Exact Three-Dimensional Elasticity Solution for Buckling of Beam of a Circular Cross-Section *

Transcription

1 ans. Japan Soc. Aeo. Space Sc. Vol., No., pp., Exact hee-dmensonal Elastcty Soluton fo Bucklng of Beam of a Ccula Coss-Secton * oshm AKI ) and Kyohe KONDO ) ) Aeospace Company, Kawasak Heavy Industes, Ltd., Kakamgahaa, Gfu 8, Japan ) Depatment of Aeonautcs and Astonautcs, he Unvesty of okyo, okyo 8, Japan An exact thee-dmensonal elastcty soluton fo bucklng of a ccula coss-secton beam wth one end fxed and the othe end fee s obtaned. A compason s made usng the appoxmate soluton by Kadomateas and an exact two-dmensonal soluton by Chattopadhyay and Gu. he bucklng loads of both exact solutons ae n good ageement wth each othe, but the bucklng load usng the appoxmate soluton s slghtly hghe than the exact solutons. he exact solutons suppot Engesse s fomula athe than Hangx s fomula fo the bucklng load of a beam wth tansvese shea effect. Key Wods Bucklng Analyss of Beam, hee-dmensonal Elastcty, ansvese Shea Effect Nomenclatue a adus of ccula coss-secton beam A coss sectonal aea of beam C ;C ; components of stffness matx C stffness matx E; E L ; Young s modulus E ;E ; ; ; components of Geen stan tenso E Geen stan tenso F defomaton gadent tenso G; G L ;G shea modulus G ; G ; G base vectos afte defomaton n cylndcal coodnates ; ; base vectos befoe defomaton n cylndcal coodnates I unt matx l; m; n components of nomal vecto of a suface I n ðþ modfed Bessel functon of the fst knd k tansvese shea facto L length of beam L effectve column length c ; E ; bucklng loads ; ; stess vectos n cylndcal coodnates ; ; cylndcal coodnates poston vecto of a pont befoe defomaton R poston vecto of a pont afte defomaton S F bounday whee extenal foce condton s specfed S U bounday whee dsplacement s specfed t ; t ; t components of extenal foce vecto pe aea actng on mechancal bounday suface compessve stess defned as total foce on the uppe end dvded by ntal (undefomed) he Japan Socety fo Aeonautcal and Space Scences + esented at the th JSASS/JSME/JAXA Stuctues Confeence, August,, Kumamoto, Japan Receved Apl ; fnal evson eceved June ; accepted fo publcaton July. Coespondng autho, tak_toshmkh.co.jp coss-sectonal aea u dsplacement vecto u; v; w dsplacement components n cylndcal coodnates ; L ; ; osson s ato adus of gyaton ; ; ; ; ; components of Kchhoff stess tenso Kchhoff stess tenso n cylndcal coodnates Subscpts befoe bucklng afte bucklng E Eule En Engesse H Hangx K Kadomateas K ak and Kondo. Intoducton he study of elastc beam bucklng has a long hstoy and many studes ae stll beng conducted n ths feld. he effect of tansvese shea stffness on the bucklng load of a beam s an mpotant topc. As efeed to n moshenko and Gee, ) bucklng load fomulas by Engesse and Hangx ae poposed fo a beam wth tansvese shea defomaton. he Engesse fomula was poposed fo beams, and the Hangx fomula was fst poposed fo bucklng loads of col spngs and ubbe ods. hee has been a contovesy about whch fomula s sutable fo beams wth low tansvese shea stffness, such as lattce beams and sandwch beams. he Engesse and Hengx fomulas wee deved usng one-dmensonal theoes, and they ae based on dffeent assumptons n the defnton of axal and shea foces actng on the beam coss-secton. wo-dmensonal analyss o theedmensonal analyss of the beam s necessay to judge whch s coect. Chattopadhyay and Gu ) deved an exact twodmensonal soluton fo the elastc bucklng of a ectangula othotopc plate. hey dd not menton the Engesse and

2 ans. Japan Soc. Aeo. Space Sc., Vol., No., Hangx fomulas n the pape. Kadomateas ) conducted thee-dmensonal elastcty analyss fo bucklng of a beam wth a ccula coss-secton. He calculated the bucklng load of a beam made of othotopc mateal and compaed the esult wth bucklng loads usng the Engesse and Hangx fomulas. He concluded that hs elastcty soluton s close to the Hangx fomula than the Engesse fomula. Kadomateas pape encouages poponents of the Hangx fomula, and ths s a eason why the Hangx fomula was ecogned as the coect fomula. Bažant ) explaned that the dffeence between the Engesse and Hangx fomulas s attbuted to dffeent fnte stan measues, and the Hangx fomula s moe convenent than the Engesse fomula. Bažant changed hs poston to suppot the Engesse fomula n the papes on bucklng of a sandwch beam.,) He concluded that analyss and bucklng test data of the sandwch beam suppot that the Engesse fomula s coect. hee stll exst some poponents of the Hangx fomula,,) but t may be concluded that the contovesy has ended. 8,9) he concluson s that the Engesse fomula s applcable to beams wth weak tansvese shea stffness ncludng lattce beams and sandwch beams, and the Hangx fomula s applcable only to col spngs and ubbe beangs. hs pape pesents an exact soluton of thee-dmensonal elastcty fo bucklng of a ccula coss-secton beam. he exact thee-dmensonal soluton s compaed to an exact two-dmensonal soluton poposed by Chattopadhyay and Gu. hen, the bucklng loads of beams made of othotopc mateal ae compaed to the bucklng loads deved usng the Engesse and Hangx fomulas. It s shown n ths pape that the soluton by Kadomateas s an appoxmate soluton, and the exact soluton suppots applcablty of the Engesse fomula to the bucklng load of beams wth a tansvese shea effect.. Fomulaton Compesson bucklng of a ccula coss-secton beam wth one end fxed and the othe end fee (see Fg. ) s analyed usng a thee-dmensonal elastcty theoy... Govenng euatons ont A defned by poston vecto n the cylndcal coodnates moves to a new poston A, whch has a poston vecto R as shown n Fg.. oston vecto R s expessed usng the base vecto befoe defomaton and dsplacement components. R ¼ u ¼ bu v wc ðþ Base vectos afte defomaton ae, G G ¼ R ¼ F G whee F s the defomaton gadent tenso. v w F ¼ v v u w v w ðþ Stess vectos ; ; ae expessed by the Kchhoff stess tenso as follows. G ¼ ¼ F ðþ Sym G G he eulbum euaton s, F b ch ¼ Befoe defomaton d A d dd ðþ ðþ u d G d G d L R A x a O y x O dd y G d d ( d) Afte defomaton Fg.. fee. Ccula coss-secton beam wth one end fxed and the othe end Fg.. Infntesmal paallelepped and stess vectos n cylndcal coodnates. JSASS

3 ans. Japan Soc. Aeo. Space Sc., Vol., No., whee H ¼ v u v v v v v v v u w w w he Geen stan tenso s expessed as E ¼ F F I E ¼ E ðþ Sym whee 8 E ¼ v ¼ v v v v u ¼ w w E ¼ v u 8 ¼ v w w w v w v 8 E ¼ w v u v E 9 w ; v w w v v w v u w 9 ; ðþ 9 ; v he stess-stan elatonshp fo tansvesely sotopc mateal s wtten as, ð8þ ðþ ¼ C C C C C C ¼ C C Sym ¼ CE ðþ C C E E E whee ðþ and E ðþ ae Kchhoff stess and Geen stan expessed n vecto fom, espectvely. he stffness matx C s expessed usng engneeng elastc popetes as follows. E ðþ ¼ C ðþ L ¼ Sym L E L L E L L E L G L G L G ð9þ ðþ ðþ he defnton of the engneeng elastc popetes of the tansvesely sotopc mateal s shown n Fg.. he followng elatonshp exsts between the popetes. L E L ¼ L ; G ¼ he geometcal bounday condton s, ð Þ ðþ b u v wc¼bu v w c on S U ðþ whee S U s the suface on whch dsplacements ae specfed. he mechancal bounday condton s, b l m ncf ¼bt t t c on S F ðþ JSASS

4 ans. Japan Soc. Aeo. Space Sc., Vol., No., whee S F s the suface on whch the suface foces ae specfed... e-bucklng euatons and bucklng euatons Defomaton s expessed by the sum of the pe-bucklng defomaton b u v w c and bucklng defomaton b u v w c. b u v wc¼bu v w cb u v w c ðþ Fom Es. () and (9), we obtan, E ¼ E E ðþ ¼ CE ðþ E ðþ ¼ ðþ ðþ Fom Es. () and (), ðþ ðþ F ¼ F F ; H ¼ H H ðþ hen, the eulbum euaton s expessed as follows. F 8 9 F F > > b ch ¼ > G L G b ch >; he bounday condtons ae expessed as follows. L Fg.. Defnton of elastc popetes of the tansvesely sotopc mateal. H ð8þ b u v w cbu v w c¼bu v w c on S U ð9þ b l m nc F b l m nc F F n ¼bt t t c on S F ðþ n L E L n L. Bucklng Analyss A bucklng analyss s conducted accodng to the pocedue gven n Kadomateas. ) he dffeence s that we consde pe-bucklng defomaton whle Kadomateas dd not... e-bucklng state he stess condton unde unfom compesson as shown n Fg., s expessed as follows. ¼ ðþ p Fom the mechancal bounday condton at ¼ L,weobtan, p w ¼ ðþ whee s the compessve stess defned as total foce on the uppe end dvded by the ntal (undefomed) coss-sectonal aea. Defomaton befoe bucklng s expessed as follows. ¼ u ¼ p ffffffffffffffffffffffffffffffffff c p w ¼ p ffffffffffffffffffffffffffffffffff c p whee C C " # c C C ¼ c C C C C C C C C C C C C C ðþ ðþ Kadomateas dd not consde these euatons... Bucklng state Substtutng E. () fo the eulbum euaton (E. (8)), we obtan the followng euaton. 8 9 > > p u p v p w u ( ) u u u > u u w w w >; w JSASS 8

5 8 9 > > ¼ > >; Fom the geometcal bounday condton (E. (9)), ðþ b u v w c¼b c on S U ðþ Fom the mechancal bounday condton (E. ()), b l m nc p u u p v u w w p w w ¼b c on S F ðþ Fom E. (), the condton fo the extenal foce fee on the uppe end s expessed as follows. p p v p w u w ¼b c at ¼ L ð8þ Fom the condton fo the extenal foce fee on the suface of the beam, the followng stesses ae eo. ¼ ; ¼ ; ¼ on ¼ a ð9þ Fom E. (), ðþ ¼ CE ðþ ðþ Substtutng E. () fo E. (), the followng euatons ae obtaned. hese euatons (E. ()) coespond to E. () n Kadomateas. ) u C u 8, C p u C C ans. Japan Soc. Aeo. Space Sc., Vol., No., C C 9 u ; v ðc C Þ u v B w u Cð A C C Þ w ¼ C C v v v v C 8, 9 C p u v ; C C u w B u Cð A C w C Þ w C w w 8, 9 C p w ; u B w Cð A C C Þ ðc C Þ u w u v ¼ ¼ ðþ Followng the pocedue of Kadomateas, ) we consde two goups of solutons fo the eulbum euaton (). Rgd-body dsplacement s not consdeed hee because gd-body dsplacement does not affect the bucklng load. he fnal bucklng soluton s the supeposton of the two goups of the solutons.... Fst goup soluton he followng functons satsfy the eulbum euaton, E. (). u ¼ ; v ¼ ; w ¼ k ðþ whee k s a constant. Substtutng E. () fo E. (), we obtan, C 8 w 9, > C kb u Cð A C C Þp u > > >; ¼ ðþ JSASS 9

6 ans. Japan Soc. Aeo. Space Sc., Vol., No., 8 u 9 fo. > > kc B w Cð A C C Þ > >; ð; ; Þ ¼ C ðcos ÞI s cos ð ¼ ; Þ ðþ... Second goup soluton 8, 9 k C p w he followng functons satsfy the eulbum euaton, E. (). ¼ ðþ ; Fom Es. () and (), s ¼ ðþ whee s s the soluton of the followng euaton. hs euaton coesponds to E. (b) n Kadomateas. ) C C s 8, ðc C Þ C C p w 8, 9 C C p u ; 8, C p u ¼ 9 ; 98, ; C p w s 9 ; ðþ hen, k ae expessed wth s as follows. hs euaton coesponds to E. (c) n Kadomateas. ) k ¼ C s C w u p u ðc C Þ ð ¼ ; Þ ðþ he followng functon s consdeed as the soluton of E. (). u ¼ ; Substtutng E. () fo E. (), v ¼ ; w ¼ ðþ ¼ ðþ whee s expessed as follows. ¼ C p u C C ðþ he followng functon s consdeed as the soluton of E. (). ð; Þ ¼ Z ðþb ð Þsn ðþ Fom Es. () and (), d B db d d B ¼ ð8þ he soluton of E. (8) s as follows. B ¼ C I ¼ ð9þ Substtutng Es. (9) and (9) fo E. (), we obtan. ð; ; Þ ¼ C cos I sn ðþ Deflectons ae calculated fom Es. () and (), u ¼ v ¼ ðþ ð; ; Þ ¼ Z ðþa ðþcos ð ¼ ; Þ ð8þ Assumng Es. (9) and (), we obtan E. () fom E. (). Z ðþ¼cos ¼ d A da d s s d s A ¼ s ð9þ ðþ ðþ he soluton of E. () s as follows. A ¼ C I s ðþ whee I s s the modfed Bessel functon of the fst ode of the fst knd. Fnally we have the followng expesson w ¼ k Fom the bounday condton at the uppe end (E. (8)), p w w ¼ at ¼ L ðþ It s shown that w and n E. () ae popotonal to cos, and E. () s satsfed wth the followng condton. ¼ L ðþ Usng the above euatons, the stesses ae expessed as follows. JSASS

7 ans. Japan Soc. Aeo. Space Sc., Vol., No., ( ) C ðc C Þ u I 8 * + 9 C u ðs Þ I ðs Þ s I ðs Þ * + ¼ > C C u s > I ðs Þ C w k I ðs Þ > >; cos cos ( ) C ðc C Þ u I 8 * + 9 C u ðs Þ I ðs Þ s I ðs Þ * + ¼ > C C u s > I ðs Þ C w k I ðs Þ > >; cos cos ( ) C u ðs Þ I ðs Þ ¼ C ( ) C w cos cos k I ðs Þ ( ) C u I I ¼ C ( ) C u w k I ðs Þ sn sn C u I ( ) ¼C C u w k ( ) I ðs Þs I ðs Þ cos sn ¼ C C u ( ) I C I ( ) C s I ðs Þ sn cos ðþ Usng the bounday condton on the fee suface of the cylnde (E. (9)), we obtan the followng euatons. " # C ðc C Þ u a I a ( ) C u s I ðs aþ s a I ðs aþ ( ) X C C u s a I ðs aþ ¼ C w fk I ðs aþg ¼ " # C I a a I a " # X s C ¼ a I ðs aþ ¼ " # C u a I a "( ) X C u w k ¼ ( )# a I ðs aþs I ðs aþ ¼ ðþ If we delete u and w n E. (), we obtan the same euatons as E. (8) n Kadomateas. ) hs means that the bucklng load epoted by Kadomateas s not an exact soluton but an appoxmate one. Euaton () s smultaneous euatons fo C, C, and C. We euate the detemnant of the coeffcents of E. () as eo, and we obtan the egenvalue, p c. he bucklng stess c s calculated fom E. ()... Numecal calculaton pocedue of bucklng stess Fst, assume a value of p c, then calculate s usng Es. () and (). Next, calculate k usng E. (8) and usng E. (8). s, k, and ae complex numbes n geneal. Substtute these numbes n E. (), the detemnant of the coeffcents of E. () can be evaluated. Fnd the value of p c wth whch the detemnant becomes eo usng solve of MS-Excel. JSASS

8 ans. Japan Soc. Aeo. Space Sc., Vol., No., y lane stan condton.. D exact soluton epoted by ak and Kondo D appoxmate soluton epoted by Kadomateas D exact soluton epoted by Chattopadhyay and Gu h l x c E.8. Isotopc mateal Fg.. Analyss model of D exact soluton. ). he values of the modfed Bessel functon of the fst knd ae calculated usng the numecal ntegaton of the followng fomula. I n ðþ¼ Z e cos cosðnþd ðþ whee s complex vaable and n s ntege.. Numecal Results.. Effect of slendeness ato on the bucklng load Compasons of the exact thee-dmensonal soluton of the pesent pape, the appoxmate soluton epoted by Kadomateas ) and a two-dmensonal exact soluton epoted by Chattopadhyay and Gu ) ae conducted below. he elatonshp between the slendeness ato L and bucklng load s calculated fo the beam made of sotopc mateal. Snce the soluton by Chattopadhyay and Gu s based on a plane stan poblem (see Fg. ), the coecton facto fo osson s ato s appled to the moment of neta of the beam coss-secton. he followng euatons ae used to the calculate the slendeness ato L. Fo the two-dmensonal soluton by Chattopadhyay and Gu L ¼ l ffffffffffffffffffffffffffffffffffffffffff ðþ h Fo the thee-dmensonal solutons by Kadomateas and that n the pesent pape L ¼ L a ð8þ he bucklng load esults epoted by Kadomateas ae taken fom ables and of hs pape. Fgue shows the compason. he exact solutons fo two-dmensonal and thee-dmensonal elastcty ae almost eual, but the bucklng loads epoted fo the appoxmate soluton of Kadomateas ae lage than the exact solutons. he dffeence s lage n the egon of the small slendeness ato... Effect of shea stffness on bucklng load he effect of shea stffness on the bucklng load s compaed n ths secton. wo types of beam bucklng load fomula ncludng tansvese shea effect ae poposed. ) Engesse s fomula. Slendeness ato, Lʹ Fg.. Compason of bucklng loads D exact soluton, D appoxmate soluton and D exact soluton. able. Mateal popetes. ) Isotopc Mateal Mateal mateal Glass/Epoxy Gaphte/Epoxy E L (Ga) (Ga) L G L (Ga) 9.. C (Ga)... C (Ga) C (Ga) C (Ga)... C (Ga) 9... En ¼ E E GA e Hangx s fomula sffffffffffffffffffffffffffffffffffffffff E GA e H ¼ GA e whee E s Eule bucklng load, ð9þ ðþ E ¼ EI ðþ L I s the moment of neta of the coss-secton. GA e ¼ kga, k s tansvese shea coecton-facto. k ¼ 8 fo the ccula coss-secton, k ¼ fo the ectangula coss-secton. ) L s the effectve length of the column. Axal defomaton s not consdeed n Es. (9) and (). As the slendeness ato becomes smalle, the effect of axal defomaton on the bucklng load becomes sgnfcant. he bucklng fomulas, ncludng both tansvese shea and axal defomaton, ae as follows. JSASS

9 ans. Japan Soc. Aeo. Space Sc., Vol., No., able. Compason between Kadomateas soluton and that of the pesent pape. Isotopc mateal Kadomateas ak and Kondo La E G k E EA EGA e En E H E E K En K H K E K En K H K (Ga) (Ga) E. () E. () Mateal Glass/Epoxy Kadomateas ak and Kondo La E G k E EA EGA e En E H E E K En K H K E K En K H K (Ga) (Ga) E. () E. () Mateal Gaphte/Epoxy Kadomateas ak and Kondo La E G k E EA EGA e En E H E E K En K H K E K En K H K (Ga) (Ga) E. () E. () Engesse s fomula,) E En E EA E EA E EA E GA e En E En E ¼ ðþ Hangx s fomula ) vffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff u t E E GA e EA H ¼ E E ðþ E GA e EA E EA s a paamete of the axal stffness and E GA e s a paamete of the shea stffness. hese paametes fo a ccula coss-secton beam, wth one end fxed and the othe end fee (Fg. ), ae expessed as follows. E EA ¼ a L ; E ¼ Ea GA e kgl ðþ... Dscusson on able of Kadomateas pape ) Kadomateas analyed the bucklng loads of beams made of thee mateals sotopc mateal, glass/epoxy and gaphte epoxy. He compaed the bucklng loads wth Engesse s and Hangx s fomulas, Es. (9) and (). Because the compason ncludes beams wth a small slendeness ato, t s necessay to use Es. () and () athe than Es. (9) and (). Compasons between thee-dmensonal elastcty solutons and Es. () and () ae conducted usng the mateal popetes shown n able. he esults ae shown n able and Fgs. to 8. As Kadomateas stated, hs soluton (appoxmate soluton) s n bette ageement wth Hangx s fomula than Engesse s fomula, whle the exact soluton of the pesent pape s n bette ageement wth Engesse s fomula. It s notewothy that the appoxmate soluton fo the sotopc mateal beam s n bette ageement wth both beam fomulas than the exact soluton (Fg. ). he eason fo ths s that the effect of shea s small and the effect of slendeness ato s sgnfcant fo sotopc mateal. Fo othotopc mateals, whch ae soft n tansvese shea, the effect of shea defomaton becomes moe domnant than axal defomaton, conseuently, ageement between the exact soluton and Engesse s fomula s bette than that wth sotopc mateal. JSASS

10 ans. Japan Soc. Aeo. Space Sc., Vol., No., able. Compason of bucklng loads, and effect of tansvese shea stffness. c E Slendeness ato G L E L E EA E GA e Engesse E. (9) Hangx E. () ak and Kondo Kadomateas Hangx/Kadomateas Engesse/Kadomateas Hangx/ak and Kondo Engesse/ak and Kondo.. Hangx/Kadomateas Engesse/Kadomateas Hangx/ak and Kondo Engesse/ak and Kondo c, beam c, elastcty c, beam c, elastcty.8. Isotopc mateal.8. Mateal Gaphte/Epoxy L a 8 L a Fg.. Compason of bucklng loads of sotopc mateal D exact soluton, D appoxmate soluton and beam fomulas. Fg. 8. Compason of bucklng loads of mateal D exact soluton, D appoxmate soluton and beam fomulas.... c, beam c, elastcty.8. Hangx/Kadomateas Engesse/Kadomateas Hangx/ak and Kondo Engesse/ak and Kondo Mateal Glass/Epoxy c E...8. Engesse Hangx D appoxmate soluton by Kadomateas D exact soluson by Chattopadhyay and Gu D exact soluton by ak and Kondo.. 8 L a Fg.. Compason of bucklng loads of mateal D exact soluton, D appoxmate soluton and beam fomulas.... Effect of tansvese shea stffness he esults of Fgs. to 8 nclude both shea and axal defomaton effects. It s necessay to elmnate the effect of axal defomaton n ode to evaluate the coectness of the beam fomulas. A beam wth a hgh slendeness ato and weak shea stffness s analyed. It s assumed that the slendeness ato s and that the hypothetcal othotopc mateal has the followng mateal popetes. Compason of bucklng loads, and effect of tansvese shea stff- Fg. 9. ness... Othotopc mateal Slendeness ato Lʹ E GA e E L ¼ Ga; ¼ Ga; L ¼ ; ¼ G L ¼ { Ga he esults ae shown n able and Fg. 9. he values of c E usng Engesse and Hangx fomulas, as shown n able, ae based on Es. (9) and (), espectvely, and JSASS

11 ans. Japan Soc. Aeo. Space Sc., Vol., No., the values ae almost eual to the values fom Es. () and () because the effect of axal defomaton s neglgble. he values obtaned usng Engesse s fomula ae on the cuve of thee-dmensonal and two-dmensonal exact solutons. hs esult confms that Engesse s fomula s the coect beam fomula, and not Hangx s fomula. he theedmensonal appoxmate soluton epoted by Kadomateas s close to Hangx s fomula, and ths s the eason why Kadomateas eoneously concluded that Hangx fomula s the coect fomula.. Conclusons () An exact thee-dmensonal elastcty soluton fo bucklng of a beam wth a ccula coss-secton s deved. he bucklng load deved usng the thee-dmensonal soluton s n good ageement wth the two-dmensonal soluton epoted by Chattopadhyay and Gu. () he exact thee-dmensonal elastcty soluton confms that Engesse s fomula s coect fo the bucklng load of beam wth a tansvese shea effect. Hangx s fomula should not be appled to beams. () It s shown that Kadomateas soluton ) s an appoxmate soluton. Refeences ) moshenko, S.. and Gee, J. M. heoy of Elastc Stablty, Second Edton, Dove ublcatons, Inc., Mneola, New Yok,. ) Chattopadhyay, A. and Gu, H. Exact Elastcty Soluton fo Bucklng of Composte Lamnates, Compos. Stuct., (99), pp ) Kadomateas, G. A. hee-dmensonal Elastcty Soluton fo the Bucklng of ansvesely Isotopc Rods he Eule Load Revsted, J. Appl. Mech., (99), pp.. ) Bažant, Z.. Shea Bucklng of Sandwch, Fbe Composte and Lattce Columns, Beangs, and Helcal Spngs aadox Resolved, J. Appl. Mech., (), pp. 8. ) Bažant, Z.. and Beghn, A. Whch Fomulaton Allows Usng a Constant Shea Modulus fo Small-Stan Bucklng of Soft-Coe Sandwch Stuctues? J. Appl. Mech., (), pp ) Bažant, Z.. and Beghn, A. Stablty and Fnte Stan of Homogened Stuctues Soft n Shea Sandwch o Fbe Compostes, and Layeed Bodes, Int. J. Solds Stuct., (), pp. 9. ) Attad, M. M. Global Bucklng Expements on Sandwch Columns wth Soft Shea Coes, Elect. J. Stuct. Eng., (), pp.. 8) Blaauwendaad, J. Shea n Stuctual Stablty On the Engesse-Hangx Dscod, J. Appl. Mech., (), pp ) ak,. Note on Compesson Bucklng of Sheaable Beam, oc. of th JSASS/JSME/JAXA Stuctues Confeence,, pp. (n Japanese). ) Washu, K. Vaatonal Methods n Elastcty and lastcty, egamon ess, Oxfod, 98. ) Iwakua,. and Kuansh, S. How Much Contbuton Does the Shea Defomaton Have n a Beam heoy? oc. JSCE,, I- (98), pp.. ) Kondo, K. Bucklng of Extensble Sheaable Elastca, oc. of th JSASS/JSME/JAXA Stuctues Confeence, 8, pp. (n Japanese).. Ogasawaa Assocate Edto JSASS