Buckling of Stiffened Panels 1 overall buckling vs plate buckling PCCB Panel Collapse Combined Buckling

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1 Buckling of Stiffened Pnels overll uckling vs plte uckling PCCB Pnel Collpse Comined Buckling Vrious estimtes hve een developed to determine the minimum size stiffener to insure the plte uckles while the stiffener remins stright. this is equivlent to insuring tht plte uckling occurs efore overll uckling. Timoshenko does so y clculting k in σ cr k π D nd oserving the t vlue of γ which results in criticl stress ove tht which will cuse plte uckling lone. where; flexurl_rigidity_of_comined_section EI x γ γ where; flexurl_rigidity_of_plte D I x is the inerti of the plte with the ttched plte ssocited with individul stiffener. Bleich pg 365, 367, for pltes with longitudinl stiffness determines minimum γ to insure the plte uckles efore the stiffener (overll uckling). unfortuntely Bleich uses different rtios thn Hughes. Bleich uses B where Hughes uses. for the following: A x γ B Bt EI x DB.,.... κ, n γ o,, n κ, n 6 π n, tnh κ ( n) κ, n n n π tn κ, n κ, n + 6 n γ omx, n π mx 4 n (,, ) (,, 3) γ o,, γ o γ o γ o (,, ) 4 γ o (,, ) γ o (,, 3) γ omx (, ) 5 5

2 curve fit: one stiffener, comining Hughes nd Bleich terms, N= stiffener.8 γ x, γ x, min γ x, γ x, γ x, Hughes: 3..4 slightly modified in terms 4 3 γ omx (, ) γ x (, ) Fig. 8, Bleich 5 to move to Bleich reltionships in Hughes terms (rtios) to mtch text: where: A x Π γ B t EI x D one stiffener.8 Π γ x Π, ( +.5 ) γ x Π, Π Π two stiffeners 43.5 Π 3 γ x Π, 88 6 γ x Π, + 36Π Π, γ x Π, γ x Π, γ x Π, γ x Π, Π Π

3 comintion if N γ Π,, N γ x Π, =, min, min γ x ( Π, ) γ x Π, γ x Π, 5 minimum rtio of EI/D (A/t =.) 4 gmm = EI/D pnel spect rtio = /B stiffeners stiffener A more direct pproch is to clculte the overll uckling stress nd insure it is lrger thn the plte criticl stress. The overll uckling stress is the vlue t which the stiffeners rech criticl stress, modeling ech stiffener s column of stiffener with ttched (portion) of plte with some equivlent slenderness rtio. λ L_over_ρ eq We will continue to model the plte filure s grdul filure i.e the center of the plte "fils" in uckling while the outer section remins effective t n effective redth prdoxiclly, the column is "stiffer" when the plte flnge ( ) is reduced for rtios typicl of ship structure: let's first evlute the plte nd column criticl stresses for short pnel. As we ssumed in plte uckling (nd ending) the width is such tht we cn model slice independently: the column is stiffener with n ttched plte of width the plte is width some typicl scntlings: stiffener we A w.8 stiffener flnge A f.85 stiffener re A s A w + A f stiffener depth d 5 pnel length L 96 redth etween stiffeners 3 set ove t =.5 3

4 clculte in terms of, initilly = moment of inerti using to clculte rdius of gyrtion: A s + t A e C A e ( ) A w A w + A f t 3 4 ( A e ( )) I e A e ( ) d C ( ) ρ e ( ) I e A e column criticl stress: σ e_cr ( ) π E ; initil vlue σ e_cr ( ) = 4934 L ρ e ( ) plte criticl stress: σ _cr t initil vlue σ _cr ( ) = E things re ok s column > plte => plte "fils" first. now consider incresing stress eyond σ _cr ( ) nd plte grdully fils reducing effective redth. Note tht we re using n ssumption due to von Krmn, tht the "filed" center region hs no compressive stress while the outer regions re fully effective t σ e defined from force equilirium s σ σ e ( ) now consider wht hppens to the vlues of criticl stress s effective redth is reduced. the definitions ove re still ctive; repeted here for info: 5.. these vlues re not "short" ut with pproprite scntlings the result is the. 5 sme. this set of vlues ssumes yield stress is > 3x^4 or so. criticl stress column: σ e_cr plte: σ _cr 3.65 E π E L ρ e t effective redth column (stiffener with plte) plte lone 4

5 even though the numers ove represent long column (plte), the nlysis is only pproprite for short pnels. to nlyze long pnels correction is needed to λ = L_over_ρ to ccount for the tendency of the plte to deflect with more thn one hlf sine wve mking the column slightly stiffer (smller slenderness rtio λ). Τhe correction is s follows: B C π γ x or C π whichever is less. γ x ( ) + + γ x ( ) EI x ( ) D e nd L_over_ρ eq L = C π then σ ρ( ) e_cr ( ) π E L C π ρ e this dds non linerity to the prolem so n itertive method is used to mke the initil comprison of whether the column criticl stress is greter thn the plte criticl stress or to solve for the common criticl stress. The text proposes n itertive method similr to tht elow for PCCB to determine the common vlue of criticl stress. After doing so it is necessry to ring this stress (sed on effective redth) ck to pplied (verge) stress; gin using sttics σ t + A x = σ e t + A x nd σ redth tht the itertion converges. t + A x σ ( t + A x ) e_cr ( ) t the effective Some clcultions to vlidte the column ecoming "stiffer" with less effective plte. Think of the loction of the neutrl xis or the rdius of gyrtion s the plte is reduced. With wide plte flnge the neutrl xis is close to the plte. The rdius of gyrtion which plys in the criticl stress vries s follows: (recll lrger ρ => lrger criticl stress: ρ e ( ) ρ e.5 σ e_cr ( ) σ e_cr t A s now for our design rules for PCCB Pnel Collpse Comined Buckling: t A s 5

6 - PCCB - Pnel Collpse Comined Buckling function of, grphicl pproch HSF tp input input dt ignore numericl vlues shown here stiffener BSF 3.94 SDEPTH 5. TSF.5 TSW.7 plte try this with SDEPTH = 6, to see yield effect generl prmeters: mteril 8 5 t.5 L σ Y 8 3 L = 96 υ.3 N E D Et 3 υ HSW SDEPTH TSF A w ( SDEPTH TSF) TSW A f BSFTSF A s A w + A f HSW = 4.79 A w =.8 A f =.85 A s =.66 d SDEPTH TSF t + B ( N + ) A p t t A s = 7.53 A s t d = 5.4 B = 75 A p =.5 =.3 prmeters tht re functions of let:.7657 s plce to strt nd for printing purposes. Cn lso use e directly nd compre with r ( ). If r ( )/ < strt use result until converges. A s + t A e C ( ) A e ( ) A w A w + A f t 3 4 I e ( A e ( )) A e ( ) d A e ( ) =.3 C ( ) =.9 I e ( ) = 5.87 ρ e ( ) ρ e ( ) =.5 I e A e γ x ( ) υ γ x ( ) = 8.7 for itertion C r ( ) π t r ρ e ( ) 3( υ ) r ( t) 3 I e ( ) C π min B γ x ( ) ( ) + + γ x C ( ) C π = r =.7657 Checks?? rt rt =.766 = 9.4 6

7 criticl stress reltionships: stiffener & plte s column effective portion of plte originl plte σ ecr ( ) π E C π L ρ e σ ecr_pl 4 π D t σ o 3.6E σ o = 484 t OR... σ o 4 π D t σ o = 484 σ ecr = 735 σ ecr_pl = 738 for plotting to determine intersection;, σ ecr.. 5 σ ecr_pl ( ) σ Y σ o now oseving intersection: rt = 9.4 ssuming itertion complete nd mtched ove. 7

8 stiffener & plte s column σ ecr ( ) σ ecr = 738 prtil sfety fctor π E C π L ρ e effective portion of plte σ ecr_pl σ ecr_pl 4 = 735 π D t trnsltion ck to pplied stress σ xcr t t + A s + A s σ ecr σ xcr = 5793 γ C.5 σ C input is it slender?? λ>?? σ C γr PCCB γ C σ xcr λ πρ e ( ) σ Y E λ =.5 γr PCCB =.58 8