Explicit expression for effective moment of inertia of RC beams

Transcription

1 54 Explii xprssion for ffiv momn of inria of RC bams Absra Dflion is an imporan dsin paramr for sruurs subjd o srvi load. This papr provids an xplii xprssion for ffiv momn of inria onsidrin rakin, for uniformly disribud loadd rinford onr (RC) bams. Th proposd xplii xprssion an b usd for rapid prdiion of shor-rm dflion a srvi load. Th xplii xprssion has bn obaind from h raind nural nwork onsidrin onr rakin, nsion siffnin and nir praial ran of rinformn. Thr sinifian sruural paramrs hav bn idnifid ha ovrn h han in ffiv momn of inria and hrfor dflion. Ths hr paramrs ar hosn as inpus o rain nural nwork. Th rainin daa ss for nural nwork ar nrad usin fini lmn sofwar ABAQUS. Th xplii xprssion has bn validad for a numbr of simply suppord and oninuous bams and i is shown ha h prdid dflions hav rasonabl auray for praial purpos. A snsiiviy analysis has bn prformd, whih indias subsanial dpndn of ffiv momn of inria on h sld inpu paramrs. Kywords Conr rakin; dflion; fini lmn analysis; momn of inria; nural nwork; rinford onr; nsion siffnin. K.A. Pal a A. Bhardwaj b S. Chaudhary A.K. Napal d a,b Rsarh Sholar, Civil En. Dp., IIT Dlhi, Nw Dlhi, India. Assoia Profssor, Civil En. Dp., MNIT Jaipur, Jaipur, India. d Dora Chair Profssor, Civil En. Dp., IIT Dlhi, Nw Dlhi, India. Corrspondin auhor: a iid.kashyap@mail.om b ank.bhardwaj@mail.om shaudhary.@mni.a.in d aknapal@ivil.iid.a.in hp://dx.doi.or/.59/ Rivd In rvisd form.7.4 Apd Availabl onlin 3..4 Nomnlaurs A, A ara of op and boom rinformn, rspivly s BD, widh and dph of bam B, D widh and dph of flan f f B, D widh and dph of wb w w E, E modulus of lasiiy of onr and sl, rspivly I s ffiv momn of inria I, I momn of inria of ross and fully rakd ransformd ross sion, rspivly I j r h j inpu paramr L lnh of bam

2 543 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams M, M O r bias minimum momn a whih h rakin aks pla a a ross-sion in h bam and applid (lasi) momn, rspivly oupu paramr bias of hiddn or oupu nuron dexp, dfem, d NN mid-span dflion from xprimns, FEM, and nural nwork/xplii xprssion, rspivly d, ffiv onr ovr a op and boom. rspivly b f nsil srnh of onr ' f h h k k hiddn nuron m onsan q r wn, ylindrial omprssiv srnh of onr a 8 days numbr of inpu paramrs numbr of hiddn nurons uniformly disribud load and modular raio, rspivly w r minimum load a whih h rakin aks pla in h bam rakin ih w jk, wih of h link bwn I j and h k w wih of h link bwn h k and O ho k,, r u, Subsrip j k o Suprsrip ho ih rakin srain, and maximum nsil srain of onr, rspivly prna nsion and omprssion rinformn, rspivly inpu nuron numbr hiddn nuron numbr or funion numbr oupu nuron numbr onnion bwn hiddn and oupu layrs onnion bwn inpu and hiddn layrs INTRODUCTION Dflion is an imporan paramr o hk h srviabiliy riria of sruur. Th shor rm dflion is nrally alulad usin ffiv momn of inria of nir span a srvi load. Th quaions for ffiv momn of inria, availabl in liraur, ar mainly basd on wo approahs: (i) sprins in paralll and (ii) sprins in sris (Kalkan, ). Th siffnsss of h unrakd and rakd porions ar avrad in h sprins in paralll approah (Branson, 965; Al-Zaid al., 99; Al-Shaikh and Al-Zaid, 993; SAA-AS 36, 994; TS 5, ; CSA- A3.3, 4; ACI 38, 5; AASHTO, 5), whras h flxibiliis of h unrakd and rakd porions ar avrad in h sprins in sris approah (Ghali, 993; CEN Eurood, 4; Bishoff, 5; Bishoff and Sanlon, 7; Bishoff, 7). Considrin paralll sprins approah firs, h followin quaion of ffiv momn of inria ( I ) in rms of fully rakd and unrakd momn of inria was oriinally proposd by Branson (965) for simply suppord bams as m m m Mr Mr Mr I I I I ( I I ) I M M M r r r () Lain Amrian Journal of Solids and Sruurs (5) 54-56

3 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 544 whr, M r = minimum momn a whih h rakin aks pla a a ross-sion in h bam; M = applid (lasi) momn alon h span; = momn of inria of h ross ross sion; I r = momn of inria of h fully rakd ransformd ross sion and m =onsan. Eq. () was drivd mpirially basd on h xprimnal s rsuls of simply suppord ranular rinford onr (RC) uniformly loadd bams wih nsion rinformn, =.65% and raio of momn of inria of h fully rakd ransformd ross sion and momn of inria of h ross ross sion, I r =.45 a maximum applid (lasi) momn qual o.5 M r (Branson, 965). Eq. () has bn adopd in many inrnaional sandards and ods (SAA-AS 36, 994; TS 5, ; CSA-A3.3, 4; ACI 38, 5; AASHTO, 5) o alula I and hrfor dflion, akin m = 3. Som rsarhrs (Bishoff, 5; Gilbr, 999; Sanlon al., ; Gilbr, 6) found ou ha Eq. () wih m = 3 alulas ffiv momn of inria auraly in as of mdium o hih nsion rinformn( > %), whil i ovrsimas ffiv momn of inria for low nsion rinformn ( < %). Al-Zaid al. (99) xprimnally provd ha h valu of m in Eq. () dpnds on h loadin onfiuraions and susd m =.8 (in Eq. ()) for uniformly disribud load whn M.5 M r. Th valu of m was found o han from abou 3 o 4.3 for modraly-rinford onr bams ( =.%, I r =.34) in h ran of Mr M.5M r. Al-Shaikh and Al- Zaid (993) prformd xprimns on mid-span poin loadd bams wih varyin rinformn. Th valus of m was found o vary from abou.8 o.5 for lihly rinford bams ( =.8%, I r I =.) in h ran of.5 Mr M 4 M r, whil for h havily rinford bams ( = %, Ir I =.44), m varid in a ran of.9 o.3. Thy also susd m 3.8 inorporain rinformn ff in Eq. () for poin loadd bams. Al-Zaid al. (99); Al- Shaikh and Al-Zaid (993) also proposd o alula I basd on rakd lnh inorporain rinformn and loadin ffs rspivly. Nx, onsidr h sprins in sris approah. Th modls basd on his approah (Bishoff, 5; Bishoff and Sanlon, 7; Bishoff, 7) ak ino aoun nsion siffnin ff in onr for alulain I. Th dflions obaind by h xprssion proposd by Bishoff (5) hav bn found in ood armn wih xprimnal dflions for lihly rinford bams ( < %) (Gilbr, 6; Bishoff and Sanlon, 7). Kalkan () found ou ha h xprssions ivn by Eq. () and Bishoff (5) sima dflions of modraly-rinford o hihly-rinford onr bams ( >%) auraly on usin h xprimnal valu of rakin momn whih, howvr, is diffiul o obain for ah and vry as. I is obsrvd from h rviw ha no sinl approah or modl is dirly appliabl for h nir ran of praial rinformn. Thrfor, dvlopmn of an approah for rapid simaion of h mid-span dflions in uniformly disribud loadd RC bams onsidrin nir praial ran of rinformn a srvi load is dsirabl. Th approah should b simpl o us rquirin a minimal ompuaional ffor bu mus iv auray ha is apabl for praial appliaions. Th appliaion of nural nwork an b suh an alrna approah. For nraion of rainin daa for nural nworks, fini lmn hniqu may b usd. Nowadays, nural nworks ar bin xnsivly applid in h fild of sruural ninrin. Som of h rn appliaions of nural nworks in h fild of sruural ninrin inlud Lain Amrian Journal of Solids and Sruurs (5) 54-56

4 545 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams prdiion of im ffs in RC frams (Maru and Napal, 4), prdiion of dama dion in RC framd buildins afr arhquak (Kanwar al., 7), sruural halh moniorin (Min al., ; Kaloop and Kim, 4), bndin momn and dflion prdiion in omposi sruurs (Chaudhary al., 7, 4; Pndharkar al., 7,, ; Tadss al., ; Gupa al., 3), prdiin h rp rspons of a roain omposi dis oprain a lvad mpraur (Gupa al., 7), opimum dsin of RC bams subjd o os (Sarkar and Gupa, 9), sai modl idnifiaion (Kim al., 9), rspons prdiion of offshor floain sruur (Uddin al., ), prdiion of dflion in hih srnh slfompain onr dp bams (Mohammadhassani al., 3a; 3b) and prdiion of nry absorpion apabiliy and mhanial propris of fibr rinford slf-ompain onr onainin nano-silia parils (Tavakoli al., 4a; 4b). Ths sudis rval h srnh of nural nworks in prdiin h soluions of diffrn sruural ninrin problms. This papr prsns an alrnaiv approah for simain ffiv momn of inria whih is nihr sprin in paralll nor sprin in sris approah. Nural nwork modl is dvlopd, a srvi load, for prdiin ffiv momn of inria (and dflion), in a RC bam onsidrin nir praial ran of nsion and omprssion rinformn, nsion siffnin and flxural onr rakin. Th daa ss for rainin, validain and sin ar nrad usin fini lmn modls. Th fini lmn modls hav bn dvlopd in ABAQUS () sofwar and validad wih h xprimnal rsuls availabl in liraur. Explii xprssion has bn obaind basd on dvlopd nural nwork modl whih an b usd in dsin offis by praiin ninrs. Th proposd nural nwork/xplii xprssion has bn validad for a numbr of simply suppord and oninuous RC bams. Snsiiviy analysis has bn prformd o undrsand h influn of rlvan paramrs on ffiv momn of inria. FINITE ELEMENT MODEL AND ITS VALIDATION Th fini lmn modl (FEM) has bn dvlopd usin h ABAQUS () sofwar. Th bam has bn modlld usin B lmns (-nod linar Timoshnko bam lmn). Undr srvi load, h srss-srain rlaionship of onr is assumd o b linar in omprssion. Conr has bn onsidrd as an lasi marial in nsion bfor rakin and sofnin bhaviour is assumd afr rakin (Fiur ). Furhr, a srvi load, h srss in rinformn is assumd o b in h linar ran. Th sl rinformn has bn mbddd ino h onr usin REBAR opion in whih a prf bond is onsidrd bwn sl rinformn and onr. In ordr o onsidr rakin and nsion siffnin, h smard rak modl has bn usd. Tnsion siffnin has bn dfind usin pos-failur srss-srain daa proposd by Gilbr and Warnr (978). A hih shar siffnss has bn assumd o nl h shar dformaions. Th rsuls of FEM hav bn ompard wih h xprimnal rsuls (mid-span dflions of h bam undr inrasin uniformly disribud load, w, afr h rakin of h onr) rpord by Al-Zaid al. (99) for a simply suppord bam (VB) wih.5 m lar span (ffiv span =.6 m) and ross-sional dimnsions B D mm (Fiur ). Th ' ohr propris onsidrd ar: ylindrial omprssiv srnh of onr a 8 days, f = 38. Lain Amrian Journal of Solids and Sruurs (5) 54-56

5 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 546 N/mm ; modulus of lasiiy of onr, E =.96 4 N/mm ; modulus of lasiiy of sl, E s = 5 N/mm ; nsil srnh of onr, f = 3.47 N/mm ; rakin momn, M r = 5. knm; ara of op rinformn, A s = mm and ara of boom rinformn, A = 4. mm. Th ffiv onr ovr a op ( d ) and a boom ( db ) hav bn akn as 3 mm and 33 mm rspivly. Fiur : Tnsion siffnin modl. As d D RC ranular ross-sion A B db Fiur : Ranular ross-sion. In ordr o dfin h smard rak modl, h absolu valu of h raio of uniaxial nsil srss a failur o h uniaxial omprssiv srss a failur is akn as.9. Th srain a rakin, r is akn as. and in viw of low/modra nsil rinformn, A = 4. mm (=.%), h plasi srain is ( u r ) akn as.4. For onvrn, abou 6 lmns ar rquird whn rakin is onsidrd (Pal al., 4). Rsuls from h dvlopd FEM and xprimns ar ompard in Fiur 3. Clos armn is obsrvd bwn h rsuls from FEM and xprimns. Nx, h rsuls hav bn ompard wih xprimnal rsuls rpord by Washa and Fluk (95) for four ss of ranular ross-sional (Fiur ) simply suppord bams: A,A4; B,B4; C,C4; D,D4 subjd o uniformly disribud loads a srvi load. Two bams in a s ar idnial. Th ross-sional propris, marial propris, span lnhs and uniform Lain Amrian Journal of Solids and Sruurs (5) 54-56

6 547 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams disribud loads of all four bams hav bn ivn in Tabl. Addiionally, Es has bn assumd as 5 N/mm. E and f ar akn in aordan wih ACI 38 (5). Th midspan dflions obaind from h FEM ( d FEM ) ar in los armn wih h rpord xprimnal dflions ( dexp ) as shown in Tabl. Th fini lmn modls an hrfor b usd for nraion of daa ss. Uniformly Disribud Load (kn/m) 5 5 FEM Exprimnal Mid-span Dflion Fiur 3: Comparison of mid-span dflions of bam VB. Bams B D M (knm) Propris w (kn/m) ' f (N/mm ) L d d b A =A s (mm ) Mid-span dflions d EXP d FEM A-A B-B C-C D-D Tabl : Propris of simply suppord bams wih ranular ross-sions, onsidrd for validaion of FE modl. 3 SAMPLING POINTS AND DATA SETS For dvlopmn of nural nwork, sinifian paramrs nd o b idnifid. Eq. () shows ha I xpliily dpnds on I r and Mr M a. I is assumd ha I dpnds on and (prna omprssion rinformn) also. Th valu of howvr dpnds on and rans from. o ( n ) n, whr, n = modular raio. Th valu of Ir I in urn also dpnds on h ombinaions of and. Considr a ypial bam ross-sion as shown in 4 5 Fiur ( B = 3 mm; D = 7 mm; E =.73 N mm ; E s =. N mm ; d = 3 mm and d b = 33 mm). For his bam, h variaions of and Ir I wih ar shown in Fiur 4. Th paramr Mr Ma dpnds on h load and momn only. Takin h abov obsrvaions ino aoun, is also onsidrd as an inpu paramr alonwih Ir I and Mr M. Th samplin poins of h paramrs onsidrd for daa nraion ar shown in Tabl. I may b nod ha h ombinaions of samplin poins ak Lain Amrian Journal of Solids and Sruurs (5) 54-56

7 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 548 ino aoun h diffrn valus of h (fourh and lf ou paramr), orrspondin o ah valu of. Considrin h quaion 4 dfem 5wL 384E I, h oupu paramr I I is 4 obaind as 5wL 384E d FEM..5 I r /.5 r = r r (n-)/n r (%) Fiur 4: Variaion of and I r wih. Inpu Paramrs Samplin poins I r r M M As pr inrmn in FE analysis Tabl : Inpu paramrs and samplin poins TRAINING OF NEURAL NETWORK Nural nwork has bn dvlopd for h prdiion of ffiv momn of inria in RC bams. Th nural nwork hosn is a s of mulilayrd fd-forward nworks wih nurons in all h layrs fully onnd in h fd forward mannr (Fiur 5). Th rainin is arrid ou usin h MATLAB Nural Nwork oolbox (9). Simoid funion (losi) is usd as an aivaion funion and h Lvnbr-Marquard bak propaaion larnin alorihm (rainlm) is usd for rainin. Th bak propaaion alorihm has bn usd sussfully for many sruural ninrin appliaions (Maru and Napal, 4; Kanwar al., 7; Gupa al., 7; Pndharkar al., 7; ; ; Chaudhary al., 7; 4; Sarkar and Gupa, 9; Gupa and Sarkar, 9; Min al., ; Tadss al., ; Mohammadhassani al., 3a; Gupa al., 3) and is onsidrd as on of h ffiin alorihms in ninrin appliaions (Hsu al., 993). On hiddn layr is hosn and h numbr of nurons in h layr is didd in h larnin pross by rial and rror. Lain Amrian Journal of Solids and Sruurs (5) 54-56

8 549 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams Hiddn nurons h I h I Inpu nurons I j w ih j,k h k ho w k, O Oupu nuron I h r Fiur 5: Confiuraion of a ypial nural nwork. Diffrn ombinaions of samplin poins of h inpu paramrs and h rsulin valus of h oupu paramrs ar onsidrd in ordr o rain h nural nwork. Eah suh ombinaion of h inpu paramrs and h rsulin oupu paramrs ompriss a daa s. Th oal numbr of daa ss onsidrd for h rainin, validain and sin of h nwork ar Normalisaion faors ar applid o inpu and oupu paramr o brin and wll disribu hm in h ran. No bias is applid o h inpu and oupu paramrs. Normalisaion faors of 4,, 7 and 3 ar applid o inpu paramrs, I r, Mr M and oupu paramr I rspivly. 7% daa ss ar usd for rainin and h rmainin daa ss ar dividd qually in h validain and sin ss. For h rainin, svral rials ar arrid ou wih diffrn numbrs of nurons in h hiddn layr sarin wih a small numbr of nurons in h hiddn layr and prorssivly inrasin i, and hkin h man squar rrors (MSE) for h rainin, validain and sin. Th numbr of nurons in h hiddn layr is didd on h basis of h las man squar rrors (MSE) for h rainin as wll as validain and sin. Car is akn ha h man squar rror for s rsuls should no inras wih h numbr of nurons in hiddn layr or pohs (ovrrainin). Th final onfiuraion (numbr of inpu paramrs - numbr of nurons in h hiddn layr - numbr of oupu paramrs) of NN is Th rsponss of proposd nural nwork modl o prdi ffiv momn of inria for rainin, validain, and sin ar shown in Fiurs 6(a)-() rspivly. Th proposd nural nwork modl ahivd ood prforman as h sin daa poins ar mosly on quiy lin (Fiur 6()). Th saisial paramrs i.. man squar rror (MSE), roo man squar rror (RMSE), man absolu prna rror (MAPE), avra absolu dviaion (AAD), orrlaion Lain Amrian Journal of Solids and Sruurs (5) 54-56

9 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 55 offiin (R ) and offiin of variaion (COV) (Sozn al., 4; Azmahullah al., 5) of rainin, validain and sin daa ss ar shown in Tabl 3. All h paramrs india a ood armn. Saisial Daa ss paramrs Trainin Validain Tsin MSE RMSE MAPE AAD R COV Tabl 3: Saisial paramrs of nural nwork Prdid I / Trainin Equiy Lin Prdid I / Validain Equiy Lin Tar I / (a) Tar I / (b) Prdid I / Tsin Equiy Lin Tar I / Fiur 6: Rspons of nural nwork modl in prdiin I () I : (a) rainin; (b) validain; and () sin. 5 EXPLICIT EXPRESSION FOR PREDICTION OF EFFECTIVE MOMENT OF INERTIA For h as of praiin ninrs and usrs, simplifid xplii xprssion an b dvlopd for h prdiion of ffiv momn of inria. Th xplii xprssion rquirs h valus of inpus, wihs of h links bwn h nurons in diffrn layrs, and biass of oupu nurons (Tadss al., ; Gupa al., 3). Lain Amrian Journal of Solids and Sruurs (5) 54-56

10 55 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams As sad arlir, h simoid funion (losi) has bn usd as h aivaion funion. Th oupu O (Fiur 5) may hrfor b obaind as blow (Tadss al., ; Gupa al., 3): O r ho wk, biaso H k k q ih k j,k j k j H w I bias (3) () whr, q and r ar h numbr of inpu paramrs and h numbr of hiddn nurons h rspivly; bias k and bias o ar h bias of k hiddn nuron ( h k ) and h bias of oupu ih ho nuron rspivly; w j,k and wk, ar h wih of h link bwn I j and h k and h wih of h link bwn hk and O rspivly. Th wihs of h links and biass of h oupu nurons for NN ar lisd in Tabl 4. Connion Inpu o Hiddn Wih/ Bias ih, k Numbr of h hiddn layr nuron ( k ) w ih, k w ih 3, k w bias k ho Hiddn o Oupu w k, Tabl 4: Wih valus and biass of nural nwork. Th valu of I is qual o d-normalizd oupu O. Th ffiv momn of inria I ho may b obaind from Eq. () by puin h valus of wk, from Tabl 4 as I I H H H 3 H 4 H 5 H 6 (4) whr, H, H, H3, H4, H5 and H 6 may b obaind from Eqs. (5)-() by usin h wihs and biass (Tabl 4) as H Ir I. Mr M.386 (5) H Ir I.83 Mr M (6) H Ir I Mr M (7) H Ir I Mr M 7.94 (8) H Ir I 5.68 Mr M (9) H Ir I.89 Mr M.966 () Lain Amrian Journal of Solids and Sruurs (5) 54-56

11 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 55 6 VALIDATION OF NEURAL NETWORK/ EXPLICIT EXPRESSION Th dvlopd nural nwork/xplii xprssion is validad for a numbr of simply suppord and oninuous bams wih a wid variaion of inpu paramrs. Th rsuls (mid-span dflions), obaind from h proposd nural nwork/xplii xprssion ar ompard wih h xprimnal rsuls for simply suppord bams availabl in liraur and wih h FEM rsuls for oninuous bams. 6. Simply suppord bams Firs, h rsuls hav bn ompard wih xprimnal rsuls rpord by Washa and Fluk (95) for ss of simply suppord bams wih ranular ross-sion (Fiur ): A,A5; B,B5; C,C5; D,D5 subjd o uniformly disribud loads, and dsinad, hr, as VB- VB4, rspivly. Two bams in a s ar idnial. Th dails of h bams ar ivn in Tabl 5 5. Addiionally, E s is assumd as.5 N/mm. E and f ar akn in aordan wih ACI 38 (5). Th mid-span dflions obaind from h proposd xplii xprssion ( d ) NN ar shown in Tabl 5 alon wih h rpord xprimnal mid-span dflions ( d ) EXP. Th valus obaind from h proposd xplii xprssion ar in rasonabl armn wih h rpord xprimnal valus of mid-span dflions. Bams B D M w (knm) (kn/m) Propris ' f (N/mm ) L d d b A s (mm ) A (mm ) Mid-span dflions d EXP d NN VB (A,A5) VB (B,B5) VB3 (C,C5) VB4 (D,D5) Tabl 5: Propris of simply suppord bams wih ranular ross-sions, onsidrd for validaion of h xplii xprssion. Nx, h rsuls hav bn ompard wih xprimnal rsuls rpord by Yu and Winr (96) for simply suppord bams wih T ross-sion (Fiur 7): A-; B-; C-; D-; E-; F- subjd o uniformly disribud loads, and dsinad, hr, as VB5-VB, rspivly. Th ross-sional and marial propris of h bams ar ivn in Tabl 6. Th mid-span dflions obaind from h proposd xplii xprssion ( d ) NN ar shown in Tabl 6 alon wih h rpord xprimnal mid-span dflions ( d ) EXP. Aain, h valus obaind from h proposd xplii xprssion ar in rasonabl armn wih h rpord xprimnal valus of mid-span dflions. Th rsuls (mid-span dflions) obaind from h proposd nural nwork/xplii xprssion nd o b ompard wih h fini lmn rsuls for lihly rinford simply suppord bams ( % ) also. Considr a.65 m lon simply suppord bam VB wih ranular ross-sion (Fiur ) subjd o uniformly disribud load. Th ohr propris ' ar: B = mm; D = 5 mm; f = 7.9 N/mm ; E s =.5 5 N/mm ; A s = 4 mm ; A = Lain Amrian Journal of Solids and Sruurs (5) 54-56

12 553 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 7 mm ; d d b = 35 mm. E and f ar akn in aordan wih ACI 38 (5). Mid-span dflions, for bam VB ar obaind from h proposd xplii xprssion, FEM and ACI 38 (5) for varyin maniud of uniformly disribud loads, w and shown in Fiur 8. Th midspan dflions obaind from h proposd xplii xprssion and FEM ar los for h ran of h load onsidrd. Th diffrn bwn FEM and proposd xplii xprssion is.9% as ompard o 6.8% diffrn bwn FEM and ACI 38 (5) a 4w r,( w r = rakin uniformly disribud load). Df Bf As d Dw RC T ross-sion A db Bw Fiur 7: T ross-sion. Paramrs VB5 (A-) VB6 (B-) Propris of bams VB7 (C-) VB8 (D-) VB9 (E-) VB (F-) B f D f B w D w d d b A (mm ) s A (mm ) ' f (N/mm ) E (N/mm ) E (N/mm ) s w (N/mm) f (N/mm ) L Mid-span dflions d EXP d NN Tabl 6: Propris of simply suppord bams wih T ross-sions, onsidrd for validaion of h xplii xprssion. Lain Amrian Journal of Solids and Sruurs (5) 54-56

13 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams Uniformly Disribud Load (kn/m) 5 5 FEM Explii xprssion ACI 38-5 (5) Mid-span Dflion Fiur 8: Comparison of mid-span dflions of bam VB. Considr anohr simply suppord bam VB subjd o uniformly disribud load wih h sam ross-sional (Fiur ) and marial propris as ha of bam VB xp A. Th valu of A is now assumd as 9 mm. Th los armn is obsrvd bwn h mid-span dflions obaind from h proposd xplii xprssion, FEM and ACI 38 (5) as shown in Fiur 9. 3 Uniformly Disribud Load (kn/m) FEM Explii xprssion ACI 38-5 (5) Mid-span Dflion Fiur 9: Comparison of mid-span dflions of bam VB. 6. Coninuous bams In ordr o valida h proposd xplii xprssion for a oninuous bam, rsuls from h xplii xprssion ar also ompard wih FEM and ACI 38 (5) rsuls for a. m wo qual span uniformly disribud loadd oninuous bam (VB3) wih ranular ross-sion ' (Fiur ). Th ohr propris ar: B = 5.4 mm; D = 3. mm; f = 4. N/mm ; Es.7x 5 N/mm ; As A = mm ; d d b= 5 mm. E and f ar akn in aordan wih ACI 38 (5). Th mid-span dflions obaind from h proposd xplii xprssion and FEM ar los for h ran of h load onsidrd (Fiur ). Th diffrn bwn FEM and proposd xplii xprssion is 5.34% as ompard o 8.5% diffrn bwn FEM and ACI 38 (5) a 5w r. Lain Amrian Journal of Solids and Sruurs (5) 54-56

14 555 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 6 Uniformly Disribud Load (kn/m) FEM Explii xprssion ACI 38-5 (5) Mid-span Dflion Fiur : Comparison of mid-span dflions of bam VB3. Similarly, anohr. m wo qual span oninuous bam VB4 wih ranular rosssion (Fiur ) subjd o uniformly disribud load has bn onsidrd. Th rosssional and marial propris ar akn sam as ha of bam VB3 and only A and A s ar inrasd o mm. Th los armn is obsrvd bwn h mid-span dflions obaind from h proposd xplii xprssion, FEM and ACI 38 (5) as shown in Fiur. 6 Uniformly Disribud Load (kn/m) FEM Explii xprssion ACI 38-5 (5) Mid-span Dflion Fiur : Comparison of mid-span dflions of bam VB4. 7 SENSITIVITY ANALYSIS Th proposd xplii xprssion shows saisfaory prforman on validaion wih xprimnal rsuls availabl in liraur and FEM rsuls. A snsiiviy analysis is arrid ou nx o apur h influn of individual inpu paramrs on oupu paramr usin h proposd xplii xprssion. Th ff of inpu paramrs, Ir I, Mr M alon wih addiional paramrs, n on oupu paramr I I is sudid. Only on paramr (h paramr undr onsidraion) is varid a a im, kpin h ohr paramrs onsan. 7. Eff of As sad arlir, has bn onsidrd as h inpu paramr in h prsn sudy. Fiur shows h variaion of I wih rsp o for various valus of I r, kpin h valu of Lain Amrian Journal of Solids and Sruurs (5) 54-56

15 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 556 M r M onsan as.5. Rih influn of on I I is sn in Fiur. Thouh, h ff is sinifian for all valus of, h ff of lowr valus of is mor sinifian in as of hihr I I. r I /.5.5 I r / r (%) Fiur : Variaion of I wih rsp o. 7. Eff of I r Th variaion of I wih rsp o I r for various valus of is shown in Fiur 3. Th valu of Mr M is kp onsan as.5. Th ff is sinifian only for lowr valus of I r in as of low. Howvr, h ff xnds of h ran onsidrd for I r in as of hihr valus of. Th ff of I r is sinifian for all valus of I /.5.5 r (%) I r / Fiur 3: Variaion of I wih rsp o I r. 7.3 Eff of Mr M As sad arlir, Mr M has bn onsidrd as h inpu paramr affin I I. Th variaion of I wih rsp o Mr M for diffrn valus of is shown in Fiur 4(a). Th valu of I r is kp onsan as.5. Similarly, Fiur 4(b) shows h variaion of h raio I wih rsp o Mr M for diffrn valus of I r. Th valu of is kp onsan as.5. As xpd, h ff of Mr M is sinifian durin rakin ( M r M ) and h valu of I I inrass wih inras in valu of Mr M up o.. Th ff is mor for hihr valu of. Lain Amrian Journal of Solids and Sruurs (5) 54-56

16 557 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 3.5 I /.5.5 r (%).5 I /.5 I r / M r / M M r / M (a) (b) Fiur 4: Variaion of I wih rsp o Mr M for diffrn (a) valus, and (b) I r valus. 7.4 Eff of Fiur 5 shows h variaion of I wih rsp o for diffrn valus of. Th valu of M r M is kp onsan as.5. Th valu of I I is found o inras wih h inras in valu of. A sinifian variaion is obsrvd in as of hihr valu of..5 I /.5 r (%) r (%) Fiur 5: Variaion of I wih rsp o. 7.5 Eff of n Th variaion of I wih rsp o n for diffrn valus of is shown in Fiurs 6(a)-(b) for and ( n ) Mr M is kp onsan as.5. Th naur of plo hans from onav o onvx wih inras in. 8 CONCLUSIONS An xplii xprssion has bn proposd for h prdiion of ffiv momn of inria (and dflion) onsidrin onr rakin, nsion siffnin and nir praial ran of rinformn a srvi load. A s of hr paramrs (, I r, Mr M ) has bn idnifid Lain Amrian Journal of Solids and Sruurs (5) 54-56

17 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 558 ha ovrn h han in I I and hrfor dflion. Usin h samplin poins of hs paramrs and h validad FEM, h daa ss hav bn nrad for rainin, validain and sin of nural nwork. Th xplii xprssion has bn dvlopd from h raind nural nwork. Th proposd xplii xprssion has bn validad for a numbr of simply suppord and oninuous bams and i is found ha h prdid dflions hav rasonabl auray for praial purpos. Snsiiviy analysis has bn arrid ou o apur h influn of individual inpu paramrs on oupu paramr. Th ff of h inpu paramrs, Ir I, Mr M on oupu paramr I is sudid usin h proposd xplii xprssion. Th lowr valus of ar found o hav mor sinifian ff on I I. Th ff of Mr M is found o b sinifian durin rakin ( M r M ) and h valu of I I is found o inras up o. wih inras in valu of Mr M. Th ff of and n is found o b lss sinifian and an b inorporad hrouh I r I /.5.5 r (%) 4.75 I /.5.5 r (%) n (a) n (b) Fiur 6: Variaion of I wih rsp o n for (a) and (b) ( n ) n. Th mhodoloy prsnd hrin an b furhr dvlopd for bams wih poin loads. Th ff of shar dformaion may b inorporad in fuur sudis by onsidrin span o dph raio of bam as an inpu paramr. Similarly, a of loadin and hararisi omprssiv srnh of onr an also b onsidrd as inpu paramrs o aoun for shrinka rakin in fuur sudis. Rfrns ABAQUS 6., Sandard usr s manuals, USA,. Al-Shaikh, A.H., Al-Zaid, R.Z., (993). Eff of rinformn raio on h ffiv momn of inria of rinford onr bams. ACI Sruural Journal 9(): Al-Zaid, R.Z., Al-Shaikh, A.H., Abu-Hussin, M.M., (99). Eff of loadin yp on h ffiv momn of inria of rinford onr bams. ACI Sruural Journal 88(): Amrian Assoiaion of Sa Hihway and Transporaion Offiials (AASHTO), (5). AASHTO LRFD brid dsin spifiaions (SI unis), USA. Amrian Conr Insiu (ACI) 38, (5). Buildin od rquirmns for sruural onr (ACI 38-5) Lain Amrian Journal of Solids and Sruurs (5) 54-56

18 559 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams and ommnary (ACI 38R-5), USA. Azmahullah, M.C., Do, M.C., Dolalikar, P.B., (5). Nural nworks for simaion of sour downsram of a ski-jump buk. Journal of Hydrauli Eninrin ASCE 3(): Bishoff, P.H., (5). Rvaluaion of dflion prdiion for onr bams rinford wih sl and fibr rinford polymr bars. Journal of Sruural Eninrin ASCE 3(5): Bishoff, P.H., (7). Raional modl for alulain dflion of rinford onr bams and slabs. Canadian Journal of Civil Eninrin 34(8): 99-. Bishoff, P.H., Sanlon, A., (7). Effiv momn of inria for alulain dflions of onr mmbrs onainin sl rinformn and fibr-rinford polymr rinformn. ACI Sruural Journal 4(): Branson, D.E., (965). Insananous and im-dpndn dflions of simpl and oninuous rinford onr bams, HPR Rpor No.7(), Alabama Hihway Dparmn, Burau of Publi Roads, Alabama: -78. Canadian Sandards Assoiaion (CSA) A3.3, (4). Dsin of onr sruurs, Canada. Chaudhary, S., Pndharkar, U., Napal, A.K., (7). Bndin momn prdiion for oninuous omposi bams by nural nworks. Advans in Sruural Eninrin (4): Chaudhary, S., Pndharkar, U., Pal, K.A., Napal, A.K., (4). Nural nworks for dflions in oninuous omposi bams onsidrin onr rakin. Iranian Journal of Sin and Thnoloy: Transaions of Civil Eninrin 38(C + ): 5-. Europan Commi for Sandardizaion (CEN) Eurood BS EN 99--, (4). Dsin of onr sruurs-par -: Gnral ruls and ruls for buildins, Blium. Ghali, A., (993). Dflion of rinford onr mmbrs: A riial rviw. ACI Sruural Journal 9(4): Gilbr, R.I., (999). Dflion alulaion for rinford onr Sruurs- Why w somims i wron. ACI Sruural Journal 96(6): Gilbr, R.I., (6). Disussion of Rvaluaion of dflion prdiion for onr bams rinford wih sl and fibr rinford polymr bars by P.H. Bishoff. Journal of Sruural Eninrin ASCE 3(8): Gilbr, R.I., Warnr, R.F., (978). Tnsion siffnin in rinford onr slabs. Journal of Sruural Division ASCE 4(): Gupa, M.K., Sarkar, K., (9).Modlin of sion fors in a oninuous bam usin arifiial nural. Journal of Sruural Eninrin SERC 35(6): Gupa, R.K., Pal, K.A., Chaudhary, S., Napal, A. K., (3). Closd form soluion for dflion of flxibl omposi brids. Prodia Eninrin 5: Gupa, V.K., Kwara, N., Ray, S., (7). Arifiial nural nwork modlin of rp bhavior in a roain omposi dis. Eninrin Compuaions 4(): Hsu, D.S., Yh, I.C., Lian, W.T., (993). Arifiial nural dama dion of xisin sruur, in: Pro. 3 rd ROC and Japan Sminar on Naural Hazards Miiaion Confrn. Kalkan, İ., (). Dflion prdiion for rinford onr bams hrouh diffrn ffiv momn of inria xprssions. Inrnaional Journal of Eninrin Rsarh Dvlopmn (): 7-8. Kaloop, M.R., Kim, D.K., (4). GPS-sruural halh moniorin of a lon span brid usin nural nwork adapiv filr. Survy Rviw 6(334): 7-4. Kanwar, V., Kwara, N., Aarwal, P., (7). Dama dion for framd RCC buildins usin ANN modllin. Inrnaional Journal of Dama Mhanis 6(4): Kim, D.K., Kim, D.H., Cui, J., So, H.Y., L, Y.H., (9). Iraiv nural nwork sray for sai modl idnifiaion of an FRP dk. Sl and Composi Sruurs 9(5): Maru, S., Napal, A.K., (4). Nural nwork for rp and shrinka dflions in rinford onr frams. Lain Amrian Journal of Solids and Sruurs (5) 54-56

19 K.A. Pal al. / Explii xprssion for ffiv momn of inria of RC bams 56 Journal of Compuin in Civil Eninrin ASCE 8(4): MATLAB 7.8, (9). Nural nworks oolbox usr s uid, USA. Min, J., Park, S., Yun, C.B., L, C.G, L, C., (). Impdan-basd sruural halh moniorin inorporain nural nwork hniqu for idnifiaion of dama yp and svriy. Eninrin Sruurs 39: -. Mohammadhassani, M., Nzamabadi-Pour, H., Jumaa, M.Z., Jaml, M., Arumuam A.M.S., (3). Appliaion of arifiial nural nworks (ANNs) and linar rrssions (LR) o prdi h dflion of onr dp bams. Compurs and Conr (3): Mohammadhassani, M., Nzamabadi-Pour, H., Jumaa, M.Z., Jaml, M., Hakim, S.J.S., Zarar, M., (3). Appliaion of h ANFIS modl in dflion prdiion of onr dp bam. Sruural Eninrin and Mhanis 45(3): Pal, K.A., Chaudhary, S., Napal, A.K., (4). Analyial-numrial produr inorporain rakin in RC bams. Eninrin Compuaions 3(5): Pndharkar, U., Chaudhary, S., Napal, A.K., (7). Nural nwork for bndin momn in oninuous omposi bams onsidrin rakin and im ffs in onr. Eninrin Sruurs 9(9): Pndharkar, U., Chaudhary, S., Napal, A.K., (). Nural nworks for inlasi mid-span dflions in oninuous omposi bams. Sruural Eninrin and Mhanis 36(): Pndharkar, U., Chaudhary, S., Napal, A.K., (). Prdiion of momns in omposi frams onsidrin rakin and im ffs usin nural nwork modls. Sruural Eninrin and Mhanis 39(): Sarkar, K., Gupa, M.K., (9). Comparaiv sudy of opimum dsin approahs and arifiial nural nwork basd opimum dsin of a sinly rinford onr bam. Journal of Sruural Eninrin SERC 36(5): Sanlon, A., Orsak, D.R.C., Bunr, D.R., (). ACI od rquirmns for dflion onrol: A riial rviw. ACI Spial publiaions 3-: -4. Sozn, A., Arakliolu, E., Ozalp, M., Kani, E.G., (4). Us of arifiial nural nworks for mappin of solar ponial in Turky. Applid Enry 77: Sandards Assoiaion of Ausralia (SAA) AS 36, (994). Ausralian sandard for onr sruurs, Ausralia. Tadss, Z., Pal, K.A., Chaudhary, S., Napal, A.K., (). Nural nworks for prdiion of dflion in omposi brids. Journal of Consruional Sl Rsarh 68(): Tavakoli, H.R., Omran, O.L., Kuanai, S.S., Shiad, M.F., (4). Prdiion of nry absorpion apabiliy in fibr rinford slf-ompain onr onainin nano-silia parils usin arifiial nural nwork. Lain Amrian Journal of Solids and Sruurs (6): Tavakoli, H.R., Omran, O.L., Shiad, M.F., Kuanai, S.S., (4). Prdiion of ombind ffs of fibrs and nano-silia on h mhanial propris of slf-ompain onr usin arifiial nural nwork. Lain Amrian Journal of Solids and Sruurs (): Turkish Sandards Insiu TS 5, (). Rquirmns for dsin and onsruion of rinford onr sruurs, Turky. Uddin, M.A., Jaml, M., Razak, H.A., Islam, A.B.M., (). Rspons prdiion of offshor floain sruur usin arifiial nural nwork. Advand Sin Lrs 4(): Washa, G.W., Fluk, P.G., (95). Eff of omprssiv rinformn on h plasi flow of rinford onr bams. ACI Journal 49(): Yu, W.W., Winr, G., (96). Insananous and lon-rm dflion of rinford onr bams undr workin loads. ACI Journal 57(): 9-5. Lain Amrian Journal of Solids and Sruurs (5) 54-56