = ield stress collpse. Ultimte strength Lrge mrgin eteen uckling nd collpse Plte strength ithout elding (σ r 51 =0 Post-uckling stress distriution

52 1.6 Ultimte Strength of Pltes Pltes Without Residul Stress Pltes of intermedite slenderness (1<β<.4 Buckling stress ield stress For rigorous nlsis, elsto-plstic lrge deflection theor to e used. As pplied stress ineses mgnifiction of the initil distortion loss of stiffness some locl ield stress redistriution ielding of the sides sudden collpse. Pitched roof : llos lrge il shortening ith minimum strin energ. Tpicl post uckling ehvior

53 1.6 Ultimte Strength of Pltes Pltes Without Residul Stress Sturd pltes (1>β The initil distortion is smller nd the mgnifiction is less ecuse the elstic uckling stress is ver lrge. Pltes cn r lod equl to the full sqush lod σ,u = σ Y. After the pek lod, the lod ring cpcit remins pproimtel constnt up to ver lrge strins. Plte strength ithout elding (σ r =0

54 1.6 Ultimte Strength of Pltes Pltes With Residul Stress Deprture from linerit occur t the stress hich is less σ less thn for stress-free plte. Sturd plte (1>β : no lod shedding, ut lrge reduction in stiffness regrded collpse. Intermeditel slender nd slender plte (1<β : the loss of ultimte strength σ r Idelized residul stress distriution r Y = t Middle prt σ + σ r = σ Y Edge prt σ - σ Y = σ Y Plte strength ith elding (σ r =0.1σ Y

55 1.6 Ultimte Strength of Pltes Effects of Other Prmeters Restrint t Sides Clmping the sides of plte inese the elstic uckling stress 75%, hoever, the inese in uckling stress even in slender plte 10% t most. Stiffeners surrounding the pnel is not clmped edge this restrint cn e ignored. Initil Deformtion The effect of initil deformtion removes shrp knuckle in curve of σ nd ε. The inesing lterl deflection cuses progressive reduction in the in-plne stiffness of the plte. Hoever, the ultimte strength is slightl deesed. Sher stress In plne sher stress tends to loer the resistnce to longitudinl compression. Reduced ield stress r τ σ Y r 1 3 Y eq 3 Y 55

56 1.6 Ultimte Strength of Pltes Ultimte Strength of Uniil Loded Pltes Plting of uniill loded, longitudinll stiffened, initil deformtion (δ p < 0.βt, residul stress (σ r 0.1σ Y side constrined to remin stright ut free to pull in For sturd plte, first loss of stiffness is tken s collpse. For pltes of greter slenderness : loss of stiffness is grdul. Secnt modulus rtio T E E s , 1 Design curves of ultimte strength nd secnt modulus 56

57 Curves for ultimte strength of pltes Ultimte Strength of Pltes Ultimte Strength of Uniil Loded Pltes Fulkner s formul for the ultimte strength of unedded pltes : good greement ith etensive eperimentl dt., u 1 Y The effect of residul stress strength reduction fctor R r rets Rr 1 Ets 1 (1.5 E E Y for.5 E ts E nd for 1.0 E ts 0 Restrined : the sides remin stright nd do not pull in. Unrestrined : oth tpes of trnsverse deformtions cn occur. Stress relived (σ r =0, verge elding (σ r 0.1σ r, hevil elded (σ r 0.33σ r