Some recent advances in the concepts of plate-effectiveness evaluation

Transcription

1 Thin-Wlled Structures 46 (008) Some recent dvnces in the concepts of plte-effectiveness evlution Jeom Kee Pik Deprtment of Nvl Architecture nd Ocen Engineering Pusn Ntionl University 30 Jngjeon-Dong Geumjeong-Gu Busn Repulic of Kore Aville online 18 Mrch 008 Astrct The effectiveness of plte elements in continuous plte structure cn e reduced y lterl deflections tht primrily occur due to lterl pressure loding or uckling. It is recognized tht deflected pltes cn e modeled s equivlent flt (undeflected) pltes ut with reduced effectiveness. A numer of the concepts to evlute the effectiveness of deflected pltes hve een developed in the literture. The present pper surveys recent dvnces in the concepts of plte-effectiveness evlution in terms of effective redth effective width nd effective sher modulus. Some closed-form expressions of the plte effectiveness re reviewed. The present pper is dedicted to Prof. J. Rhodes who mde gret chievement nd contriution to the relted res noted ove mong others. r 008 Elsevier Ltd. All rights reserved. Keywords: Plte elements; Deflected pltes; Buckled pltes; Plte effectiveness; Effective redth; Effective width; Effective tngent modulus; Effective sher modulus 1. Bckground to the concepts of plte-effectiveness evlution Plte elements in continuous plte structure cn deflect for vrious resons: welding used for ttching stiffeners to the pltes cn induce initil deflection lterl pressure loding cn cuse lterl deflection nd xil compression or edge sher ctions cn give rise to uckling tht susequently results in deflection. The memrne stress distriution inside deflected plte is not uniform nymore in contrst to flt or undeflected plte. The ehvior of plte with lterl deflection exhiits geometricl nonlinerity tht is distinct from Hook s lw the reltionship etween memrne stress nd strin is liner. It is cler with certinty tht much more efficient procedure of computtions cn e developed for nonliner structurl ehvior ssocited with geometricl nonlinerity if the ehvior of deflected plte cn e treted s liner prolem even fter the inception of plte deflections. This issue could e resolved y dopting the concept of n equivlent flt plte i.e. without lterl deflection ut with reduced plte effectiveness. This mens tht the Tel.: ; fx: E-mil ddress: jeompik@pusn.c.kr deflected plte is virtully delt with s n undeflected or flt plte ut the plte effectiveness hs een reduced. Since the oject plte is now flt plte i.e. without lterl deflection the memrne stress distriution inside the plte must e uniform nd thus Hook s lw is pplicle so tht the computtionl procedure for the plte ehvior cn ecome much simpler. In terms of pplying the equivlent flt plte concept the fundmentl question is how to evlute nd identify the reduction of plte effectiveness in the deflected plte. It is redily understood tht the plte effectiveness will e function of lterl deflection mening tht the plte effectiveness could e progressively vried (decresed) s the lterl deflection increses. Since the lterl deflection is function of pplied ctions (lods) the plte effectiveness must essentilly e function of pplied ctions. When norml stress is predominnt inside the deflected plte deflected plte is virtully modeled s n equivlent flt plte ut with reduced plte redth. There re two different terms to mke distinction etween the types of ctions in the evlution of plte effectiveness [1]. The term effective redth or effective flnge width is typiclly used when the lterl deflection is cused y out-of-plne or lterl ctions such s lterl pressure lods in ssocition with sher-lg effect. The term effective width is typiclly used when the deflection occurs y uckling under /$ - see front mtter r 008 Elsevier Ltd. All rights reserved. doi: /j.tws

2 1036 J.K. Pik / Thin-Wlled Structures 46 (008) predominntly xil compressive lods. Therefore the evlution of plte effectiveness corresponds to the identifiction of plte redth or plte width for plte elements deflected y ctions tht cuse norml stresses. On the other hnd plte uckled y predominntly edge sher ctions exhiits nonliner distriution of sher stress inside the plte. In this cse the deflected plte cn e modeled s n equivlent flt plte ut with reduced sher modulus of the plte []. Therefore the evlution of plte effectiveness now corresponds to the identifiction of the effective sher modulus for plte elements uckled y edge sher. The prolem of the effective width for steel plting in compression ws initilly rised y John [3] nvl rchitect who investigted the strength of ship which hd roken into two pieces during hevy wether presumly s result of high stress induced y sgging moment. He pointed out tht the light plting of the deck nd topsides could not e considered s fully effective under compression. To ccount for this effect in the clcultion of section modulus of the ship he reduced the thickness of the plting keeping the stress (which could e clculted without considering uckling) unchnged. A pioneer of using n nlyticl pproch for the plte effective width is Bortsch [4] who employed n pproximte formul of the effective width for the prcticl prolems relting to ridge engineering. The modern er in the effective width concept ws strted y von Krmn [5] who developed generl method to solve the prolem theoreticlly nd introduced for the first time the term effective width. He clculted the stress distriution of twodimensionl prolems using the stress function pproch to evlute the effective width. A remrkle dvncement of the Krmn method ws chieved y Metzer [6] who studied the effective flnge width or effective redth of simple ems nd continuous ems. In the 1930s lrge series of compression tests on steel pltes were undertken y Schumn nd Bck [7]. Bsed on the test results they noted tht the uckled steel plte my ehve s if only prt of its width is effective in crrying lods. By pplying the effective width concept this phenomenon ws investigted theoreticlly y von Krmn et l. [8] who otined the first effective width expression of plting p which ws lter shown to e equivlent to e = ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi s cr =s Y is the full plte width e the effective plte width s cr the plte uckling stress nd s Y the mteril yield stress. Since the modern er of the effective width concept ws opened y von Krmn et l. [8] the concept hs een recognized s n efficient nd ccurte pproch to compute the post-uckling ehvior of plte in compression. Prof. Rhodes hs provided gret contriution to the res on effectiveness evlution of thin pltes fter uckling mong other res. His ppers [9 11] extensively reviewed the history of plte post-uckling nlysis nd gve expressions for the effective widths of pltes under vrious oundry conditions nd lods ner uckling. A comprison with vrious design formule of the effective widths ws lso mde in his ppers. Since then the concepts of plte-effectiveness evlution hve in fct een dvnced to lrge extent nd widely pplied to strength ssessment of thin-wlled structures such s ships offshore structures lnd-sed structures nd erospce structures to nme few. Plte elements in thin-wlled structures re often sujected to comined lods such s ixil compressive lods sher nd lterl pressure lods nd thus vrious lod components in type nd mgnitude should e delt with s prmeters of influence in the evlution of plte effectiveness. Friction relted initil imperfections in the form of initil deflection nd welding residul stresses tht lwys develop in plte elements of welded structures will lso ffect the plte effectiveness. In recent yers the ultimte limit stte pproch hs een more extensively pplied for structurl design nd strength ssessment [1]. This is ecuse the ultimte limit stte is much etter sis for design nd strength ssessment thn the llowle working stress nd lso it is not possile to determine true mrgin of sfety s long s the ultimte limit stte remins unknown. In this regrd it is lso importnt to evlute the effectiveness of plte elements t ultimte limit stte. The im of the present pper is to survey some recent dvnces in the concepts of plte-effectiveness evlution. While the present pper will focus on steel pltes simply supported t ll (four) edges tht re well dopted for idelizing plte elements in continuous stiffened plte structures for mrine pplictions it will ddress the concepts nd ssocited expressions of plte effectiveness s functions of comined lod components nd friction relted initil imperfections. A concept of the effective sher modulus is lso presented which is useful for nlysis of the post-uckling ehvior of pltes uckled in edge sher. For the effects of plte edge conditions the reders my refer to the ppers of Rhodes [9 11].. The concept of effective redth When lterl pressure ctions re pplied stiffened plte structure s shown in Fig. 1 is often idelized y plte stiffener comintion model s representtive s shown in Fig.. It is importnt to relize tht the ttched plting of the plte em comintion model does not work seprtely from the djcent memers nd it is restricted from deforming sidewys while the stiffener flnge my e free to deflect verticlly nd sidewys. When stiffened plte structure is idelized s n ssemly of plte em comintion elements therefore one of the primry questions is to wht degree nd extent the ttched plting reinforces the ssocited strut-we. Troitsky [1] reviewed vrious methods to derive nlyticl formultions of the sher-lg oriented effective redth for wide flnged ems (plte em comintions). In the present pper n nlyticl formultion of the effective

3 J.K. Pik / Thin-Wlled Structures 46 (008) B L Fig. 1. A continuous stiffened plte structure. y 0 Fig.. A plte stiffener comintion model. σ x σ x mx redth for plte em comintion under predominntly sher lg or ending on wide flnge [1] is presented. Following the coordinte illustrted in Fig. 3 the effective redth (or width) e cn e given y R = = e ¼ s x dy (1) s x mx s x is the non-uniform memrne stress nd s x mx the mximum memrne stress t plte/we junctions. It is evident from Eq. (1) tht the plte effective redth cn e defined once the non-uniform memrne stress distriution is known. The clssicl theory of elsticity [13] cn e pplied to compute the distriution of memrne stress s x s follows [1]: s x ¼ p py py C 1 sinh o o o þðc 1 þ C Þ cosh py o e e Fig. 3. Effective redth or width of the ttched plting in plte em comintion model. sin px o () x C 1 ¼ C 3 sinh p o C ¼ C 3 1 n 1 þ n sinh p o p p cosh o o o 3 n C 3 ¼ E 0 sinh p 1 p o ð1 þ nþp o o is the deflection wve-length depending on rigidities of the stiffener nd the type of lod ppliction tht is often tken s o ¼ L for stiff trnsverse frmes n Poisson s rtio 0 ¼ u 0 ðp=oþ u 0 the mplitude of the xil displcement function. By sustituting Eq. () into Eq. (1) the effective redth e cn e clculted s follows: 4o sinh ðp=oþ e ¼ pð1 þ nþ½ð3 nþ sinhðp=oþ ð1þnþðp=oþš. (3) The effective redth normlly vries long the spn of plte em comintion ut for prcticl design purposes it my e tken to hve the smllest vlue which occurs t the loction the mximum longitudinl stress develops. Since e must e smller thn Eq. (3) my e pproximted to e 8 e ¼ >< 1:0 for o p0:18; 0:18 >: ð=lþ for o 40:18: Fig. 4 plots Eq. (4) (pproximte formul) y comprison with Eq. (3) (exct solution). It is considered tht Eq. (4) is ccurte enough for prcticl design purpose. 3. The concept of effective width After uckling under xil compressive ctions the memrne stress distriution inside the uckled plte is non-uniform. Fig. 5 shows typicl exmple of the xil memrne stress distriution inside plte under predominntly longitudinl compressive loding efore nd fter uckling occurs. It is seen tht the memrne stress distriution in the loding (x) direction cn ecome non-uniform s the plte (4)

4 1038 J.K. Pik / Thin-Wlled Structures 46 (008) deflects. It is interesting to note tht the memrne stress distriution in the y-direction lso ecomes non-uniform s long s the unloded plte edges remin stright while no memrne stresses will develop in the y-direction if the unloded plte edges move freely in plne. It is noted tht the condition of unloded edges of plte elements in stiffened plte structure is more likely to remin stright. The mximum compressive memrne stresses re developed round the plte edges tht remin stright while the minimum memrne stresses occur in the middle of the plte memrne tension field is formed y the plte deflection since the plte edges remin stright. For the nlysis of plte post-uckling ehvior under xil compression there re three different spects regrding the term plte effective width nmely the effective width for strength the effective width for stiffness nd the effective tngent modulus. e / Exct solution Approximte formul /L Fig. 4. Vrition of the effective redth versus the rtio of stiffener spcing to the em spn when o ¼ L Effective width for strength Immeditely fter uckling of perfect plte under xil compression the mximum stress ecomes lrger thn the verge stress. It my e pprent in this cse tht the rtio of effective width to full width is the sme s the rtio of the verge stress to the mximum stress s follows: R = = e ¼ s x dy ¼ s xv (5) s x mx s x mx s xv is the verge stress. The ultimte strength of plte pproximtely corresponds to the pplied lod t which the mximum memrne stress reches the mteril yield stress. Since the effective width in terms of the mximum memrne stress is useful in predicting the ultimte strength of plte it is termed the effective width for strength. Although the originl von Krmn p effective width expression of pltes i.e. e = ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi s cr =s Y is considered resonly ccurte for reltively thin pltes it is found to e optimistic for reltively thick pltes with initil imperfections. In this regrd Winter [14] modified the von Krmn eqution s follows: rffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffi e ¼ s cr s cr 1 C (6) s mx s mx s mx is the pplied mximum (edge) stress which my e tken s s mx ¼ s Y s Y the mteril yield stress nd C the constnt tht is often tken s 0.5 ut it is lso tken s 0.18 or 0. in some design codes. One of the most typicl effective width expressions for compressive strength of long pltes which re often employed y merchnt ship clssifiction societies is given in the following form: 8 e ¼ >< 1:0 for o1; C 1 C >: for X1; C 1 C re the constnts pffiffiffiffiffiffiffiffiffiffiffiffi depending on the plte oundry conditions ¼ð=tÞ s Y =E the plte redth (or stiffener spcing) t the plte thickness nd E the elstic modulus. Bsed on the nlysis of ville experimentl (7) y σ xmx y y σ xv σ xmin σ = 1 σ = 1 xv xv σ x dy σ x dy 0 0 x σ xv σ xmx x σ xmin σ xv σ = 1 xv σ x dy 0 σ ymin σ ymx x Fig. 5. Memrne stress distriution inside the plte under predominntly longitudinl compressive lods: () efore uckling () fter uckling with unloded edges moving freely in plte nd (c) fter uckling with unloded edges remining stright.

5 dt for steel pltes with initil deflections t moderte level ut without residul stresses Fulkner [15] proposes C 1 ¼.0 nd C ¼ 1.0 for pltes simply supported t ll (four) edges or C 1 ¼.5 nd C ¼ 1.5 for pltes clmped t ll edges. It is importnt to relize tht Eqs. (6) nd (7) re in fct likely to e the effective widths when the plte just reched the ultimte limit stte. In other words they do not descrie the vrition of the plte effectiveness s the externl ctions increse lthough the plte effective width must vry with increse in the pplied ctions. Under xil compressive lods perfect plte i.e. without initil deflection cn uckle when the verge stress s xv reches the uckling stress s xe tht is given y s xe ¼ p D m t þ (8) m D ¼ Et 3 =½1ð1 n ÞŠ nd m is the uckling hlf wve numer tht is n integer stisfying the following condition: p p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mðm þ 1Þ. (9) The mximum stress s x mx of the plte fter uckling cn e clculted y [1] s x mx ¼ 1 s xv þ (10) m 4 1 ¼ 1 þ 4 ðm 4 = 4 þ 1= 4 Þ m p D ¼ ðm 4 = 4 þ 1= 4 Þ t m þ 1. Therefore the effective width cn e determined from Eq. (5) together with Eq. (10). It is lso interesting to note tht steel plte elements used for mrine structures nd lnd-sed structures typiclly hve initil deflections induced y welding during friction process. In this cse the mximum stress s x mx of the imperfect plte under xil compression is given y [1] s x mx ¼ s xv m p EA m ða m þ A om Þ 8 (11) A m is the mplitude of dded deflection of the plte A om the mplitude of initil deflection corresponding to uckling mode m s defined in Eq. (9). The mplitude A m of dded deflection of the plte cn e otined s solution of the following third-order eqution nmely C 1 A 3 m þ C A m þ C 3A m þ C 4 ¼ 0 (1) C 1 ¼ p E 16 m 4 3 þ 3 ; C ¼ 3p EA om 16 m 4 3 þ 3 J.K. Pik / Thin-Wlled Structures 46 (008) C 3 ¼ p EA om 8 þ p D t m 4 3 þ 3 m þ m m m C 4 ¼ A om s xv. þ m s xv Fig. 6 shows the vrition of the plte effective width s function of xil compressive lods nd initil deflection. The effective widths t the ultimte strength given y Eq. (7) for simply supported pltes re lso compred in Fig. 6 denoted y the Fulkner formul for different plte slenderness rtio or plte thickness. It is seen from Fig. 6 tht the plte effective width decreses s the xil compressive lods increse. When very little initil deflection exists the plte effective width ruptly decreses immeditely fter the inception of uckling. However the effective width for the imperfect plte with lrge initil deflection decreses from the very eginning with increse in the xil compressive lods. As would e expected the plte effective width t the ultimte strength must e different depending on the level of initil deflection. However Eq. (7) does not tke into ccount the effect of initil deflection s prmeter of influence. Pik nd Thymlli [1] presented the formultions of the effective width of plte elements under comined ixil compression nd lterl pressure lods together with initil imperfections in the form of initil deflection nd welding residul stress. Fig. 7 shows the vritions of the plte effective width s function of initil deflection residul stress xil compression nd lterl pressure. In this figure s rcx nd s rcy re compressive residul stresses in the plte length nd redth directions respectively. e Fulkner formul with β = 1.5 Fulkner formul with β = 3.0 :β = 1.5 :β = 3.0 :Ultimte strength = σ rcx = σ rcy = Fulkner formul t ultimte strength: σ π = 4 D σy xe β = t e 1 t E = β β σ Y= 313.6MP A om /t = 5β A om /t = 0.1β A om /t = 0.3β σ xv /σ xe Fig. 6. Vrition of the plte effective width s function of xil compressive lods together with initil deflection.

6 1040 J.K. Pik / Thin-Wlled Structures 46 (008) e Fulkner formul with β = 3.0 Fulkner formul t ultimte strength: e 1 = β β σ xv /σ xe Fulkner formul with β = 1.5 A om /t = 5β A om /t = 0.1β A om /t = 0.3β : β = 1.5 : β = 3.0 : Ultimte strength = σ rcx /σ Y = 0.1 σ rcy = σ π D σ xe = 4 β = Y t t E σ Y = 313.6MP of xil strin long the plte/we junctions i.e. e xv ¼ e x t y ¼ 7/ nd u is the end displcement. In this cse the effective width cn lso e evluted from Eq. (5) ut replcing s x mx y the xil strin of Eq. (13). An effective width for stiffness i.e. sed on the verge xil strin my e used to chrcterize the overll stiffness of uckled plte under predominntly xil compression. The closed-form expression of the effective width for stiffness representing the in-plne effectiveness of the uckled plte cn e derived y e ¼ e qs 1 x mx (14) qs xv is the effective width for stiffness. For perfect plte i.e. without initil deflection the mximum stress cn e clculted from Eq. (10). In this cse the effective width for stiffness cn e otined s follows: e Wter hed = 0 m 3.. Effective width for stiffness Fulkner formul with β = 1.5 Wter hed = 0 m Wter hed = 40 m Fulkner formul with β = 3.0 : β = 1.5 : β = 3.0 : Ultimte strength Wter hed = 10 m = Wter hed = 0 m σ Fulkner formul t rcx /σ Y = 0.1 σ rcy = ultimte strength: σ π D σ xe = 4 e 1 β = Y t t E = β β σ Y = 313.6MP A om /t = 0.1β σ xv /σ xe Fig. 7. () Vrition of the plte effective width s function of xil compressive lods together with initil deflection nd welding residul stress. () Vrition of the plte effective width s function of xil compressive lods together with initil deflection welding residul stress nd lterl pressure lods. The tendency of incresing the verge strin with the verge stress is of course greter fter uckling thn tht efore uckling. As long s the plte/we junction remins stright the verge vlue of the mximum memrne stress long the plte edges my e otined for unixilly compressed plte s follows: s x mx ¼ E xv ¼ E u L (13) e xv is the verge xil strin of the ttched plting which my pproximtely e tken s the verge vlue e ¼ 1. (15) 1 The effective width for stiffness cn lso e expressed s function of verge strin. The reltionship etween the mximum stress nd verge strin is given s long s the unloded edges remin stright nmely s x mx ¼ E xv (16) or considering Eq. (10) together with Eq. (13) the following eqution is otined: 1 s xv þ ¼ E xv (16) e ¼ s xv ¼ (16c) s x mx 1 E xv Fig. 8 shows the vrition of the plte in-plne stiffness in terms of the reltionship etween verge stress nd verge strin. Immeditely fter uckling the plte inplne stiffness decreses significntly. The stiffness reduction tends to depend on the plte spect rtio Effective tngent modulus The plte stiffness ginst xil compression is reduced immeditely fter uckling. While this ehvior my e chrcterized y the effective width for stiffness it is sometimes of interest to know the mgnitude of the tngent stiffness or the slope of the verge stress strin curve fter uckling which cn mthemticlly e computed y qs xv =q xv in the post-uckling regime. The tngent stiffness fter uckling is termed the effective tngent modulus or the effective Young s modulus E. Using this formultion the rtio of the compressive stiffness fter uckling to tht efore uckling is given y E =E. For perfect plte simply supported t four edges it is known tht E =E 0:5 fter uckling. As long s unloded edges remin stright so tht some trnsverse stresses re

7 J.K. Pik / Thin-Wlled Structures 46 (008) / = 3.4 / = / =.5 E * /E 0.6 σ xv σ xe developed long unloded edges it is recognized tht E =E corresponds to qs xv =qs x mx while the former is lwys greter thn the ltter when unloded edges re free to move in plne susequent to no stresses long them [9]. For plte under xil compression Eq. (16) cn e rewritten s follows: xv ¼ 1 E s x mx ¼ 1 s xv or s xv ¼ e E e E xv. (17) The incrementl form of Eq. (17) is given y D xv ¼ 1 qs x mx Ds xv or E qs xv Ds xv ¼ qs 1 x mx ED xv (18) qs xv the prefix D represents the increment of the vrile. The numericl pproch is often more pertinent for computtion of qs x mx =qs xv with infinitesiml stress vritions round s xv. For perfect plte i.e. without initil deflection the following equtions re then otined nmely s xv ¼ 1 1 ðe xv Þ Ds xv ¼ E 1 D xv ¼ E D xv ε xv ε E Fig. 8. The verge stress versus strin reltionships for perfect plte under unixil compression in the elstic regime (e xe ¼ verge xil strin t s xv ¼ s xe ). " # E ¼ E m 4 ¼ E 1 þ 1 m 4 þ 4 = 4 ¼ effective Young s modulus ðtngent modulusþ fter uckling. (19) (19) 0.3 It is evident from Eq. (19) tht the tngent modulus of the uckled plte does not chnge with the pplied lods while it is function of the plte spect rtio. Fig. 9 shows the vrition of the tngent modulus of the uckled plte s function of the plte spect rtio. It is seen tht the effective tngent modulus vries in cyclic pttern with regrd to men equl to E =E ¼ 0:5 nd for shorter pltes the effect of the spect rtio is more significnt. 4. Effective sher modulus While the concept of effective width is imed t the evlution of in-plne stiffness of plte elements uckled in compression Pik [] suggested new concept of the effective sher modulus to evlute the effectiveness of plte elements uckled in edge sher. The effective sher modulus concept is useful for computtion of the postuckling ehvior of plte girders under predominntly sher forces. The sher strin distriution in plte element is not uniform fter sher uckling. By tking into ccount the lrge deflection effect the sher strin g in plte element uckled y edge sher cn e clculted s follows: g ¼ 0 qu qy þ qv qx / Fig. 9. Vrition of the reduced tngent modulus fter uckling s function of the plte spect rtio. þ qw qx qw qy þ qw qw 0 qx qy þ qw 0 qx qw qy (0) u v is the xil displcements in the x- nd y-directions respectively w the dded deflection w 0 the initil deflection. The first rcketed term on the righthnd side of the ove eqution represents the memrne sher strin component nd the second term indictes the dditionl sher strin component due to lrge deflection effects. The sic ide of either the effective width or the effective sher modulus concepts is to model the deflected plte s n equivlent flt (undeflected) plte ut with reduced (effective) in-plne stiffness. In this regrd the

8 104 J.K. Pik / Thin-Wlled Structures 46 (008) memrne sher strin component g m of the uckled plte must e defined s follows: g m ¼ qu qy þ qv qx ¼ t qw qw G qx qy þ qw qw 0 qx qy þ qw 0 qw (1) qx qy G ¼ E/[(1+n)] ¼ sher modulus. The memrne sher strin t ny point inside the ucked plte cn e computed seprtely using refined methods such s semi-nlyticl methods or finite element methods (FEMs). Once the memrne strin distriution inside the uckled plte is computed the men memrne sher strin g v cn e otined s n verge of sher strins over the entire plte s follows: g v ¼ 1 Z Z g m dx dy. () 0 0 As long s the plte edges remin stright the verge sher stress my equl to the sher stress t the plte edges i.e. t ¼ t v. Therefore the effective sher modulus G e representing the effectiveness of the plte uckled in edge sher cn e defined y G e ¼ t v. (3) g v An empiricl expression of the effective sher modulus for plte elements uckled y edge sher hs een developed y curve fitting sed on numericl computtions vrying influentil fctors such s the plte spect rtio nd initil imperfections s follows: ( G e G ¼ c 1V 3 þ c V þ c 3 V þ c 4 for Vp1:0; d 1 V (4) þ d V þ d 3 for V41:0; c 1 ¼ 0:309W 3 0 þ 0:590W 0 0:86W 0 c ¼ 0:353W 3 0 0:644W 0 þ 0:70W 0 c 3 ¼ 0:07W 3 0 þ 0:134W 0 0:059W 0 c 4 ¼ 0:005W 3 0 0:033W 0 þ 0:001W 0 þ 1:0 k t 4 þ 5:34 for X1 k t 5:34 þ 4:0 for o1. When the plte hs no initil deflection Eq. (4) is simplified to 8 t v 1:0 for p1; G >< t E e G ¼ >: 0:015 t v 0:118 t v þ 1:103 t E t E for t v t E 41: (6) Fig. 10 plots the vrition of the effective sher modulus of plte uckled y edge sher with vrying the edge sher stress nd initil deflection. It is seen tht the effective sher modulus of perfect plte i.e. without initil deflection decreses ruptly fter the inception of sher uckling. Also the initil deflection reduces the effective sher modulus of the plte element under edge sher. 5. Reltionships etween memrne stresses versus strins For the nlysis of nonliner ehvior for deflected pltes it is required to identify the reltionship etween memrne stresses nd strins [116]. The formultion of such reltionship in the pre-ultimte strength regime is distinct from tht in the post-ultimte strength regime Pre-ultimte strength regime The memrne strin components of deflected or uckled plte elements under comined ixil lods edge 1.04 d 1 ¼ 0:007W 3 0 þ 0:015W 0 0:018W 0 þ 0:015 d ¼ 0:0W 3 0 þ 0:006W 0 þ 0:075W 0 0: W o = 0 d 3 ¼ 0:008W 3 0 þ 0:05W 0 0:130W 0 þ 1:103 G e /G 0.96 W o = 0. V ¼ t v t E 0.9 W o = 0.5 W 0 ¼ w 0pl. t w 0pl is the mximum initil deflection nd t E the elstic sher uckling stress of the plte which is given y p E t t E ¼ k t (5) 1ð1 n Þ τ v /τ E Fig. 10. Vrition of the effective sher modulus of plte with increse in edge sher.

9 J.K. Pik / Thin-Wlled Structures 46 (008) sher nd lterl pressure cn e given y xv ¼ 1 E ðs x mx ns yv Þ yv ¼ 1 E ð ns xv þ s y mx Þ (7) (7) g v ¼ t v (7c) G e s x mx s y mx re the mximum memrne stresses in the x- ory-direction G e the effective sher modulus s defined in Eq. (4). Since s x mx s y mx nd G e re nonliner functions with regrd to the corresponding verge stress components Eq. (7) indictes set of nonliner reltionships etween memrne stresses nd strins. The incrementl form of the memrne stress strin reltionship is relevnt y differentiting Eq. (7) with regrd to the corresponding verge stress components s follows: D xv ¼ 1 qs x mx Ds xv þ qs x mx n Ds yv (8) E qs xv qs yv D yv ¼ 1 qs y mx n Ds xv þ qs y mx Ds yv (8) E qs xv qs yv Dg v ¼ 1 1 t v qg e Dt v. (8c) G e G e qt v The mtrix form of Eq. (8) is given y >< Ds xv D >= >< xv >= Ds yv >: >; ¼½DŠB D yv >: >; (9) Dt v Dg v 3 B A 0 ½DŠ B 1 ¼ 6 B 1 A A 1 B A B ¼ stressstrin 0 0 1=C 1 mtrix of the plte in the post-uckling regime; with A 1 ¼ 1 qs x mx E qs xv A ¼ 1 qs x mx n E qs yv B 1 ¼ 1 qs y mx n E qs xv B ¼ 1 qs y mx E qs yv C 1 ¼ 1 1 t v qg e. G e G e qt v When no uckling hs occurred in the perfect plte element i.e. without initil imperfections [D] B mtrix in Eq. (9) will of course ecome 3 1 n 0 ½DŠ B ¼ E n n 4 1 n 5. (30) Post-ultimte strength regime In the post-ultimte regime the internl stress will decrese s long s the xil compressive displcements continully increse. In this cse the verge memrne stress components my e clculted in terms of the plte effective width or length s follows: s xv ¼ e su x mx (31) s yv ¼ e su y mx (31) e is the effective width e the effective length nd s u x mx su y mx the mximum memrne stresses of the plte in the x- ory-directions immeditely fter the ultimte strength is reched i.e. s u x mx ¼ s x mx t s xv ¼ s xu the ultimte compressive strength in the x-direction or s u y mx ¼ s y mx t s yv ¼ s yu the ultimte compressive strength in the y-direction. The effective width or length of the plte in the postultimte strength regime my e defined s follows: e ¼ s xv s (3) x mx e ¼ s yv s (3) y mx the sterisk represents vlue of the plte in the post-ultimte regime. The effects of initil imperfections nd Poisson s rtio re negligily smll in the post-ultimte regime. While the plte effective width will of course decrese in the postultimte regime s long s the xil compressive displcements increse it is ssumed tht the reduction tendency of the plte effective width or length is similr to tht in the pre-ultimte regime. In this cse s x mx nd s y mx in Eq. (3) cn e determined y s x mx ¼ E xv ¼ s xv s xe (33) s y mx ¼ E yv ¼ s yv s ye (33) s xe nd s ye re the elstic compressive uckling stresses in the x- or y-direction respectively. By sustituting Eq. (33) into Eq. (3) the plte effective width or length cn e expressed in terms of strin

10 1044 J.K. Pik / Thin-Wlled Structures 46 (008) ` t ~ ` Fig. 11. Modeling of the plte y ALPS/GENERAL (left) nd ANSYS (right). components s follows: e ¼ 1 1 þ s xe E xv (34) e ¼ 1 1 þ s ye. E yv (34) The verge stress strin reltionships in the postultimte regime cn then e derived y sustituting Eq. (34) into Eq. (31) s follows: s xv ¼ 1 1 þ s xe s u x mx E xv (35) s yv ¼ 1 1 þ s ye s u y mx E yv (35) The incrementl form of Eq. (35) is then given y Ds xv ¼ su x mx s xe E xv s ycr D xv (36) Ds yv ¼ su y mx E D yv. (36) yv On the other hnd the sher stress strin reltionship in the post-ultimte regime is given y Dt v ¼ G e Dg v (36c) G e is the tngent sher modulus in the post-ultimte regime which is often supposed to e G e ¼ 0 when the unloding ehvior due to sher is not very significnt. In comined lod cses the verge stress strin reltionship of the plte in the post-ultimte regime is therefore given from ll together with Eq. (36) s follows: >< Ds xv D >= >< xv >= Ds yv >: >; ¼½D pš U D yv >: >; (37) Dt v Dg v 3 A ½D p Š U ¼ 6 0 A ¼ stressstrin 0 0 A 3 mtrix of the plte in the post-ultimte regime; with A 1 ¼ su x mx A ¼ su y mx A 3 ¼ G e. s xcr E xv s ycr E yv 5.3. An illustrtive exmple An imperfect rectngulr plte under unixil compressive ctions is considered. The dimension of the plte is = (mm) Young s modulus (E)= N/mm yield stress (s Y )=35.8 N/mm nd Poisson s rtio (n)=0.3 while plte thickness (t) is vried. Initil deflection of the plte is w 0 ¼ 0:05t sin px sin py nd no residul stress is considered to exist. The plte is considered to e simply supported long ll (four) edges keeping stright. The reltionship etween verge stresses nd strins ws implemented into ALPS/GENERAL [17] tht is computer progrm for progressive collpse nlysis of plted structures using idelized structurl unit method (ISUM) [116]. Fig. 11 shows the nlysis models y ALPS/ GENERAL nd nonliner FEM [18]. For the ANSYS nonliner FEM nlysis qurter of the plte is tken s the extent of the nlysis. Fig. 1 compres the progressive collpse ehvior of the plte under xil compressive ctions indicting tht the ISUM solutions re in good greement with more refined FEM results. 6. Concluding remrks It hs een recognized tht the concepts of idelizing deflected pltes s equivlent flt (undeflected) pltes ut with reduced plte effectiveness re very useful for the nlysis of nonliner ehvior of pltes deflected y lterl loding or uckling. In this cse key issue is how to identify the effectiveness of the deflected pltes. A numer

11 J.K. Pik / Thin-Wlled Structures 46 (008) σ x /σ Y t = 0mm t = 15mm t = 5mm : ISUM (ALPS/GENERAL) : FEM (ANSYS) = (mm) p = E = 05.8GP ν = 0.3σ = 355MP Y w opl /t = 5 ε = σ Y Y /E t ISUM FEM ISUM/FEA /t σ Y /E ε x /ε Y t = 5mm Bsed on the insights otined from the present study the following conclusions cn e drwn: (1) The term effective redth presents the plte effectiveness ssocited with sher-lg effect. () The term effective width presents the plte effective width ssocited with uckling in compressive ctions. (3) The term effective length is defined for plte uckled in trnsverse compression while the term effective width is defined for plte uckling in longitudinl compression. (4) Effective width or length is function of vrious prmeters of influence such s pplied lods nd initil imperfections (initil deflection welding residul stress). (5) The concept of effective sher modulus is useful for the effectiveness evlution of plte uckled in edge sher. (6) The reltionships etween memrne stresses versus memrne strins for deflected pltes in pre- nd postultimte strength regimes could e derived in closed form using the concepts of effective width nd effective sher modulus. σ y /σ Y t = 0mm t =15mm : ISUM (ALPS/GENERAL) : FEM (ANSYS) = (mm) p= E=05.8GP ν=0.3σ Y =355MP w opl /t=5 ε Y =σ Y /E t ISUM FEM ISUM/FEA /t σ Y /E ε y /ε Y Fig. 1. A comprison of the progressive plte collpse ehvior under unixil compression in the x-direction otined y ISUM nd FEM. () A comprison of the progressive plte collpse ehvior under unixil compression in the y-direction otined y ISUM nd FEM. of terms such s effective redth effective width nd effective sher modulus hve een suggested in the literture for the evlution of strength nd stiffness together with their closed-form expressions. In the present pper some recent dvnces in the plteeffectiveness evlution hve een surveyed. Some useful expressions of effective redth ssocited with sher-lg effect nd effective width ssocited with uckling were presented. A new concept termed effective sher modulus is presented to evlute the post-uckling ehvior of plted uckled in edge sher. The reltionships etween memrne stresses versus memrne strins re presented in the pre- nd post-ultimte strength regimes. Acknowledgments The present work ws undertken t the Ship nd Offshore Structurl Mechnics Lortory Pusn Ntionl University which is Ntionl Reserch Lortory funded y the Kore Science nd Engineering Foundtion (Grnt no. RoA ). The uthor is plesed to cknowledge the support of Lloyd s Register Eductionl Trust through the Reserch Centre of Excellence t PNU. References [1] Pik JK Thymlli AK. Ultimte limit stte design of steel-plted structures. Chichester UK: Wiley; 003. [] Pik JK. A new concept of the effective sher modulus for plte uckled in sher. J Ship Res 1995;39(1):70 5. [3] John W. On the strins of iron ships. Trns R Inst Nvl Archit 1877;18: [4] Bortsch R. Die mitwirkende plttenreite. Der Buing 191;3:66 7 [in Germn]. [5] von Krmn T. Die Mittrgende reite. Beitrge zur Technischen Mechnik und Technischen Physik August Foppl Festschrift. Berlin: Julius Springer; 194. p [in Germn]. [6] Metzer W. Die Mittrgende reite. Disserttion der Technischen Hochschule zu Ache. Munchen: R. Oldenurg 199. p. 1 1 [in Germn]. [7] Schumn L Bck G. Strength of rectngulr flt pltes under edge compression. NACA Technicl Report No. 356 Ntionl Advisory Committee for Aeronutics Wshington DC [8] von Krmn T Sechler EE Donnell LH. Strength of thin pltes in compression. Trns ASME 193;54(5):53 7. [9] Rhodes J. Effective widths in plte uckling. In: Developments in thin-wlled structures. London: Applied Science Pulisher; 198. p [chpter 4]. [10] Rhodes J. Buckling of thin pltes nd memers nd erly work on rectngulr tues. Thin-Wlled Struct 00;40:

12 1046 J.K. Pik / Thin-Wlled Structures 46 (008) [11] Rhodes J. Some oservtions on the post-uckling ehvior of thin pltes nd thin-wlled memers. Thin-Wlled Struct 003;41: [1] Troitsky MS. Stiffened pltes: ending stility nd virtions. Amsterdm: Elsevier Scientific Pulishing Compny; [13] Timoshenko SP Gere JM. Theory of elstic stility. nd ed. New York: McGrw-Hill; [14] Winter G. Strength of thin steel compression flnges. Reprint No. 3 Engineering Experimentl Sttion Cornell University New York [15] Fulkner D. A review of effective plting for use in the nlysis of stiffened plting in ending nd compression. J Ship Res 1975;1(1):1 17. [16] Pik JK Thymlli AK. Ship-shped offshore instlltions: design uilding nd opertion. Cmridge UK: Cmridge University Press; 007. [17] ALPS/GENERAL. A computer progrm for progressive collpse nlysis of generl types of plted structures Ship nd Offshore Structurl Mechnics L. Pusn Ntionl University Kore. [18] ANSYS. User s mnul (version 1). Cnonsurg PA: ANSYS Inc. 006.