TMR4205 Buckling and Ultimate Strength of Marine Structures

Transcription

1 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of TMR5 Buckling nd Ultite Strength of Mrine Structures Chpter 3: Buckling of Stiffened Pltes Professor Jørgen Adhl MTS-5..8

2 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of COTTS 3. BUCKLIG OF STIFFD PLATS 3. Introduction Locl Plte Buckling lstic Buckling of Initill Perfect Pltes Correction for Plsticit Post-Buckling Cpcit of Pltes ffective Width Concept The Influence of Boundr Conditions The Influence of Initil Deflections The Influence of Residul Stresses Siple odel for post-uckling cpcit Mrguerre s lrge deflection equtions for pltes The Influence of Coined Loding Buckling of Stiffened Pltes Collpse Modes Idel lstic-plstic Strut Anlsis ffective Width Method According to Fulkner Interction Beteen Copression nd Lterl Pressure Initil Yield Method (DV Clssifiction ote 3.) Tripping of Stiffeners Grillge Buckling lstic Anlsis References... 3

3 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of 3. BUCKLIG OF STIFFD PLATS 3. Introduction Stiffened pltes re frequentl used s structurl coponents in rine structures. Tpicl eples re the hull girder nd superstructure of ship, the pontoons of sei-suersile nd the deck of offshore pltfors. The in tpe of fring sste found in hull girders consists of reltivel closel spced longitudinl stiffeners ith ore idel spced hevier girders in the trnsverse direction. This is illustrted for otto/side structure in Figure 3-. The in purpose of the pltes is to trnsfer the hdrosttic lods (the difference eteen eternl nd internl pressure) to the stiffeners, hich gin, through e ction, trnsfer the lods to the trnsverse girders. These re prts of the trnsverse fres of the hull girder. Fro the verticl girders the lods re introduced s erne stresses in the side. The side ill lso e sujected to hdrosttic lods. In generl the otto plte ill, in ddition to the hdrosttic pressure, e sujected to iil in-plne lods cused longitudinl ending of the hull girder nd fro the hdrosttic pressure on the sides s illustrted in Figure 3-. It is ver difficult to perfor rigorous nlsis of such pnels sujected to siultneous ction of lterl pressure s ell s in-plne lods. For design purposes, the prole is often split such tht the criticl lod is first deterined for ech of the lods cting lone. The criticl lod for the coined loding is found ens of soe interction forul. Preters of jor iportnce for the ehviour of stiffened pltes re:- length/idth rtio of the pnel stiffener geoetr nd spcing spect rtio for plte eteen stiffener plte slenderness residul stresses initil distortions oundr conditions tpe of loding

4 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of Figure 3- Stiffened Pnels in Botto Structure, (Interction Beteen Glol nd Locl Lods). The possile filure odes of stiffened pnel under longitudinl copression e clssified s follos: Plte uckling nd ultite collpse, hich ens tht the iu plte lod is eceeded nd is folloed unloding of the plte, leding to collpse of the stiffened pnel efore significnt ield occurs in the stiffeners. Interfre fleurl uckling of the longitudinl stiffeners ith ssocited plting. This tpe of filure involves ielding of the stiffeners, hich is ccelerted loss of stiffness due to uckling or ielding of the plte. Restrined torsionl uckling of stiffeners (see Section.8), hich is due to elstic or elstoplstic loss of stiffness depending on the slenderness of the stiffeners, the rottionl restrint provided the plting, nd the initil out-of-shpe. Overll grillge uckling, hich involves ending of trnsverse girders s ell s longitudinl stiffeners. Most structures re designed to prevent overll grillge uckling. Therefore, this filure ode is unlikel ecept for lightl stiffened pnels found in superstructure decks. For short pnels, locl plte uckling e the criticl ode. It ill e shon, hoever, tht pltes, depending on the oundr conditions, possess significnt reserve strength s indicted in Figure 3-c. This reserve strength e tken into ccount in the design of stiffened pnels lloing the plte to defor into the post-uckling region, ut it hs to e ssured tht locl uckling does not occur frequentl. Repetitive uckling nd strightening under cclic loding led to (etree) lo ccle ftigue filure.

5 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 5 of Ail lod perfect plte iperfect plte Deflection (c) Lod-deflection ehviour of plte eleent. Figure 3- Buckling of Short Pnel. The post-uckling ehviour of the plte depends hevil on the oundr conditions. Iportnt interctions cn occur eteen stiffeners nd the plte. Torsionll fleile stiffeners ill tist in ccordnce ith the plte uckling ode s shon in Figure 3-. In the cse of hev stiffeners (Figure 3-) considerle redistriution of lods is possile in the post-uckling phse, hich gives n incresing lod-crring cpcit s indicted in Figure 3-c. For long pnel inter-fre fleurl uckling of the stiffener ith ssocited plte flnge ecoes potentil filure ode. Pnels ith hev stiffeners ill follo colun ode of collpse (Figure 3-3). The ssocited lod-deflection chrcteristic is shon in Figure 3-3. Hoever, if uckling occurs ith the stiffeners in copression, fleile stiffeners ill e susceptile to restrined torsionl uckling. The interction eteen the to filure odes led to drtic unloding in the post-collpse region (Figure 3-3c). One ight suggest tht the optiu of designing pnel is to require equl cpcit ginst locl uckling nd inter-fre fleurl uckling, (see Figure 3-). Oing to interction effects eteen the to odes, the elstic uckling lod is reduced s copred to the uler lod. The optiu pnel is lso ver iperfection sensitive (Figure 3-).

6 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 6 of A vst ount of reserch hs een crried out on the ehviour of stiffened pltes. Anltic ork hs inl delt ith idel structures. Hoever, the developent of non-liner coputer progrs hs rendered possile in-depth studies of the effect of different iperfections. A revie is given in /7./. Hoever, for design purposes siplified procedures clirted ginst eperients or nuericl studies, should e ville. In the reinder of this section soe of the ost iportnt ethods re presented. Ail lod perfect pnel iperfect pnel () Hev stiffeners. Deflection Ail lod perfect iperfect Deflection (c) Torsionl uckling of stiffeners. Figure 3-3 Buckling of Long Pnel. Ail lod perfect pnel uler Lod iperfect pnel Deflection () Lod-deflection ehviour. Figure 3- Buckling of Optiu Pnel.

7 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 7 of 3. Locl Plte Buckling The clssicl pproch to elstic plte uckling proles is either solving the differentil eqution of equiliriu or ppling energ ethods. 3.. lstic Buckling of Initill Perfect Pltes Solution of the differentil eqution The procedure for clculting the elstic uckling lod is illustrted for n initill plne plte sujected to in-plne unifor copression. The equiliriu eqution for plte is given /7./ q (3.) D here the plte stiffness is given, nd, The quntities, 3 t D (3.) ( ) ν ( ) (3.3) t t t (3.) re the erne stress resultnts. For uniil copression, siple supports nd no eternl lod, qution (3.) tkes the for, (3.5) D The criticl lod results fro the solution of the differentil eqution. The folloing displceent function stisfies qution (3.5) nd the oundr conditions, n C n sin sin (3.6) here nd n re nuer of hlf ves in the - nd -directions, (see Figure 3-5). The solution is given the epression, t ( ν ) k (3.7) here k is fctor depending on the plte spect rtio, (see Figure 3-6).

8 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 8 of o Figure 3-5 Sipl Supported Plte Sujected to Unifor Copression. o k fctor 8 6 3,5,5,5 3 3,5,5 5 5,5 Aspect rtio (/ ) Figure 3-6 Buckling Coefficient versus Plte Aspect Rtio. Solution ens of energ ethod Alterntivel, the energ ethod e pplied in the se s deonstrted for colun uckling in Section 5.. The elstic strin energ cused ending defortion of the plte t the criticl lod is given, U D ( ) ( v) dd (3.8) It cn e shon tht the ter in the lst rcket in qution (3.8) disppers if either of the to conditions re stisfied long the oundries; i.) ii.) n (3.9)

9 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 9 of here denotes the differentition in the edge norl direction. The epression for the n potentil of eternl copressive lod reds H dd (3.) Then, the totl potentil energ ecoes, Π U H (3.) The criticl lod is no found ppling the principle of iniu potentil energ, i.e. setting the vrition; Π δ Π δ Ci i,,..., n (3.) Ci here C i denotes the function plitudes in the selected displceent field,. In order to stisf the ove condition, ll the derivtives hve to vnish. The ccurc of the energ ethods, to predict the criticl lod, depends on the selected displceent functions. At lest, the principl (essentil) oundr conditions should e stisfied. Generll, the energ ethod ields criticl lod greter or equl to the ect solution. The idel criticl lod cn generll e ritten s, t k (3.3) ( ν ) here k is fctor ccounting for the spect rtio, (/), the oundr conditions, nd lod the condition. In Figure 3-6, the uckling coefficient k hs een plotted ginst the spect rtio for sipl supported plte sujected to unifor copression. It ppers tht the iniu uckling stress occurs hen the length is ultiplu of the idth. For interedite vlues the nuer of ves is incoptile ith the plte length, hence rising the uckling lod soeht. In prctice, hoever, this dditionl strength is not tken into ccount. In Figure 3-7 nd Figure 3-8, the uckling coefficients re tulted for vrious lod nd oundr conditions.

10 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of.) on-unifor copression ψ ( ) ψ 8. k ψ. ( ψ : k ).) Pure ending ψ ( ) k c.) ( < ψ < ) k ψ ψ ψ τ d.) Pure sher k 5.3 e.) on-unifor copression ψ ( ) ψ. k ψ. ψ : k f.) Pure ending ψ ( ) 3 : 3 > : 6 8 g.) ( < ψ < ) k ( ψ ) k ψk ψ ( ψ ) k :Cse ith ψ e k :Csef f e f ψ Figure 3-7 The Buckling Coefficients for Vrious Lod Conditions.

11 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of 3 k k k k.3 k..3 k k k 7. k. 5 k length of pnel, - idth of pnel. /. Free edge Sipl surpotted edge Clped edge Figure 3-8 The Buckling Coefficient for Vrious Boundr Conditions. 3.. Correction for Plsticit For pltes ith lo idth to thickness rtio, qution (3.3) predict criticl stress in ecess of the ield stress, (see Figure 3-9), hich is unphsicl. Vrious ethods eist to ccount for plsticit effects. A convenient technique for odifing the elstic criticl stress due to plsticit is the φ-ethod, here the elstic-plstic uckling stress is given, cr φ Y (3.) here φ is n epiricl function relted to the structurl slenderness. Severl preters e used, ut the ost generl esure is the reduced slenderness rtio. Y λ (3.5)

12 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of Vrious epressions for φ, eists. One ethod is to ccount for elsto-plstic effects ens of n ellipticl interction eqution; It is seen tht, cr cr hen Y hen << Y Hence, the forul converges to the correct solution for oth:- stock eers nd slender eers. Solving for cr, e otin, Y cr φ (3.6) λ λ Another ell-knon solution is the so clled Johnson-Ostenfeld forul, λ, λ φ (3.7), λ λ In the cse of coined loding, s shon in Figure 3-, the ove procedure e pplied provided tht n equivlent stress nd n equivlent elstic uckling stress re defined. The requireent is tht the utiliztion for the equivlent stress should e equl to the utiliztion for the coined loding. This is convenientl epressed the folloing interction forul, c e e c c c τ τ (3.8) here,, nd τ re the elstic uckling stresses hen the corresponding stress coponent cts lone, nd e is the equivlent elstic uckling stress corresponding to the equivlent stress e. It is nturl to use the Von Mises stress, e τ (3.9)

13 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of, Stress rtio ΗΗcΗ Y,8,6,, Johnson-Ostenfeldt DnV, SLS uler ffective idth DnV, ULS,5,5,5 3 3,5 Reduced slenderness rtio Η Figure 3-9 lsto-plstic Buckling Curves. τ Figure 3- Coined Loding. The equivlent reduced slenderness rtio to e used in the ove odifiction for plsticit cn then e epressed s, c c c c Y Y τ λ e e e (3.) τ The eponent c depends on the plte spect rtio. Squre pltes tend to e ore sensitive to coined loding thn long pltes, ecuse the to uckling odes coincide for i-il

14 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of copression. Therefore, liner interction is often used for squre pltes nd n elliptic interction for long pltes. DV clssifiction note 3. specifies the folloing reltionship, c, > (3.) The ove design procedure is intended for checks in the serviceilit liit stte (SLS). In ost cses, plte uckling does not represent the ultite cpcit of the plte. If uckling does not represent serviceilit prole, for eple ecessive defortion for prcticl use of the structure or lo ccle ftigue repeted uckling nd strightening of the plte, the ultite cpcit e tken s Y ult,. < λ 5. (3.) λ The to criteri re copred in Figure 3-. For ver slender pltes the ultite strength is significntl lrger thn uckling strength. The post-uckling strength reserves re lso utilized in the design of plte/stiffener s descried in the susequent sections.,, Stress rtio c / Y,8,6,, SLS, FLS ULS,,,5,,5,,5 3, 3,5 Reduced slenderness rtio _ Figure 3- Ultite Strength versus Buckling Strength Of Pltes.

15 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 5 of 3.3 Post-Buckling Cpcit of Pltes 3.3. ffective Width Concept Slender pltes cn crr lod sustntill in ecess of ht is predicted elstic theor provided tht their unloded edges re constrined to rein stright. As result of lrge lterl deflections, erne stresses developes in the trnsverse direction, hich tends to stilize the pltes. At this stge the distriution of stresses long the unloded edges is no longer unifor ut increses tords the stiffeners. According to the effective idth ethod the ultite lod is otined hen the edge stress, e, in Figure 3-, pproches the ield stress. The folloing forul hs een proposed for sipl supported pltes here the unloded edges re constrined to rein stright, (reference /3/). e β β β β (3.3) here the plte slenderness preter is given, Y β (3.) t Figure 3- Actul Stress Distriution in Copressed Stiffened Plte. qution (3.3) ccounts for resonle degree of initil deflection in the uckling ode (d o /t ~ /5) ut not residul stresses. The epression is plotted in Figure 3-9. It ppers tht the effective idth forul predicts considerle post-uckling reserve strength for slender pltes. Hoever, this dditionl cpcit is reduced considerl if residul stresses re tken into ccount (see Section 3.3.). The post-uckling strength is norll not tken into ccount hen designing pltes for ships nd offshore structures, since this ould led to flutter the plte ech tie the uckling lod is eceeded. This is n undesired effect. Hoever, in the nlsis of coined stiffener-plte

16 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 6 of filure, the effective plte flnge is often ssessed ens of qution (3.3), (see Section 3..). The epressions ove hold true for plte loded on its short edge. For copressive lods on the long edge,, the folloing effective idth forul hs een proposed //. e , α Y β αβ β (3.5) 3.3. The Influence of Boundr Conditions The ctul oundr conditions ill in ost cses differ fro the idelized cses shon in Figure 3-8. It is in generll ccepted tht the oundr conditions of the loded edges do not hve significnt influence on the ultite strength nd it is usul to odel these edges s sipl supported. girder stiffner F D C Figure 3-3 Vrious Boundr Conditions for Plte leents in Stiffened Pnel. A B The jor influence stes for the conditions t the unloded edges. These re dependent on the ctul loction of the plte field in the pnel. With reference to Figure 3-3, plte F cn e considered s restrined, plte B s constrined, nd plte A s unrestrined. In the restrined cse the edges rein undistorted hile in the constrined cse trnsverse displceents re lloed ut the edges re forced to rein stright. In the unrestrined cse the edges re copletel free ith respect to trnsverse displceent. The difference in oundr conditions, eteen pltes B nd F, is cused the spect rtio. The closeness of the trnsverse girders t F does not llo trnsverse displceents, hile tht esil occur t the id-section of plte B. Generll, soe degree of elstic rottionl- nd trnsverse restrint on the plte fro the djcent stiffeners re present. Their effect depends on the reltion eteen stiffener diensions nd plte thickness. Results fro nuericl nd eperientl studies hve shon tht in-plne restrint cn hve strengthening effect of 5-5%, depending on the plte slenderness nd the gnitude of initil iperfections. The effect is ore pronounced for slender pltes nd interedite vlues of iperfections, δ o, (see Figure 3-). The rottionl restrint shos soeht stronger influence, ith strengthening effect of -5%.

17 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 7 of Figure 3- ffect of In-Plne Restrint on Pltes in Copression The Influence of Initil Deflections The effect of iperfections on the ultite strength of pltes depends strongl on their shpe. In ost theoreticl studies, initil deflections hve een ssued to hve the se shpe s the uckling ode hich gives reduction in ultite stress. Hoever, the elding process norll introduces n overll clindricl deflection. This out-of-ode deflection hve stiffening effect on pltes s shon in Figure 3-5. Hoever, this increse in ultite strength is of little prcticl use since it is folloed ore violent nture of unloding. It hs een clied tht onl the Fourier coponent of the deflected shpe hich coincides ith the uckling ode hs significnt influence on the ultite strength s illustrted in Figure 3-6, (reference /6/). Sttisticl nlsis of esureents of plte distortions shos tht the plitude of the uckling coponent is out hlf of the iu distortions. Vrious foruls re ville for predicting the iu distortion. The folloing reltion hs een used, δ o C C3, > (3.6) t t t here, s proposed Crlsen nd Czujko /6/, C.6 nd C The reduction in ultite strength hs een found to e lost linerl dependent on the gnitude of initil distortions /6/ ut ore sophisticted foruls lso eists /7/.

18 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 8 of Figure 3-5 Lod-nd Shortening Curve for Rectngulr nd Qudrtic Plte ith Single Hlf-Sine-Wve Initil Iperfection; dges Restrined.

19 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 9 of Figure 3-6 The ffect of Buckling Mode Coponents on Plte Strength /6/ The Influence of Residul Stresses The eld induced residul stress pttern in stiffened pnel is shon in Figure 3-7 (see lso Section.). The nlsis odel consists of tension lock in ield t the stiffener ttchent hich is lnced zone of unifor copressive residul stresses in the centre of the plte. The gnitude of these residul stresses results fro equiliriu considertions. r Y η (3.7) η t A ide intervl of η-vlues hs een quoted. For s-elded structures, η tends to e ver high. Hoever, if the eer is suject to lternting lods the residul stresses ill e reduced fter soe ers in service due to shke-out occsionl tension lods. Fulkner /3/, hs suggested design vlues of η eteen 3 nd.5 for ctul ships.

20 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of ηt - ηt Idelized ηt Tension Y Rel Copression r t Figure 3-7 Welding Stress Pttern in Pltes. The effect of residul stresses is to cuse loss of copressive plte stiffness s result of preture ielding in the copression zone. The gretest sensitivit to residul stresses is in the region here cr Y, hich hppens for /t rtio out 5-6, nd is ssocited in shift of copressive strin ε u t filure fro ε u < ε Y for pltes contining oderte residul stresses to ε u ε Y for pltes ith sustntil residul stresses. ( β ) r t η Rr, < β <.5 (3.8) Y β η t here t is the tngent odulus of the plte. The reduction in ultite strength due to residul stresses cn e clculted ultipling the epression in qution (3.3) ith the fctor given in qution (3.8). For design purposes, the folloing siple epression, ccounting for oth resonle initil defortions nd residul stresses, is dopted in DV Clssifiction ote 3.. e u.8.8 β Y β β (3.9) e β Siple odel for post-uckling cpcit Pltes loded into the post-uckling region cn crr lods sustntill in ecess of the clssicl uckling lod. This is ecuse in-plne erne forces develop in the trnsverse direction t finite deflections. In order to illustrte this siple is estlished. Consider the sipl, supported squre plte in Figure 3-8

21 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of sin sin dge trnsltes inrds z Figure 3-8 Assued displceent function in the post-uckling rnge During copression nd uckling it is ssued tht unloded edges rein stright, ut cn trnslte inrds, i.e. constrined oundr conditions. In the post-uckling rnge it is ssued tht the totl lod-crring cpcit contins to ters: ) The clssicl uckling lod, hich reins constnt during finite deflections. For squre plte the stress is given : t (3.3) ( ν ) ) An dditionl stress induced erne stresses in the trnsverse direction. This stress is non-unifor over the idth of the plte. The clcultion of the dditionl stress is sed upon the energ ethod. The folloing displceent function is ssued sin sin (3.3), sin cos (3.3) This is ect solution t initition of uckling nd is ssued to descrie the displceent in the post-criticl rnge ell.

22 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of The il erne strin for plte strip in the trnsverse direction is given ( ),,, v ε (3.33) nd the verge strin ecoes ( ) cos cos sin,,, v d v d v v ε (3.3) v represents the trnsverse displceent of the unloded plte edge. Since it reins stright v is constnt. The verge strin in trnsverse direction over the hole plte is given ( ) v, d v ε ε (3.35) With unconstrined edges the resultnt force nd hence the verge strin should e equl to zero. This ens tht the unloded edge trnsltes inrd distnce v (3.36) The resulting erne strin is therefore cos, v ε (3.37) To this strin there is ssocited trnsverse erne force given cos t t ε (3.38) The corresponding strin energ in the trnsverse direction is given

23 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of ( ), 6 cos 8 t d t dd U v ε (3.39) The potentil of the dditionl eternl lod in -direction is, 8 dd H (3.) (ote: The potentil of eternl lod is negtive since Ν is defined negtive in copression) The potentil energ ecoes H U Π (3.) The dditionl stress is deterined fro the condition ) ( U H δ (3.) or t (3.3) This ields 6 t (3.) Anlogous to the clcultion of strins in trnsverse direction the verge erne strin in the longitudinl direction cn e ritten s u ε (3.5) In this cse the folloing condition pplies t u t t ε (3.6) This ields

24 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of u 3 (3.7) Hence the vrition in -direction of the dditionl stress resultnt is given ( ) t ( ) t 6 u t cos cos (3.8) Tension Copression Copression Figure 3-9 Stress distriutions This distriution is sketched in Figure 3-9 It is oserved tht the iu copressive stress occurs t the oundries hile stress relief is cused the lrge displceents in the iddle of the plte. The iu copressive stresses occurs long the unloded edges (,) 3 (3.9) 6 Along this edge, the stress in trnsverse direction vries eteen ± (3.5) 8

25 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 5 of Introducing the en il stress This cn e ritten: (3.5) 6 3( ) 3 ± ( ) (3.5) It e ssued tht the plte ill fil once the von-mises ield criterion is violted. The criterion reds (3.53) Y This ens tht the criticl section ill e in the iddle of one of the unloded edges here the longitudinl copressive stress nd the trnsverse tensile stress ttins iu This ields (3.5) This epression cn e solved ith respect to ielding Y 3 76 / 38 λ λ (3.55) here the reduced slenderness is given ffective idth e//y,8,6,, ffective idth-fulkner Clcultion odel uler 3 5 Plte slenderness β

26 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 6 of β λ 9. Y (3.56) The post-uckling stress is plotted versus the plte slenderness fctor in Figure 3-. Figure 3- ffective idth versus plte slenderness It is oserved tht the siple clcultion odel ields criticl stress hich is significntl higher thn the uler stress for slender pltes. The odel grees ver ell ith the forul proposed Fulkner for high plte slenderness, ut ecoes too optiistic for lo slenderness. The reson for this discrepnc is the influence of initil deflections, hich ill cuse ielding t loer lod levels. Initil deflection e tken into ccount in the siple odel. We ill then hve to sutrct the initil strin free condition in the epression for the il strin, i.e. ( ) ( ) cos,,,, v d v o o v ε (3.57) This iplies tht the strin energ no tkes the for 6 t U (3.58) The potentil of the dditionl eternl lod in -direction is lso ffected the initil deflection; ( ),, 8 dd H (3.59) B iniizing the potentil energ there is otined 8 6. t (3.6) This ields 6 t (3.6)

27 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 7 of The stress in the pre-uckling rnge is lso ffected the initil deflection. Reclling qution (.) the stress resultnt cn e ritten s (3.6) The stresses long the unloded edge in the longitudinl nd trnsverse direction red ccordingl: 6 3 (3.63) ± 8 (3.6) Introducing the en il stress 6 o (3.65) e get: ± 3 3 (3.66) The von Mises ield criterion t the iddle of the unloded edges cn no e forulted ( ) ) ( ) )( (3 3 Y (3.67) here (3.68) The solution ecoes:

28 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 8 of ffective idth,8,6, Fulkner ffective idth /t..3, 3 5 Plte slenderness-β Y 3 λ 76 λ / 38 (3.69) Since e don t kno priori the gnitude of the deflection tht stisfies the ield criterion, n itertive procedure is required in order to deterine. Figure 3- shos the effective idth otined ith to different initil deflections. It ppers tht deflection plitude in the rnge of.-.3 ties the plte thickness ields quite good greeent ith Fulkner s epression ecept for sll plte slenderness. Figure 3- ffective idth ith initil deflection Mrguerre s lrge deflection equtions for pltes In order to stud the post-uckling cpcit of pltes sujected to in-plte copression it is necessr to stud the effect of lrge deflections. In the susequent nlsis, pltes hich hve n initil iperfection in the for of doule sinusoidl ve ill e studies, see Figure 3-.

29 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 9 of sin sin Pressure, p z Figure 3- Plte ith initil iperfection Let us first consider ht hppens hen plte strip of length d ith n initil deflection chnge d is further defored s shon in Figure 3-3. d d d Figure 3-3 Defored configurtion of plte eleent It is seen tht plte stretches due to the defortion. The elongtion is given d d ( d d) d d l (3.7) or dl d (,, (,,,, ), ) (,, ) (3.7)

30 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of The pproite reltionship is otined Tlor series epnsion of the root epression dl nd is vlid for oderte rottions. It is nturl to ssocite ith strin cused lterl d deflection. The totl strin should lso include the in-plte (liner) prt so tht there is otined ε u,,,, (3.7) Siilrl in the -direction ε v,,, (3.73) The corresponding ter for sher strin reds ε j ( u, v,,,,,,, (3.7) It is reclled (cfr. «Skiver og plter») tht the coptiilit requireent for erne eleent (in-plne loding) could e epressed s Φ,,,,, (3.75) o,,,, here the stress function Φ hs een introduced. It is relted to the stresses Φ,, Φ,, Φ, YY It is seen tht the ove eqution specilises to the ell-knon disk eqution Φ for no initil displceents nd no lterl defortion. The equiliriu eqution in the lterl direction is given D [ Φ (,, ) Φ, (,, ) Φ, (,, )], p (3.76) qutions 3.76 re clled Mrguerre s siultneous non-liner prtil differentil equtions for plte sujected to coined in-plne nd lterl loding. The couple oth the ending ehviour nd the erne ehviour for plte undergoing finite deflections. The re quite generl nd cover oth pltes ith initil deflection ( ) nd tht pltes ( ). In the ltter cse, if there is no lterl loding, i.e. p is specilises the ell-knon eqution [,,, ] D X (3.77)

31 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of The Influence of Coined Loding The effect of hdrosttic loding is to produce deflection of clindricl shpe eteen stiffeners. In siplified pproches, this is often treted s n dditionl initil deflection. Hoever, the occurrence of lterl lod on slender pltes cretes non-liner ehviour, hich does not necessril reflect the perfornce of initill deflected pltes. For eple, tensile stresses cused erne crring of the lterl lod ill e present t zero copressive lod. For squre pltes the initil deflection cused lterl lod is siilr to the uckling ode nd reduction of copressive strength is norll oserved. For long pltes the uckling ill tke on higher ode shpe hich leds to strengthening effect. Hoever, the post-collpse ehviour e ore violent. The lterl pressure in ship structures re norll oderte. Since ost ship pltes re long, the effect of lterl lod hs norll een neglected in plte design. As eplined in Section 3., plte eleents re soeties sujected to iil stte of stress ielding decrese in the unil lod-crring cpcit. Vrious interction tpe of foruls hve een proposed on the sis of eperientl nd nuericl studies. The folloing is due to Fulkner //. u u (3.78) here u nd u denote the uniil ultite copressive stresse in the il nd trnsverse direction, respectivel. An lterntive interction curve hs een proposed Vlsgård /8/. u.5 u u u (3.79) For the ore cople tpe of loding involving i-il copression, nd sher coined ith in-plne ending, Hrding nd Doling /9/ suggested u u u τ τ u (3.8) It should e noted tht for the cse of pure i-il loding, qution (3.8) represents still different interction forul.

32 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of. u /.8.6. / β /, β 3 /, β /3, / u Figure 3- Interction curve for i-il copression ccording to DnV Clss.ote 3. Severl of the interction curves tht hve een proposed cn e incorported in the eqution, c ' ' u u c (3.8) u u u u here the coefficients c nd c vr ccording to the plte spect rtio. For long pltes (/ > 3), it sees resonle to ssue tht c nd c.5 independent of the plte slenderness. For squre pltes, the interction depends hevil on the slenderness rtio. On the sis of curve fitting results of nuericl studies, DV clssifiction note 3. specifies, c c 3.e.35β (3.8) It is oserved fro Figure 3- tht stock pltes re less influenced i-il copression. The filure criterion pproches the von Mises ield criterion. Slender, squre pltes eperiences significnt interction. Long pltes re uch less influenced ecuse the filure odes re not coptile. Hence, the folloing reduction fctors re otined ' u u ' e e u.5 u, 3 (3.83) ' u u ' e e.5c u (.5 ) c, u

33 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 33 of here, e u Y (3.8) For interedite vlues of the spect rtio, interpoltion is used. In the cse tht the plte is in tension in the -direction, the ultite strength in -direction is equl to the ield stress, so tht the folloing interction is dopted, ' ' Y u u u u (3.85) Solving this ields, Y Y e e u u 3 ' ' (3.86) If the plte is sujected to sher, further reduction is needed, ' " τ Y u u 3 (3.87) or, 3 ' " ' " τ e e u u (3.88) The totl effective idth fctor is epressed s the product of the effective idth in the - direction nd the respective odifiction for trnsverse stress nd sher. This ields, e e e, e e e ' " " (3.89) If the stiffener fils tords the plte, such tht the plte ends in tension, the ove foruls re too conservtive. DV specifies the folloing effective idth forul, (Tension),.. ' e β (3.9)

34 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of 3. Buckling of Stiffened Pltes 3.. Collpse Modes A stiffener ith its ssocited plte flnge is convenientl odelled s n equivlent ecolun s shon in Figure 3-5. The folloing in tpes of collpse re distinguished, (see Figure 3-6):- i.) Fleurl uckling; - tords the stiffener, i.e. plte induced filure - tords the plte, i.e. stiffener induced filure ii.) Tripping sides of stiffener. Figure 3-5 quivlent Be-Colun Model of Stiffened Plte. Figure 3-6 Interfre Collpse Modes in Stiffened Pltes. 3.. Idel lstic-plstic Strut Anlsis An pproite solution for the collpse lod is given the intersection point of the loddeflection curves clculted for n idel elstic colun nd perfectl plstic colun. The elstic lod-deflection curve for pinned e-colun ith sinusoidl initil deflection of plitude δ o s derived in Section.3, δ e δ o (3.9) here is the uler uckling lod. The perfectl plstic solution cn e epressed s, M δ P (3.9)

35 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 35 of here the ending oent, M, nd the il force,, ust stisf the plstic interction curve of the cross-section. This depends on the direction of ending. Consider the stiffener cross-section shon in Figure 3-7 here the effective plte flnge re, A e, is greter thn the e re, A W. It is ssued tht the resultnt il force cts through the elstic neutrl is, G. For ending tords the plte, the plstic neutrl is is ssued to e t the intersection eteen the plte nd the stiffener, hich gives liner interction forul, M M P P (3.93) For ending tords the stiffener the interction forul reds, M M P z P h (3.9) here the lst ter on the left hnd side is defined in Figure 3-7. It ppers tht sll il lod is fvourle ith respect to the oent cpcit. The elstic solution, given qution (3.9), long ith the plstic solution for ending tords the plte, qutions (3.9-9) re plotted in Figure 3-7. The collpse lod, interpreted s the intersection eteen the to curves, re shon to gree ell ith results fro finite eleent nlsis. h G O O A Tords plte Tords stiffner - Y Y M z M/M p Tords plte Tords stiffner u/ Y Plstic Anlsis lstic Anlsis l/i l/i7 l/i A e Y - Y / p o/l - Figure 3-7 lstic-plstic Strut Anlsis of Plte-Stiffener ffective Width Method According to Fulkner This ethod, proposed Fulkner //, is sed on the elstic criticl lod for strut ith pinned ends, I' e (3.95) l ( A Ae) odified for plsticit ccording to the Johnson-Ostenfield forultion, qution (3.7) e λ, λ (3.96) The ultite strength is reduced to ccount for loss of plte stiffness,

36 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 36 of A Ae u e (3.97) A A p The effective oent of inerti of the stiffener is clculted for tngent (reduced) effective idth of the plte given, e (3.98) e here e is the edge stress (Figure 3-). The effective idth of the plte is given qution (3.3) hich ccounts for initil deflections. The effective idth should e reduced the R r, R, nd R τ, hich represent, respectivel, the effects of residul stresses, i-il loding, nd sher stresses. R r is given qution (3.8). Fro qution (3.3) results, nd, R. 5 u (3.99) u τ R τ (3.) τ An itertive procedure is required for clculting the correct vlue of e /, ut usull fe itertions re necessr. In coprtive stud crried out Guedes Sores //, it is concluded tht oth siplified ethods (Section 3..3; ffective idth, nd Section 3..5; Initil ield) predict the collpse lod resonl ell. The consistenc of the predictions is lso good, shoing coefficients of vrition in the order of %. The initil ield ethod generll underpredicts the strength soeho. Hence, sfet rgin is iplicitl incorported. Hoever, this is not the cse for the effective idth ethod hich should e used ith eplicit sfet fctors. 3.. Interction Beteen Copression nd Lterl Pressure For oderte lterl lods, the criticl uckling ode ill e the one ith lternting uckling in the djcent spns, nd the uckling lod is not influenced the pressure of the lterl lod. Hoever, s the lterl lod increses eond certin level, the uckling ode ill shift to the one here ll spns o fro the pressure side. Filure is then plte-induced nd the stiffeners e ssued to e clped. Conservtivel, liner interction forul is used for this filure ode //. q q u (3.) here the rigid-plstic collpse lod for three-hinge e echnis for the stiffener is given,

37 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 37 of q ( ) 8 Z t Z c (3.) l u When clculting the section odulus of the stiffeners, Z t nd Z c, ccount should e tken for the effective idth of the plte flnge in tension (Z t ) s ell s in copression (Z c ) Initil Yield Method (DV Clssifiction ote 3.) The uckling check in stiffened pltes is sed upon e colun pproch, cr Y (3.3) here is the il stress, cr is the criticl stress for plte/stiffener in pure copression, is the design ending stress, Y is the ield stress, nd is the uler uckling stress for plte/ stiffener. The criticl stress for pure copression is deterined in the se s descried for coluns, tht is, cr cr eq A (3.) Y cr Y W here, eq.5l, is n equivlent iperfection ccounting for the true out-of-strightness nd the effect of friction stresses, nd l is the eer length. Introducing the fctor, eq A z µ.5l (3.5) W i here z is the distnce fro the neutrl is to the stress point in question, the criticl il stress coes out to e, Y cr µ λ ( µ λ ) λ λ (3.6) here the reduced slenderness is defined, Y ie λ, (3.7) l e The effective rdius of grtion is defined, I e ie (3.8) A et

38 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 38 of The effective oent of inerti cn e ritten s, A I e I e A (3.9) et here e is the eccentricit of the stiffener (ithout plte flnge) to the plte flnge, (confer Figure 3-8), I is the oent of inerti of the stiffener ithout plte flnge, e is the effective idth of the plting clculted s descried in Section 3.3.5, nd t is the plte thickness. For plte induced filure there is shift of the neutrl is due to loss of effective idth. This cuses n etr eccentricit for the plte/stiffener hich hs to e tken into ccount. This is illustrted in Figure 3-8. A A z t z p t Before Buckling Buckling e After Figure 3-8 Shift of ffective eutrl Ais After Plte Buckling. The shift of neutrl is coes out to e, ( ) e t A e t z z p z p (3.) A t A t here z p is the distnce fro the plte flnge to the originl neutrl is. This shift of neutrl is hs to e dded to the equivlent initil iperfection used to clculte the ultite cpcit if the plte/stiffener, ielding eq. 5l z (3.) In the DV Clssifiction ote 3., to fctors re introduced in order to get etter greeent eteen the siple design foruls nd nuericl siultions. The initil distortion is gnified fctor of.5 for stiffener induced filure, nd z is reduced.65 for plte induced filure. The effective uckling length depends on the lterl pressure. If lterl pressure is not present, it is nturl to ssue tht the effective length is equl to the stiffener spn (fre spcing). When lterl pressure is present to filure odes cn e envisged:- setric uckling ith respect to the fre (in-out), i.e. the pressure is not sufficientl lrge to enforce uckling defortions to one side,

39 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 39 of setric uckling ith respect to the fre. Generll, the over-pressure e on either the plte side or the stiffener side. This ields four potentil uckling odes s shon in Figure 3-9. fre spcing l Trnsverse fres Long. profile of stiffner Pressure on plte side Pressure on stiffener side l e.6 l M pl /6 l e l M pl /8 Setric uckling Asetric uckling Figure 3-9 Potentil Buckling Modes for Plte/Stiffener. If the lterl pressure is lrge, the stiffener uckles setricll ith respect to the fre. This is odelled s clped end conditions. The corresponding uckling length is theoreticll.5l. In the DV Clssifiction ote 3., the effective uckling length is soeht conservtivel set to.6l. The ending oent nd the corresponding stress is clculted fro echnis pproch, ql (3.) 6 W here W, the elstic section odulus, should e clculted ith the effective plte flnge nd evluted ith respect to the plte flnge or the stiffener top (stiffener induced filure). If the hdrosttic pressure is not lrge enough, the stiffener uckles setricll ith respect to the fre. In this cse, sipl supported oundr conditions re ore relevnt nd the ending stress is given ql (3.3) 8 W The effective uckling length is equl to the stiffener spn. In ddition to the uckling check on the copressive side, the tensile side of the plte/stiffener ust e checked ith respect to ielding given the epression,

40 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of Y Y η (3.) here η is the llole usge fctor. (ote tht the ending stress is evluted on the tensile side of the plte stiffener). The chrcteristic teril strength is equl to the ield stress for plte induced filure. For stiffener induced filure possile interction ith restrined torsionl uckling hs to e tken into ccount. Hence, the chrcteristic teril strength is tken s the sller of the ield stress nd the torsionl uckling stress. In conclusion, the lterl pressure on plte side ust e perfored checked Tripping of Stiffeners The tripping prole is coplicted phenoenon to nlze nlticll. Most of the ethods do not ccount for this filure ode hich lso shos violent unloding in the post-collpse region. Hence, tripping filure is norll voided designing the stiffeners ith liited slenderness. For flt r stiffeners, the thickness-height rtio should stisf the folloing epression, h C (3.5) t here C Grillge Buckling 3.5. lstic Anlsis As previousl entioned, grillge uckling is in ost cses voided designing the structure ith sufficient rgin ginst this filure ode. It is ver coplicted to nlze the rel collpse ehviour of grillges including inelstic effects, lrge deflection, lod-redistriution effects, nd interction eteen locl nd overll instilit. Thus, ost nltic ork hs een confined to the elstic rnge. In the folloing, the energ ethod is used to clculte the elstic uckling lod of n orthogonll stiffened pnel ith pinned edges, (Figure 3-3). The displceent field is ssued to e, n C n sin sin (3.6) hich stisfies the essentil oundr conditions.

41 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of B A A s A, I Section B-B B I s Section A-A Figure 3-3 Buckling of n Orthogonll Stiffened Pnel ith Pinned dges. The elstic strin energ stored in the pltes is given, U U p q i U i q i U i (3.7) The contriution fro the plte is, U p ( ) D dd (3.8) hile the contriution fro the longitudinl stiffeners nd the trnsverse girders re respectivel, i I U d i i I U d i (3.9) (3.) The potentil energ of the eternl copressive lod is still given qution (3.). The criticl lod is found setting the vrition of the totl potentil energ equl to zero. The solution cn e ritten s follos, D i δ i δ ( t δ ) t (3.) here δ nd δ hve the folloing enings:- δ δ hen il (trnsver se) stiffe ners tke henil(trnsverse)stiffeners don' prt in de ttke prtin deflect flection ion hen longitudinl stiffeners (trnsverse girders) re not deflected. The folloing nottions re used,

42 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of i i t A s I Ds I Ds : equivlent thickness of longitudinl stiffeners (plte flnge not included) : reltive stiffness of longitudinl stiffeners (ith effective plte flnge) : reltive stiffness of trnsverse girders (ith effective plte flnge) ple 3. Wht is the necessr oent of inerti of the trnsverse girders, I, to ssure uckling of longitudinls eteen trnsverse girders? Solution: cept for ver lightl stiffened pnels, the first ter in qution (3.) cn e disregrded. Therefore, for overll uckling,,g D i i ( t t ) The iniu uckling lod is found differentiting ith respect to, ielding D ( ) [ i i ] t, g t For inter-fre fleurl uckling of stiffeners one otins,, In the liit,, g,, gives D ; i ( t t ) ( ) i i Assue tht there re to longitudinl stiffeners nd four trnsverse girders ( 5). This ields the folloing requireent to the oent of inerti of the trnsverse girders, 3 s I s 3 s.5 s 3 I I

43 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge 3 of 3.6 References. Guedes Sores, C. nd Søreide, T.H.: Behviour of Stiffened Pltes under Predoinntl Copressive Lods". Interntionl Shipuilding Progress, Vol. 3, Jn., "Buckling Strength Anlsis", Clssifiction ote o. 3., Det orske Verits, Fulkner, D.: "A Revie of ffective Plting for Use in the Anlsis of Stiffened Plting in Bending nd Copression". Journl of Ship Reserch, Vol. 9, Fulkner, D.: "Design Aginst Collpse for Mrine Structures". Interntionl Sposiu on Advnces in Mrine Technolog, Trondhei, Frieze, P.A., Doling, P.J. nd Hos, R.H.: "Ultite Lod Behviour of Pltes in Copression". Steel Plted Structures, Cros Lockood Stples, London Crlsen, C.A. nd Czujko, J.: "The Specifiction of Tolernces for Post-Welding Distortion of Stiffened Pltes in Copression". The Structurl ngineer, Vol. 56A, o.5, M Søreide, T.H. nd Czujko, J.: "Lod Crring Cpcities of Pltes Under Coined Lterl Lod nd Ail/Biil Copression". nd Interntionl Sposiu on Prcticl Design in Shipuilding, Toko/Seoul, Vlsgård, S.: "uericl Design Prediction of the Cpcit of Pltes in Biil In-Plne Copression". Coputer nd Structures, Vol., o. 5, Hrding, J.. nd Doling, P.J.: "The Bsis of the Proposed e Design Rules for the Strength to Cople dge Loding". Stilit Proles in ngineering Structures nd Coponents, Applied Science Pu., London, Rules for the Design, Construction nd Inspection of Offshore Structures, Appendi C, Det orske Verits, 977.

44 TMR5 Buckling nd Ultite Strength of Mrine Structures 3. Buckling of Stiffened Pltes Pge of IDX A setric uckling 38 B Be-Colun Model 3 iil stte of stress 3 oundr conditions 3,, 5, 7, 9, 6,, 39, C clssicl uckling lod, colun ode 5 coined loding 3,, 3 Coined Loding 3 constrined edges 5, 6, criticl lod 3, 7, 8, 9, 35, effective plte flnge 6, 35, 39, effective idth 5, 6, 8, 33, 36, 37, 38 ffective Width 5, 35 energ ethod 8, 9,, equivlent stress G grillge uckling, I Initil Deflections influence of 7 interction eqution Interfre fleurl uckling J Johnson-Ostenfeld L lterl pressure influence of 3, 3, 38, 39, locl uckling, 5 lo ccle ftigue, M Mrguerre s lrge deflection equtions 8 erne stresses 3, 5, O optiu pnel 5 P Perfect Pltes 7 Plte uckling Plte Buckling 7, 38 plte induced filure 3, 38, plte slenderness 3, 5, 6, 6, 8, 3 post-uckling cpcit, 8 potentil energ 9, 3, 6, potentil of eternl copressive lod 9 R reduced slenderness rtio, 3 reserve strength, 5 Residul Stresses influence of 9 restrined 5, 6, Restrined torsionl uckling S stiffener induced filure 3, 38, 39, stock eers strin energ 8,, 6, setric uckling 39 T Tlor series epnsion 3 Tripping 3, U uniil copression 7 V vrition 9,, 36, Von Mises stress von-mises 5