Stochastic behaviour of shallow concrete-filled steel tubular arches

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1 Sohn Co Uniiy 3d alaian Confn on h hani of S and aial 014 Sohai bhaio of allo on-filld l bla ah W Gao D W K Lo Y-L i bliaion dail Gao W W D Lo K i Y-L 014 'Sohai bhaio of allo on-filld l bla ah' in ST Smih d. 3d alaian Confn on h hani of S and aial CS3 ol. Byon Bay SW 9-1 Dmb Sohn Co Uniiy Limo SW SB: bliaion@scu i an loni oioy adminid by Sohn Co Uniiy Libay. goal i o a and h inllal o of Sohn Co Uniiy aho and ah and o ina iibiliy and ima hogh on a o ah aond h old. Fo fh infomaion la onab@.d.a.

2 3d alaian Confn on h hani of S and aial CS3 Byon Bay alia 9-1 Dmb 014 S.T. Smih d. STOCHSTC BHVOU OF SHLLOW COCT-FLLD STL TUBUL CHS W. Gao* Shool of Ciil and nionmnal ngining Th SW 05 alia..gao@n.d.a Coonding ho D. W Shool of Ciil and nionmnal ngining Th SW 05 alia. di.@n.d.a K. Lo Shool of Ciil and nionmnal ngining Th SW 05 alia. kai.lo@n.d.a Y.-L. i Shool of Ciil and nionmnal ngining Th SW 05 alia. y.i@n.d.a BSTCT Thi dy iniga non-dminii on of on-filld l bla CFST ah aoning fo im-dndn maial naini. Sohai modl a mloyd o handl h naini of final inkag ain and offiin of on o of h allo CFST ah. andomn of h aam i nd by Gaian obabiliy dnii. andom aiabl fnional momn mhod i xndd o dlo h omaional xion fo h fi o aiial momn of andom adial and axial dilamn andom axial omi fo and innal bnding momn. Th faibiliy and ffin of h nd ohai analyi aoah a dmonad by omaing h l of nmial xaml obaind fom h andom aiabl fnional momn mhod ih h on-calo imlaion mhod. KYWODS Unain and inkag ohai analyi CFST ah momn mhod. TODUCTO Con-filld l bla CFST i on y of modn l-on omoi hih nially oni of l b ih ind on o. Thi aila fom of al y o nmb of mhanial adanag hih ha moiad i aliabiliy in modn ngining aliaion ially h aln imlmnaion of CFST ah in bidg ngining. Ho on aial oblm ofn nond in al-old aliaion i h ima of nain aam inold in CFST ah aing on hi lif-im foman. Th mo di naini oigina fom h inini and inkag ff of on. Th bhaio of and inkag a ongly im-dndn hih a alo inoling high andomn Bažan and Li Thi ok i lind nd h Cai Common ibion 4.0 nnaional Lin. To i a oy of hi lin ii h://aiommon.og/lin/by/4.0/ 663

3 STOCHSTC LYSS OF CFST CHS WTH UCTTS L X dno h of all andom aiabl in a obabiliy a Ω and x i a andom aiabl of X. n hi a h nain final offiin and final inkag ain of h on o a modlld a Gaian andom aiabl hih a xd a: * 1 * h and dno h andom final offiin and andom final inkag ain ily; * and dno h man and andad diaion of h andom final * offiin and final inkag ain ily; and dno a nomal diibion. By adoing h on of ag-adjd ffi modl of on and h inkag modl inodd in C Commi 09 C Commi h andom ffi Yong modl of on o and h inkag ain a im an b xd a: g * * * * h 3 d h ; dno h nominal Yong modl of on; 0 15day fo h fi loading of h on o a h 15 h day; and d=35 day fo moi ing on. Fom io ximnal and analyial di Uy 001; Uy and Da 1997 h ff on h oall al bhaio d o h inkag of on i mh mo banial in h longidinal diion han h an diion. Thfo only h ff of on inkag in h longidinal diion on h al bhaio i onidd in hi dy. Sin h final offiin and final inkag ain a nomal andom aiabl by imlmning h laial hoy of obabiliy h man al and aian of h ffi Yong modl and inkag ain a im an b allad by: h [ g ] g f d Va [ ] [ ] * [ h ] * * * 4 h f d Va [ ] [ ] 5 dno h man and aian of h andom ffi Yong modl and inkag ain a im ily; f and f * dno h nomal obabiliy dniy fnion DF of h final offiin and final inkag ain ily. Fhmo by imlmning h hoial obabilii hoy and h andom aiabl fnional momn mhod Gao al. 010 h man al and aian of h ffi Yong modl and inkag ain a im an b xd a: g g Va 6 h * * Va Va d 7 Thfo h ffi Yong modl and inkag ain a im a andom aiabl hih ha h fom a: D D 8 h D dno a DF haaizd by h man and andad diaion of h andom aiabl. n hi dy h nial amion ha bn adod hih a 1 h dfomaion of CFST ah i lina and aify h l-bnolli hyohi; h dimnion of h o-ion i mh mall han h dimnion of h CFST ah i.. lngh adi; and 3 h l b and on CS

4 o a flly bondd. gnal al layo of yial CFST ah and a yial o-ion ha bn illad in Fig 1a. n h oodina ym oy h lina ain of any abiay oin of h o-ion of a CFST ah an b allad by: y h dno h dimnionl dilamn in h adial and axial diion h ha / / ; and dno h dilamn in h adial diion axial diion a ll a h adi of h iniial a of CFST ah ily; d /d and d /d θ dno h angla oodina of h oin in h inil axi oy. a CFST ah gnal layo b in-ndd CFST ah Fig 1. Con-filld l bla ah and i yial o-ion Fom h aly amion on h fll bonding bn h l b and on o h mhanial mmban ain of on and l m b qal o ah oh and h mhanial ain a h infa of o maial m b h am. Ho h of l and on ha diini al h a: 10 h dno h of h l and dno h of on hih i a andom aiabl. By imlmning h ial ok inil fo h long-m qilibim of h CFST ah bjd o nifomly diibd load q i qi h aiaion of h ial ok m b zo h ha: Θ 11 W V dv V dv - -Θ q d 0 fo all abiay aiaion of h admiibl dfomaion and. V and V dno h olm of h on o and l b ily; and dno h Lagang oao of imlano aiaion. Sbiing q.9-10 ino q.11 yild: Θ [ W ] - - q d 0 1 -Θ h h andom axial omi fo aking h fom a: d d 13 and h andom bnding momn: yd y d 14 h and dno h o-ional aa of h l b and on o ily; and a h ond momn of aa of h l b and on o ily. ngaing q.1 by a h diffnial qaion of qilibim fo h CFST ah in adial diion an b xd a: 3 i q and h diffnial qaion of qilibim in h axial diion i xd a: ih h ai bonday ondiion fo in-ndd ah a: 0 a 17 and i a andom aiabl hih dno h im-dndn adi of gyaion of h ffi o-ion abo i majo inil axi hih i dfind a: CS

5 18 n addiion h kinmai bonday ondiion fo h in-ndd ah a: 0 0 a 19 Thfo by oling q.15 and 16 ih h imlmnaion of h oonding bonday ondiion h andom im-dndn adial and axial dilamn fo in-ndd CFST ah a im an b xliily xd a: q 1 o o in in in o o o 0 and q ih andom aiabl hih an b xd a: ino in o o in in oin o 3oin 1 n addiion o h dilamn h innal axial fo and momn of in-ndd CFST ah a andom aiabl and an b xd a: q in o in qo o q 3 any aila im q.0-3 n h al on of in-ndd CFST ah ih h onidaion of h naini of h and inkag ff of h on o. oo all h al bhaio fomlad by q.0-3 a andom aiabl hih a fnion of h nain and inkag aam. Thfo in od o fh xnd h obabilii analyi h andom aiabl fnional momn mhod Gao al. 010 i imlmnd o alla h aiial haaii of h im-dndn nain al bhaio. By ilizing h andom aiabl fnional mhod h man al of h adial and axial dilamn of in-ndd CFST ah a im an b allad ily by: q q 1 o o in in in o o o ino in o o in in and h man of h im-dndn innal axial fo and momn an b allad by: q in o in qo o q 6 h o in o 3o in n addiion h aian of h im-dndn adial dilamn axial dilamn innal axial fo and innal momn of in-ndd CFST ah an b ily allad by: Va Va Va Va CS

6 CS h q q ] [ in o 3 33 q q ] [ in o ] in [o o in o in o in q q q - 36 o in 37 in o o o in ] [ o o in q q 38 o o in 39 and o ]o o in [ ] in in o o 1 [ 40 o in o in in in o o 1 o in o in 41 in in o o in o in 4 UCL VSTGTO Th in-ndd CFST ah on in Fig 1b i inigad o dmona h aliabiliy and ffin of h ood mhod. Th la an of h ah i L=15m and h i i f =L/10; h o adi of h o-ion i 50mm and h inn adi i 40mm. Th Yong modli of l b and on o a =00Ga and =30Ga ily. Th nain aam onidd a h final offiin and final inkag ain hih a amd a nomal andom aiabl. Th man al and offiin of aiaion COV of h andom final offiin a.9 and 0% ily; and h man al and COV of h final inkag ain a and 15% ily. Th alid load ha a magnid of q / 0.65 h / S/ and S i h ah lngh. By ing h fnional momn mhod a ll a h on-calo imlaion mhod ih and iaion a ah day of loading h im-dndn nain bhaio of h adial dilamn and axial fo a h n of h ah a allad fom 15-day o 400-day of loading. Th im-dndnofil a illad in Fig and 3 ily. Fom Fig and 3 i i la ha h im-dndnobabilii infomaion i.. man al and andad diaion of h al bhaio a any abiay loading im obaind by h fnional momn mhod i ga han h amling mhod. Th aiial haaii odd by h amling mhod old b imod if mo iaion a fomd hih qi

7 an al of nal axial fo Sandad diaion of nal axial fo an al of nal adial dilamn Sandad diaion of nal adial dilamn mo omaional ffo alhogh on-calo imlaion anno gna liabl andad diaion. idnly h ood fomlaion a aabl of diing h im-dndn obabilii ofil of h al bhaio of CFST ah. 6 x 10-3 x Fnional omn mhod on-calo Simlaion mhod on-calo Simlaion mhod on-calo Simlaion mhod Tim day Fnional omn mhod on-calo Simlaion mhod on-calo Simlaion mhod on-calo Simlaion mhod Tim day a an al b Sandad diaion Fig. Tim-dndnobabilii ofil of nal adial dilamn ni: m x Fnional omn mhod on-calo Simlaion mhod on-calo Simlaion mhod on-calo Simlaion mhod Tim day Fnional omn mhod on-calo Simlaion mhod on-calo Simlaion mhod on-calo Simlaion mhod Tim day a an al b Sandad diaion Fig 3. Tim-dndnobabilii ofil of nal axial fo ni: COCLUSOS Thi a iniga h ff of nain y im-dndn and inkag on h al bhaio of CFST ah in a obabilii mann. Th momn mhod ha bn imlmnd o xliily ala h im-dndnobabilii haaii of h al bhaio of CFST ah. By inigaing a in-ndd CFST ah bjd o nifomly diibd load h aabiliy and ffin of h ood mhodology ha bn dmonad. hl fo h CFST ah ih nain maial aam h ff d o a a no onidd in hi dy. FCS C Commi diion of C Shinkag and Tma ff in Con S mian Con ni C Doi US. Bažan Z.. and Xi Y Sohai dying and ff in on Jonal of Sal ngining SC Vol Gao W. Song C. and Tin-Loi F. 010 obabilii inal analyi fo ih nainy Sal Safy Vol Uy B. and Da S Tim ff in on-filld l box olmn in all bilding Sal Dign of Tall Bilding Vol Uy B. 001 Sai long-m ff in o on-filld l box olmn nd aind loading C Sal Jonal Vol CS

( ) ( ) ( ) 0. dt dt dt ME203 PROBLEM SET #6. 1. Text Section 4.5

( ) ( ) ( ) 0. dt dt dt ME203 PROBLEM SET #6. 1. Text Section 4.5 ME PROBLEM SET #6 T Sion 45 d w 6 dw 4 5 w d d Solion: Fis mlil his qaion b (whih w an do sin > o ansfom i ino h Cah- El qaion givn b w ( 6w ( 4 Thn b making h sbsiion (and sing qaion (7 on ag 88 of h,