MECHANICS OF STRUCTURAL INSTABILITY IN THIN-WALLED STRUCTURES

Transcription

1 MCHNICS OF STRUCTUR INSTBIITY IN THIN-WD STRUCTURS K F Chung The Hong Kong oltechnic Universit, Kowloon, Hong Kong SR BSTRCT Instilit is n iortnt rnch of structurl echnics which exines lternte equiliriu sttes ssocited with lrge defortion. Structurl instilit in one diensionl e-colu eers nd two diensionl lte eers is exined in this er. Vrious fors of uckling, nel, colu uckling, e uckling nd locl lte uckling, under different loding nd oundr conditions re illustrted to estlish the henoen s well s vrious nlsis nd design ethods for rcticl design. KYWORDS Structurl instilit, colu uckling, e uckling, locl lte uckling. 84

2 COUMN BUCKING Consider colu inned t oth ends [] s shown in Figure. defortion elstic curve of the colu quiliriu lterntive quiliriu M x Free od digr Figure Buckling of colu Tking oent out the inned suort in the free od digr, M x 0 d s M x I, dx d I dx 0 85

3 B writing k I, the ove eqution is rrnged s follows: k dx d 0 The solution of the differentil eqution is given : sin(kx) B cos(kx) C x D ling the oundr conditions t oth inned ends: d 0 when x 0 nd x dx B C D 0, nd sin (k) 0 Hence, k n nd n I. B tking n s iniu, the elstic criticl uckling lod,, is given : I The deflected she for the elstic criticl uckling of the colu is given : x sin ( ) where is constnt with n ritrr vlue. Design ethod using the slenderness of colu xnding the second oent of re, I, s the roduct of re nd the rdius of grtion, r, the elstic criticl uckling lod,, e re-resented s follows: r 86

4 Then the elstic criticl uckling strength,, is given s follows: r Re-writing the colu slenderness, is given s follows: s r, the elstic criticl uckling strength, r This eqution gives the elstic criticl uckling strength of colu in ters of its teril roert,, s well s its geoetricl roert, i.e. the colu slenderness. It is iortnt to evlute the coressive uckling strength of rel colus, c, in the resence of initil echnicl nd geoetricl ierfections, err Roertson interction forul [] is doted s follows: where c where φ φ φ is the design strength of the colu eer; ( η) η is the err fctor; 0.00 ( o ) is the Roertson constnt with ticl vlues such s.0, 3.5, 5.5, 8.0; o is the liiting slenderness; 0. The choice of the vlue of the Roertson constnt deends on the cross-sections of the colus, the xes of uckling, nd the thicknesses of the sections. 87

5 Design ethod using non-diensionlised slenderness of colu It is highl desirle to reresent the elstic criticl uckling strength of colu,, into non-diensionl rtio. Consider the elstic criticl uckling strength,, s follows: Dividing oth sides of the eqution, It is useful to estlish teril reter Y s follows: Y so tht Y Hence Y Siilrl, strength reduction fctor e estlished to llow for colu uckling in rel colus, nd nuer of non-diensionlised colu uckling curves re estlished to rovide strength reduction fctors to the design strengths of rel colus for rcticl design s shown in Figure. The following reters re doted: χ is the strength reduction fctor due to colu uckling; c is the non-diensionlised slenderness of colu; Y 88

6 . ψ.0 Mteril Yielding lstic Buckling Strength reduction fctor, Non-diensionl slenderness, Figure Non-diensionlized colu uckling curves BM BUCKING For e eer with i-setric cross-section under unifor oent, the elstic criticl uckling oent, M, is given []: M where G I H J Γ IGJ H γ GJ is the e length; is the Young s odulus; is the sher odulus; is the second oent of re out the inor xis; is the wring constnt; is the torsionl constnt; nd I I x Design ethod using the slenderness of e eer Siilr to colu eer, it is ossile to evlute the uckling oent, M, of e with the err-roertson interction forul nd the equivlent slenderness, T, of the e [] s follows: 89

7 M S x where where φ T φ T - φ T ( η ) T η T 0.00 α T ( T 0 ) α T is the Roertson constnt which is tken s 7.0; 0 is the non-diensionlised liiting slenderness; 0.4 T T u v u x r I Sx H γ υ H J I v S x υ 4 0 x is the lstic odulus out the jor xis; is the oisson s rtio. 90

8 Design ethod using non-diensionlised slenderness of e B doting the non-diensionlised slenderness of e, non-diensionlised e uckling curve e estlished s shown in Figure 3 to rovide strength reduction fctors to the design strengths or the oent ccities of rel es for rcticl design. The following reters re doted: χ T is the strength reduction fctor due to e uckling; T is the non-diensionlised slenderness of e; T Y..0 ψ T Strength reduction rtio, Mteril ielding 7.0 lstic uckling Non-diensionlised slenderness, T Figure 3 Non-diensionlised e uckling curve 9

9 OC T BUCKING UNDR UNIFORM COMRSSION Consider rectngulr lte which is sil suorted long oth the longitudinl nd the trnsverse edges [3]. The lte is under coression force of N o long the trnsverse edges. N o Figure 4 x w ocl uckling in sil suorted lte under coression ssue tht the deflected she of the lte is reresented with sine curves in oth longitudinl nd trverse directions s follows: w sin x sinn n where is constnt to e deterined. The strin energ is given : U 4 8 D n n The work done the externl force is given : N x 0 0 dw dx dx d 8 N o n B equting the work done nd the strin energ s follows: 8 N cr n 4 8 D n n 9

10 93 The elstic criticl uckling lod, N cr, is found to e s follows: N cr n n n D To otin the sllest vlue of N cr, consider onl the first ter s follows: N cr n D For the iniu vlue of N cr, n N cr D lterntivel, re-writing the forul in ter of the elstic criticl uckling strength, cr, cr t N cr or t D K t K ) υ ( where k is equl to 4 for the iniu vlue of the elstic criticl uckling strength in sil suorted ltes.

11 OC T BUCKING UNDR IN-N BNDING Consider rectngulr lte which is sil suorted long oth the longitudinl nd the trnsverse edges [3]. The lte is under n in-lne ending oent with linerl vring coression force of N o long the trnsverse edges. N o w x Figure 5 ocl uckling in sil suorted lte under in-lne ending ssue tht the deflected she of lte is reresented with sine curves in oth longitudinl nd trverse directions s follows: w n x n sin sin where is constnt to e deterined. The strin energ is given : U D 4 4 n ( n ) The work done the externl force is given : dw T N0( α )( ) dxd 0 0 dx where α for ure ending B equting the work done nd the strin energ, the elstic criticl uckling lod is given : 94

12 N cr 4 D α n [ n n n ( The derivtive with resect to is given : n ) 3 i (n i i ni ] ) D α 4 n 6 ini ( ) (N 0 ) cr { [ ]} i (n i ) ssue tht the deflected she is roxited the first three ters: w 3 x sin n n sin The iniu vlue of the elstic criticl uckling strength, cr, is given : cr N cr D or K t t ( υ ) K t The vlue of K is found to e deendent on the vlue of s follows: k

13 DSIGN MTHODS USING FFCTIV SCTIONS It is iortnt to evlute the effective section ccities of ltes undergoing locl lte uckling in the resence of initil echnicl nd geoetricl ierfections, nd codified section nlsis using n effective thickness roch or n effective width roch e doted. Bsed on the effective width roch, the effective width, eff, of lte undergoing locl lte uckling e estlished [4] s follows: eff 4 5 4[ fc/cr 0.35] where is the width of the lte nd f c is the lied coressive stress of the lte. Figure 6 lots the function of. eff ginst the vlue of. t.0 K 0.45 K eff / Figure / t Rtio of effective width to flt width of coression lte COMRSION WITH NUMRIC MODS Figures 7, 8 nd 9 illustrte locl lte uckling in rectngulr ltes under different loding nd oundr conditions. The following nlses re erfored nd surised for direct corison: nlticl nlsis using energ ethod; section nlsis using effective width roch; elstic criticl uckling nlsis using finite eleent technique [5]; nd geoetricl nd teril nonliner nlsis using finite eleent technique [5]. 96

14 Geoetr lstic criticl nlsis Geoetricl dt t.6 φ / /t 6.5 cr 30.3 kn Mteril dt Mteril nd geoetricl nonliner nlsis c lied lod, (kn) gross 44.8 kn cr 30.3 kn FM 9.3 kn Figure 7 xil shortening, c () ocl lte uckling in lte under coression. Sil suorted long four sides. cr t K ( υ ) K 4.0 cr 89.4 N/ eff gross 44.8 kn cr 30.3 kn B eff 8.9 kn FM 9.3 kn f c cr

15 Geoetr lstic criticl nlsis Free Geoetricl dt t.6 φ / /t 3.3 Mteril dt cr.8 kn Mteril nd geoetricl nonliner nlsis c 94 N/ 80 N/ 80 N/ 3 lied lod, (kn) 5 gross.4 kn 0 5 cr.8 kn 0 FM 9.3 kn xil shortening, c () cr t K ( υ ) K 0.43 cr 89.4 N/ eff gross.4 kn cr.8 kn B eff 9.45 kn FM 9.3 kn f c cr Figure 8 ocl lte uckling in lte under coression. Sil suorted long three sides nd free long one side. 98

16 M Geoetr θ Geoetricl dt t 0.6 lstic criticl nlsis M M cr kn φ / /t 6.5 Mteril dt 350 N/ 05 kn/ v Mteril nd geoetricl nonliner nlsis 0.5 o t lied oent, M (kn) M gross 0.35 kn M cr kn M FM 0.63 kn Rottion, θ M gross M cr M B 7.6 N/ Z 0.35 kn kn o Z 0.7 kn M FM 0.63 kn Figure 9 ocl lte uckling in lte under in-lne ending. Sil suorted long four sides. 99

17 Corison with gross section ccities nd effective section ccities in the resence of locl lte uckling re rovided. It should e noted tht the results fro vrious nlses re ver close ong theselves. CONCUSIONS It is iortnt to recite structurl instilit in one diensionl e-colu eers nd two-diensionl lte eers under different loding nd oundr conditions. Soe sic fetures of colu uckling, e uckling nd locl lte uckling re resented to fcilitte uckling nlsis nd design in rctice. RFRNCS. Tioshenko S nd Gere JM. Mechnics of terils. Fourth edition, The Steel Construction Institute. Steel Designers Mnul. Sixth edition, Tioshenko S nd Gere JM. Theor of elstic stilit. Second edition, The Buildings Dertent, the Governent of Hong Kong SR. Code of rctice for the Structurl Use of Steel 005. Chter Design of Cold-fored Steel Sections nd Sheet rofiles. 5. BQUS. (004). User s Mnul, Version 6.4, Hiitt, Krlsson nd Sorensen, Inc. 00