The Open Civil Engineering Journal

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1 Sed Odes fo Repts to 738 The Ope Cvl Egeeg Joual, 06, 0, The Ope Cvl Egeeg Joual Cotet lst avalable at: DOI: 0.74/ RESEARCH ARTICLE The Method of Model Appomato fo the Aalyss of Sem-gd Coecto wth Paamete Coelato Xa Dog ad Zha Wag,,* Depatmet of Cvl Egeeg, South Cha Uvesty of Techology, Guagzhou 50640, Cha State Key Laboatoy of Subtopcal Buldg Scece, South Cha Uvesty of Techology, Guagzhou 50640, Cha Receved: July, 06 Revsed: Septembe 8, 06 Accepted: Septembe, 06 Abstact: The momet-otato model s the ma study of sem-gd coecto, moeove, the sestvty coeffcets of paametes o sem-gd coecto ae the pecodto of a momet-otato model. Thee ae two ssues the cuet eseach: the defectve models would be obtaed wthout cosdeg the coelato amog the paametes of the stuctue; fo the hghe pecso sestvty coeffcets, a lage umbe of fte elemet smulatos would be coducted ad ths pocess leads to tesve computato. Accodg to these poblems, the samplg of the paametes wth cosdeg the coelato ad the appomate model whch was costucted by umbeed fte elemet smulatos ad mpoved chaotc patcle swam algothm ae poposed ths pape. Two mathematcal eample ae coducted ad the appomate models epeset a hgh degee of accuacy. A ew sestvty calculato s poposed based o the coelato paametes ad the appomate model, the momet-otato model of eteded ed plate coecto obtaed fom ths algothm has the popetes of hgh computatoal effcecy ad eplct physcal meag. Keywods: Appomate model, Coelato betwee paametes, Momet-otato model, Sem-gd coecto, Sestvty aalyss.. INTRODUCTION Beam-colum coectos, as a mpotat pat of the steel fame stuctue, have bee dvded to thee categoes accodg to the chaactestc of foce defomato: gd coecto, ped coecto, ad sem-gd coecto. I the desg of steel fame stuctue, egees desg the beam-colum coecto as so called gd ot o ped coecto to esst the lateal ad educe the lateal dsplacemet effcetly of stuctues usually, but both types ae the atfcal assumptos, pactcal egeeg, all of the coectos ae the sem-gd coectos. The vestgato o the behavo of sem-gd coecto bega 97 [], moeove, the momet-otato cuve takes o a moe mpotat ole such as lea models [], polyomal models ad epoetal models [3]. The plastc hge method has bee wdely used olea aalyss of sem-gd steel fame. Lew ad Km [4, 5] peseted the efed plastc hge aalyss model fo plae fame, Cha [6] poposed spgs--sees model wth mateal oleaty at each ed of the elemet fo sem-gd coectos. Hu shu-u [7] peseted a elemet wth spgs--sees at each ed fo sem-gd steel fame wth elemets of teal plastc hge, t could cosde the effect of sem-gd coectos ad mateal oleaty. A Satas et al. [8] peseted a med fomulato fame fte elemet wth teal sem-gd coectos fo the olea aalyss of steel stuctues. The fomulato of the elemet also cosdes the pesece of lea o olea behavo of sem-gd coecto behavo wthout addtoal * Addess coespodece to ths autho at the Depatmet of Cvl Egeeg, South Cha Uvesty of Techology, Guagzhou 50640, Cha; Laboatoy of Subtopcal Buldg Scece, South Cha Uvesty of Techology, Guagzhou 50640, Cha; Tel/Fa: ; E-mal: wagzha@scut.edu.c /6 06 Betham Ope

2 The Method of Model Appomato The Ope Cvl Egeeg Joual, 06, Volume odes ad degees of feedom. Whle the epemetal ad theoetcal modelg of sem-gd coectos have bee pefomed by eseaches, they ae almost based o the assumpto that the vaables of stuctues ae completely detemed, whch meas evaluatg the effects of vaables o the stuctues by vayg put paametes oe at a tme the holdg othe paametes at detemed values wth gog the elatoshp amog these vaables [9]. The stuctue cotas may paametes the aalyss of sem-gd coectos, thee may be coelatos betwee each paamete. These coelatos mpacted lagly o the fal output esposes of the stuctues. The wog sestvty coeffcets would be calculated by aalyzg the effect of paametc vaables o stuctues wthout cosdeg the elatvty of paametes [0]. Thee ae a few eseaches o the paametes coelato [], ad moe mpotatly, the momet-otato cuves egessed by ths assumpto would be coutetutve pactcal applcatos, so t s woth to study the sem-gd coecto based o the coelatos betwee the paametes. The aalyss of the sestvty coeffcets s to evaluate the vaatos of the stuctual espose due to the vaatos of the desg paametes [], t s the pecodto to study the momet-otato model. The pobablty desg model of fte elemet aalyss ad the basc theoy Mote Calo ae mplemeted as the basc meas to obta the sestvty coeffcets. By use of Mote Calo, a lot of compute tme ad softwae would be take to accout fte elemet smulato. Howeve, despte gowg computg powe, the softwae defects stll epeset a sgfcat weakess. Hee a dffeet way the study of the momet-otato model s poposed. Ths method cludes the samplg of coelato paametes based o the theoy of Mote Calo ad the appomate model that ams to ot depedet o the fte elemet aalyss softwae. The appomate model costtutes a fuso betwee the fte elemet smulato aalyss ad hybd eual etwok whch has bee optmzed by the method of mpoved chaotc patcle swam. Two mathematcal eamples have bee used to llustate the applcablty ad accuacy of the appomate model. Fo epesetato of the momet-otato model of sem-gd coecto, a eted ed plate coecto s mplemeted whle cosdeg the coelatos ad sestvty of paametes the lteatue. The poposed method s developed fo the aalyss of sestvty ad the momet-otato model wth effcetly.. EXPERIMENT AND FINITE ELEMENT ANALYSIS OF EXTENDED END PLATE CONNECTION.. Aalyss of Epemet Ths epemetal pogam was a eted ed plate coecto whch was subected to statc loads. Ths test specme was bolted to the colum flage ad the plate ad M0 hgh-stegth bolt was employed, the colum secto was HE340 ad the beam was IPE300. The geomety ad aagemet of the stumetato of the ot ae show Fg. () [3]. Based o the avalable test data, detaled geomety of the specme ae show Table, the aveage chaactestcs of the coecto ae set out Table, whee E s the elastc modulus, E st s the sta hadeg modulus, f y s yeld stegth, f u s the tesle stess, Ɛ e s the elastc sta, Ɛ st s the sta of the tal hadeg pot, Ɛ u s the ufom sta, Ɛ u s the ultmate sta. Fg. (). Geomety of the specme [3].

3 740 The Ope Cvl Egeeg Joual, 06, Volume 0 Dog ad Wag Table. Chaactestc values of the stuctual steels [3]. Compoets E/MPa E st /MPa f y /MPa f u /MPa Ɛ st Ɛ u Ɛ u Plate ad colum Beam web Beam flage Bolt Table. Actual geomety of the coecto [3]. Colum/mm Beam/mm h b t f t w H up H low h b t f t w L beam L load Fte Elemet Aalyss Ths beam colum ot s modelg by Asys 4, accodg to the symmety, the fte elemet model s the / stuctue of the sold model, the types of elemets cludg SOLID95, TARGE70, CONTA74 ad PRETS79. SOLID95 s a hghe-ode veso of the 3-D 8-ode sold elemet SOLID45. It ca toleate egula shapes wthout as much loss of accuacy. Each compoet of the eteded ed plate coecto was modeled by sold elemet SOLID95, all the tefaces of the cotact cludg the tefaces betwee the eteded ed plate ad the colum flage, the bolt ad the ed plate. The bolt ad colum flage wee modeled by TARGE70 ad CONTA74 elemets. The peteso foce of bolts was modeled by PRETS79 elemets. The fte elemet model cossts of 9043 SOLID95, 65 CONTA74, 30 TARGE70 ad 34 PRETS79. The aalyss of sem-gd ots s fte elemet statc, the loadg pocess s dvded to two steps: fst, the pe teso s appled at the hgh stegth bolts, ad the cocetated foce at the beam ed. The fte elemet model s show Fg. (), the stess-sta cuve of steel beam ad ed plate s show Fg. (3). The dsplacemet of DT as show Fg. () whch was to calculate the otato of the stuctue ad the epessos ae epeseted as follows [4]: Fg. (). Fte elemet model.

4 The Method of Model Appomato The Ope Cvl Egeeg Joual, 06, Volume 0 74 f f u f y Ɛ e Ɛ st Ɛ u Ɛ u Ɛ Fg. (3). Stess-sta cuves of steel [3]. 50 momet/kn.m FEM Epemet otato/mad Fg. (4). Momet-otato cuve of DT. θ δ δ δ δ acta 600 DT DT 5 b, e( DT ) = b, el ( DT ) 3 P DT Lload = EI b b 6 DT () () Whee δ Dt s the vetcal dsplacemet of DT ad δ Dt5 s the vetcal dsplacemet of DT5, δ b,el(dt) s the beam elastc deflecto at lea vaable dsplacemet tasduce of DT evaluated by equato(). Whee E b s the beam elastc modulus, I b s the momet of eta of beam, L load s the dstace betwee the load applcato pot ad the face of the ed plate, P s the load at the fee ed of beam ad DT s the dstace of DT whch ae show Fg. (). The compaso betwee epemetal esults ad fte elemet model s show Fg. (4), whee the phatom s smulated by fte elemet aalyss, the sold le s the epemetal value (umecal value efeece [3]). Ageemet appeas much close betwee the two les, the fte elemet model ca be used late the aalyss. 3. MONTE CARLO SAMPLING WITH THE CORRELATION OF THE PARAMETERS The pupose of Mote Calo samplg s to meet the equemet of accuacy pobablty, but adom samplg may leads to the paadocal coclusos. It s oted that Mote Calo samplg wth coelato paametes s accodace wth the paametes chagg.

5 74 The Ope Cvl Egeeg Joual, 06, Volume 0 Dog ad Wag 3.. Tasfomato Betwee Coelated No Nomal Vaables ad Coelated Nomal Vaables A mathematcal theoem s that the omal vaables ae depedet whe the coelato coeffcets ae zeo. No-omal adom vaables wll be coveted to omal adom vaables by the fuctos wth equal pobablty tasfomato. A adom vecto X=[X X X 3 ] ad a stadad omal dstbuto vecto Y=[Y Y Y 3 ] ae used ths tasfomato. The the elatoshp betwee adom dstbuto ad omal dstbuto ae gve by: F( ) = Φ( y) = F [ Φ( y)] =, (3) Jot pobablty desty fuctos devg fom the ules of mplct fucto devato ae epessed as follows: f ( ) f ( ) f ( ) f = y y y y ϕ ρ ( ) (, 0) ϕ( ) ϕ( ) ϕ( ) ϕ ( y, ρ ) = ep( y ρ y) T 0 0 ( π) det( ρ 0) (4) (5) whee F ( ) s a magal pobablty dstbuto fucto, F - [ ] s the vese fucto of F( ), Φ( ) s a stadad omal dstbuto fucto, ρ s the equvalet coelato coeffcet mat of Y, f ( ),=,, s the magal pobablty desty fucto coespodg wth F ( ), φ( ) s the pobablty desty fucto of stadad omal vaables, det( ) epesets the mat detemat, ( ) - s the fucto of mat veso. The coelato coeffcet ρ 0y of stadad omal adom vecto y ca be deved fom equato (6) fo the magal pobablty desty fucto of adom vaables ad ρ have bee calculated. ρ + + = µ µ ( )( ) f (, ) d d σ σ X X + + F ( ( )) ( ( )) Φ y F y µ Φ µ = ( )( ) φ( y, y, ρ 0 ) dy dy σ σ (6) Whee ρ s the coelato coeffcet betwee ad of adom vecto X, ρ 0 s the coelato coeffcet mat compoet of stadad omal adom vecto y, f X X (, ) s the ot pobablty desty fucto about the th ad th vaables. 3.. Tasfomato Betwee Coelated Nomal Vaables ad Idepedet Nomal Vaables Suppose that the coelato coeffcet mat ρ have bee calculated by the secto 3., a temedate vaable M s epeseted as M = V Λ (7) Whee Λ s a dagoal mat composed by the egevalues λ belogg to the coelato coeffcet mat ρ 0, V s a mat composed by the egevectos v coespodg wth the egevalues λ. The elatoshp betwee depedet stadad omal vaables U ad coelated stadad omal vaables Y ca be epeseted as follows: = U M Y (8) The Mote Calo samplg pocess cosdeg the coelato betwee paametes s lsted as below:. Geeate adom vaables X wth -dmeso followg omal dstbuto, assumed a computatoal accuacy θ;

6 The Method of Model Appomato The Ope Cvl Egeeg Joual, 06, Volume Calculate the coelato coeffcet mat of omal dstbuto adom vaables; 3. Tasfom coelated adom omal vaables to adom depedet vaables by the algothm of secto 3.; 4. Calculate the coelato coeffcet mat ρ of the ogal model, obta the equvalet coelato coeffcet mat ρ accodg to the lteatue [5]; 5. Geeate adom vaables followg omal dstbuto wth the coelato coeffcet ρ 0 accodg to the vese tasfomato of secto 3., get the absolute values of the dffeeces betwee coelato coeffcet mat of the geeated vaables ad ρ 0, f all the dffeeces ae less tha θ, go to step 6, othewse, go back to step ; 6. Tasfom the coelated adom omal vaables wth coelato coeffcet mat ρ 0 to the coelated adom o-omal vaables based o the equal pobablty tasfomato. The beam heght ad wdth, thckess of web ad thckess of flage whch ae belog to IPE steel ae select as the statstcal paametes. Accodg to the patal coelato aalyss by SPSS [6], we ca get the cocluso that thee ae the coelatos betwee heght ad wdth, thckess of flage ad thckess of web, the coelato coeffcets ae 0.87 ad 0.9 espectvely. 4. THE METHOD OF APPROXIMATE MODEL 4.. The Algothm of Neual Netwok wth Impoved Chaos Patcle Swam The appomate model s costucted by the feedback eual etwok whch has bee optmzed by chaos patcle swam. Feedback etwoks s a olea dyamc system wth sgfcat leag capacty by small fomato, ts state s chagg cotuously utl they each a equlbum pot. Also, they ema at the equlbum pot utl the put chages ad a ew equlbum would to be foud. Its algothm uses gadet descet seach techque to mmze a cost fucto equal to the mea squae dffeece betwee the desed ad the actual outputs [7]. I 989, Hok et al. have poved that ay complety fucto could be appomated by a feedfowad eual etwok wth a sgle hdde laye [8, 9]. Y zh-guag et al. [0] peseted a BP eual etwok model of coal seam gas pemeablty. It was bult based o the coal mechacal popetes, the accuacy of the model was tested ad the BP eual etwok model match the measued data well. Ha hog-gu et al. [] peseted a ew fleble RBF eual etwok to the poblem of the stuctue desg. The pcple of feedback eual etwok s epessed as follows: ' k = f( wk bk ), k = 0,,,, = 0 ' ' ' yl = f( wkl k b l ), l = 0,,,, m k = 0 (9) Whee y l s the etwok output, s the etwok put, k s hdde laye euo, w k s the weght of put layes to hdde layes, b k s the theshold value of put layes to hdde layes, w kl s the weght of hdde layes to output layes, b l s the theshold value of hdde layes to output layes, f s a tasfe fucto. Fo the comple optmzato poblems, the pheomeo of slow covegece ad fallg to the local eteme s ceasgly outstadg. The eual etwok whose weghts ad theshold value ae optmzed by the combato of chaotc ad patcle swam optmzato algothm would ump out of local mma, the patcles also may keep the dvesty of populato ad mpove global optmzato pefomace []. Ths algothm has bee mpoved by optmzg the patcles alteately betwee steady ad chaos [3]. Ftess fuctos accodg to the obectve fucto would be establshed to estmate whethe the patcles wee pematue covegece o ot. I ths pape, the elatve eos betwee the etwok outputs ad the actual espose ae employed as the ftess fuctos. To sue the fuctos ae oegatve, the epessos ae epeseted as Eq. (0), whee u() s the udgmet fucto of pematue covegece. ( ) = ( ( )) / ( ) = = u y f f (0)

7 744 The Ope Cvl Egeeg Joual, 06, Volume 0 Dog ad Wag The detemato about whethe the status of patcle swams s steady o chaos ca be gve as Eq. (), abs dcates the absolute value of the movemet dstace of the cuet patcle. smove = abs( ( t) ( t )) () Whee t s a umbe of the patcle evolutoay geeatos. It s cosdeed that the patcle has eached to the local optmzato whe smove o u() s less tha a lmt value. The vaables wll get to chaos by chaotc seachg. Logstc model ca be gve as: = + µ, =,,... () Whee μ s a cotol paamete decdg the state of the system, the system s totally chaotc state whe μ=.the pocedues of chaotfcato ae as follows: () Make,v mapped to the teval [-.0,.0] by Eq. (3), whee,v s the locato of patcle the vth dmeso, [a v,b v ] s the doma of,v the vth dmeso. h v, 4v, ( bv + av) = b a v v (3) () Geeate a chaotc sequece h,v,h,v,... h J,v by teatg the equato h + =-h fo tmes; (3) Map patcles of the chaotc sequece to the pmay seachg space followg the fucto,v=/4{b v + a v + h,v(b v -a v )}, chaos colum pots =(h,, h,, h,) each dmesoal chaotc sequece ca be deved fom Tet map [4]. I the patcle swam optmzato of ths pape, the patcles would be dvded to two kds, the fst kd patcles ae the weghts fom put layes to the hdde layes ad hdde layes to the output layes, the secod kd ae the theshold values fom put layes to hdde layes ad hdde layes to the output layes. These mates ae epessed as: Patcle() = [W, W ] w, w, w, w, w, w, W = wm, wm, wm, (4) Patcle() = [Z, Z ] W w+, w+, w+, m w, w, w + + +, m = w+ k, w+ k, w+ k, m z, z, z, k z, z, z, k Z = zm, zm, zmk, T (5) (6)

8 The Method of Model Appomato The Ope Cvl Egeeg Joual, 06, Volume Z z, k+ z, k+ z, k+ g z, k z, k z + +, k+ g = zmk, + zmk, + zmk, + g (7) Whee s the dmeso of put laye vaables, m s the umbe of hdde laye euos, k s the dmeso of output layes, g s the populato sze. 4.. Compaso Betwee Impoved Netwok ad Mathematcal Pecso Values To llustate the poposed methodology, two umecal eample wee valdated ths secto. Eample Cosde a o-lea fucto wth 5 ufom adom vaables. The fucto s: f( ) = c ep[ a ( p ) ] 4 5 = = (8) c =[. 3 3.], [0,],=, p = [ a ] = =,...,5 Eample Cosde a o-lea fucto wth 5 ufom adom vaables. The fucto s: = 5 = 5 ( ) = ( + + )( + + ) = = f a (9) [,], [,] a = Tables 3 ad 4 ae the compasos of the esults fou aspects fom the mea, the stadad devato, the umbe of samples, the mamum eo ad MAPE about Mote Calo, eual etwok ad mpoved eual etwok. The Mote Calo esults ae the eact solutos that obtaed fom the Eq. (8) ad Eq. (9) sepaately ad abbevated to MC. The mpoved eual etwok s the poposed appomate model. The compasos of the esults of vaable 3 mpoved etwok, geeal etwok ad ogal equato ae showed Fg. (5) espectvely. It s oted that the esults obtaed fom the appomate model have hghe pecso by oly uses 30 samples, the elatve eos of the mea ad the stadad devato ae both lowe tha 3%, MAPE less tha %. Futhemoe, geeatg the appomate models of two eamples equed oly 30 sample pots.

9 746 The Ope Cvl Egeeg Joual, 06, Volume 0 Dog ad Wag Table 3. Aalyss esults of eual etwoks fo eample. Statstc Samples umbe Mea Stadad devato Mamum eo MAPE(%) MC / / mpoved eual etwok elatve eo(%) / / / etwok elatve eo(%) / / / Table 4. Aalyss esults of eual etwoks fo eample. Statstc Samples umbe Mea Stadad devato Mamum eo MAPE(%) mpoved eual etwok / / MNN elatve eo(%) / / / ANN etwok / / / eact soluto mpoved eual etwok eual etwok f() Fg. (5). Compaso of esults. 5. CALCULATION OF STRUCTURE SENSITIVITY COEFFICIENTS The sestvty of each paamete s detemed by the combato of the pedcted esults of eual etwok ad Mote Calo whle cosdeg the samplg of elatvty accodg to pobablty desty fucto. Cosdeg the mpact of each adom vaable o stuctue espose, coelato coeffcets ae used to measue the coelato betwee adom vaables ad stuctue esposes. The coeffcets of vaato ae bought to elmate the effect of the adom vaable dmesos ad measuemet scales. The equatos ae gve by: R = = = _ ( )( f( ) f( )) _ ( ) ( f( ) f( )) (0) S u R R = ( ) γ γ = () Whee R s the Peaso coelato coeffcets, s adom vaable, s Mote Calo samplg tmes, f( ) s the _ etwok output accodg to, s the mea value of, f( ) s the mea value of f( ); γ s the coeffcet of vaato accodg to Mote Calo adom samples, u s a umbe epeseted the adom vaables total dmesos, S s the sestvty value of adom vaables, the absolute value of coeffcet eflects the fluece of adom

10 The Method of Model Appomato The Ope Cvl Egeeg Joual, 06, Volume vaables o stuctue espose, postve ad egatve eflect the degee of coelato betwee adom vaables ad stuctue espose. 6. THE MOMENT-ROTATION MODEL OF EXTENDED END PLATE CONNECTION The fte elemet model of the eted ed plate coecto whch has bee vefed secto. s used as the meta-model to costuct the appomate model. Each smulato of eteded ed plate coecto eeds about te mutes by fte elemet softwae. At least smulatos to be employed fo the sestvty coeffcets whch ae fa beyod the computg powe 50 fte elemet smulatos ae used to costuct the appomate model ad Mote Calo samples ae employed to get moe accuate esults by the poposed method. Eght dffeet put paametes wee used to model the momet-otato espose, these cluded beam heght, beam wdth, the thckess of beam web, the thckess of beam flage, the thckess of plate, the dstace betwee the mddle bolts ad the vetcal as of beam, the dstace betwee the uppe bolts ad the vetcal as of beam ad the dstace betwee bolts ad the vetcal as of beam web. The elatve eo betwee the esults of the smulatos ad the appomate models s show Fg. (6) elatve eo samples Fg. (6). Relatve eo. Table 5 dcates the sestvtes R ad R whch coclude by cosdeg elevat paametes ad wthout cosdeg elevat paametes. The sestvty coeffcet of beam wdth s wthout cosdeg the coelato of the paametes. Fom the umecal pot of vew, the elatoshp betwee beam wdth ad stffess s ot obvous. Howeve, the actual stuato s that the beam wdth s a mpotat facto to the oveall stffess of the ot. I the pocess of the stuctue secto sze chaged, the elato betwee the beam heght ad the wdth ae coespodece, wth the ceasg of the beam heght, the beam wdth s also appopate ceasg. By the way of cosdeg the coelato betwee paametes, the sestvty coeffcet becomes , that shows the beam wdth has a geat fluece o the stffess of the ot ad ths sestvty agees wth the actual facts. Table 5. Coelato coeffcets of ot. Rotato BEAM PLATE BOLT h b t t w t h h md h up V R () R () The momet-otato model s composed by the paametes of the eteded ed plate coecto, the sestvty coeffcets ad the elastc modulus. The otato at the left sde of the equato s dmesoless. I ode to make the dmesos same at both eds of the equato, the dmeso of the paametes about the eteded ed plate coecto should be L -3 fo the dmeso of the momet s ML S - ad the dmeso of the elastc modulus s ML - S -, so the sestvty coeffcets should be tasfomed to stadad coeffcets ad the tasfom pocess s epessed as Eq. (). The stadad values of paametes sestvty about eteded ed plate coecto ad the fst ode patal dffeetal values C betwee the paametes ad the otato ae show the Table 6. The model s gve by Eq. (3). Dmesos ae same at both eds of the equato, mpovg the dmesos agg fom both eds by data fttg.

11 748 The Ope Cvl Egeeg Joual, 06, Volume 0 Dog ad Wag Table 6. Itegated data. BEAM PLATE BOLT h b t t w t h h md h up V R s() C β β a R s () 3R () = R β h V M θ = E h b t t t h up w h md θ βhhup V M = C = h h E0h b t tw th h md h () (3) (4) Whee θ s the otato betwee beam ad colum, M s the momet of beam ed, h s the beam heght, b s the beam wdth, t s the thckess of beam web, t w s the thckess of beam flage, t h s the thckess of plate, h md s the dstace betwee the mddle bolts ad the vetcal as of beam, h up s the dstace betwee the uppe bolts ad the vetcal as of beam, V s the dstace betwee bolts ad the vetcal as of beam web, β s the mea coecto coeffcet about β. Fo eample, β h ca be obtaed by the Eq. (4). CONCLUSION A method of the momet otato model of sem-gd coecto has bee poposed ths pape. The ma algothm cludes Mote Calo samplg wth the coelato of the paametes, the appomate model ad the method of stochastc sestvty, moeove, the coelato of the paametes as a essetal compoet affects the whole stuctue systems. A key featue of the appomate model s the hybd eual etwok pocedue that effcetly deals wth smulato models ad pogessvely tas a set of metamodels. As pat of ths study, the effectveess of the poposed method, whch does ot eque huge fte elemet smulatos, was optmzed by the mpoved chaos-patcle swam povded a satsfactoy appoach fo hghly o-lea fuctos. Two hgh dmesoal equatos have bee used to be eamed ad ageemet appeas much close betwee these esults whch wee calculated by appomate model ad the Mote Calo esults. Futhemoe, the appomate model has the hgh effcecy fo oly dozes of samples. Fte elemet aalyss of the eted ed plate coecto s caed out by ANSYS softwae, the esults have bee compaed wth the tests esults ad epoted a hgh degee of accuacy, the mamum elatve eo s less tha %. I addto, these esults of appomate models have bee used fo sestvty aalyss momet-otato model. The calculatos of the sestvty coeffcets cosdeg paametes coelato ae moe cosstet wth the actual. The elatve momet-otato model was establshed based o the sestvty coeffcets of paametes ad stochastc aalyss. Elastc modulus was bought the egesso equato of momet otato fo the pot of mechacs to make both eds of the equato equvalet fom the dmeso, so that the egesso equato s moe tha the umecal fttg momet otato cuve. Futhemoe, the accuacy ad effcecy make the poposed methods of the appomate model ad the method of stochastc sestvty sutable to be mplemeted othe types of the steel coectos. CONFLICT OF INTEREST The authos cofm that ths atcle cotet has o coflct of teest.

12 The Method of Model Appomato The Ope Cvl Egeeg Joual, 06, Volume ACKNOWLEDGEMENTS Ths wok s suppoted by Natoal Natual Scece Foudato of Cha ude the gat umbe , REFERENCES [] F.H. Wu, Sem-gd Coecto Steel Fames., College of Egeeg, Pudue Uvesty: Pudue, 989. [] Y.L. Yee, ad R.E. Yee, "Momet-otato cuves fo bolted coectos", J. Stuct. Eg., vol., o. 3, pp , 993. [ [3] F.H. Wu, ad W.F. Che, "A desg model fo sem-gd coectos", Eg. Stuct., vol., o., pp , 990. [ [4] J.Y. Lew, D.W. Whte, ad W.F. Che, "Secod-ode efed plastc hge aalyss fo fame desg: Pat ", J. Stuct. Eg., vol. 9, o., pp , 993. [ [5] S.E. Km, ad W.F. Che, "Pactcal advaced aalyss fo baced steel fames desg", J. Stuct. Eg., vol., o., pp , 996. [ [6] S.L. Cha, ad C.Y. Yau, "Ielastc ad stablty aalyss of flebly coected steel fames by spgs--sees model", J. Stuct. Eg., vol. 0, o. 0, pp , 994. [ [7] S. Hu, "A effcet computatoal method fo sem-gd steel fame wth elemets of teal plastc hge", J. If. Comput. Sc., vol., o. 3, pp , 04. [ [8] A. Satas, A. Koseoglu, ad H.F. Ozel, "Modelg of the olea behavo of steel famed stuctues wth sem-gd coectos", I: Eghth Iteatoal Cofeece o Advaces Steel Stctues, Lsbo: Potugal, 05, pp. -8. [9] C. Xu, ad G.Z. Gete, "Ucetaty ad sestvty aalyss fo models wth coelated paametes", Relab. Eg. Syst. Saf., vol. 93, o. 0, pp , 008. [ [0] D. Xa, ad W. Zha, "Stochastc sestvty aalyss method based o coelato paametes ad hybd eual etwok", J. Buld. Stuct., vol. 36, o. 4, pp , 05. [ Chese]. [] D. Xa, W. Zha, ad J. Pa, "Stochastc sestvty aalyss fo models wth paamete coelato ad model appomato method", J. If. Comput. Sc., vol., o., pp , 05. [ [] F.W. Wag, "Optmzato ad sestvty aalyss of space fames allowg fo lage deflecto", Eg. Stuct., vol. 0, pp , 006. [3] A.M. Gão Coelho, F.S. Blaad, ad L. Smões da Slva, "Epemetal assessmet of the ductlty of eteded ed plate coectos", Eg. Stuct., vol. 6, o. 9, pp , 004. [ [4] J. Pa, Aalytcal method o sem-gd ot of famewok based o coelatos., Shatou Uvesty: Cha, 009. [ Chese] [5] A. De Kuegha, ad P.L. Lu, "Stuctual elablty ude complete pobablty fomato", J. Eg. Mech., vol., o., pp , 986. [ [6] J. Pa, ad W. Zha, "Semgd aalyss of eteded ed plate coectos based o coelato ad sestvty", J. Shagha Jaotog Uv. (Sc.), vol. 7, o. 5, pp , 0. [ [7] K.S. Al-Jab, S.M. Al-Alaw, A.H. Al-Sady, ad A.S. Aluam, "A atfcal eual etwok model fo pedctg the behavou of semgd ots fe", Adv. Steel Cost., vol. 5, o. 4, pp , 009. [8] K.M. Hok, M. Stchcombe, ad H. Whte, "Multlaye feedfowad etwok s ae uvesal appomatos", Neual Netw., vol., o., pp , 989. [ [9] M.R. Fa, M.Z. Jumaat, ad R.T. Vahd, "Nolea aalyss of load-deflecto testg of efoced oe-way slab stegtheed by cabo fbe efoced polyme (CFRP) ad usg atfcal eual etwok (ANN) fo pedcto", It. J. Phys. Sc., vol. 6, o. 3, pp , 0. [0] Guag-zh Y, ad Mg-hu L, "Model of coal gas pemeablty pedcto based o mpoved BP eual etwok", J. Cha Coal Soc., vol. 36, o. 7, pp , 03. [ Chese]. [] H. Ha, J. Qao, ad Y. Bo, "O stuctue desg fo RBF eual etwok based o fomato stegth", Acta Automatca Sca, vol. 38, o. 7, pp , 0. [ Chese]. [

13 750 The Ope Cvl Egeeg Joual, 06, Volume 0 Dog ad Wag [] G. Fe, ad H. Tog, "Paamete estmato fo system based o patcle swam optmzato", Acta. Phys., vol. 55, o., pp , 006. [ Chese]. [3] X. Xu, K. Zheg, ad L. Da, "New chaos-patcle swam optmzato algothm", J. Commu., vol. 33, o., pp. 4-9, 0. [ Chese]. [4] Dea La, Guaog Che, ad Mat Hasle, "Dstbuto of the lyapuov epoet of the chaotc skew tet map", It. J. Bfucat Chaos., vol. 9, o. 9, pp , 0. Dog ad Wag; Lcesee Betham Ope Ths s a ope access atcle lcesed ude the tems of the Ceatve Commos Attbuto-No-Commecal 4.0 Iteatoal Publc Lcese (CC BY-NC 4.0) ( whch pemts uestcted, o-commecal use, dstbuto ad epoducto ay medum, povded the wok s popely cted.