Distortional Analysis of Thin-Walled Box Girder Structure: A Comparative Study

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1 Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5): 8788 Scholarln Rsarch Insttt Jornals, (ISSN: 76) jtas.scholarlnrsarch.org Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5) 8788 (ISSN: 76) Dstortonal Analss of ThnWalld Box Grdr Strctr: A Comparatv Std Chdol, Chnn A., Osadb, Nmaonam N., and Agnam, Chwra H. Dpartmnt of Cvl Engnrng, Nnamd Aw Unvrst, Awa, Anambra, Stat Ngra Dpartmnt of Cvl Engnrng, Unvrst of Ngra, Nsa, Eng Stat, Ngra Corrspondng Athor: Chdol, Chnn A Abstract In ths papr th dffrntal qatons of dstortonal qlbrm for th analss of dobl smmtrc box grdr sctons, basd on Vlasov s thor of thnwalld box grdr strctrs, wr drvd and sd to valat th translatonaldstortonal dsplacmnts of a 5m span, smpl spportd rctanglar box grdr fram. B modfng Dbrows s dffrntal qaton for dstortonal analss of crvd thnwalld box strctr, th qvalnt qaton for straght thnwalld box grdr strctr was obtand and sd to valat th rotatonaldstortonal dformaton of th nmrcal std box grdr problm. Th prpos of th std was to obtan th translatonal and rotatonal componnts of a dobl smmtrc box grdr scton and to obtan a rlatonshp btwn th two dsplacmnt componnts. Th std was basd on th prms that f a mathmatcal rlatonshp btwn th translatonal and rotatonal componnts of th dstortonal dsplacmnt s stablshd, t wll wdn th scop of ndrstandng of th dstortonal bhavor of thnwalld box grdr strctrs. B comparng Vlasov s dstortonal dsplacmnts wth Dabrows.s dstortonal dsplacmnts t was fond that thr was no rlatonshp btwn th two dsplacmnt qantts. Kwords: Dstorton, translaton, rotaton, thnwalld box grdr, Dabrows s qaton, Vlasovs s thor. INTRODUCTION Dabrows (968), on th bass of statc qlbrm of thnwalld box strctr, obtand th followng qn. () for dstortonal analss of crvd thn walld box grdr strctrs. v M x m * () W A R whr, = Anglar dstorton n radans M x = Torsonal momnt abot xaxs m = Dstrbtd torsonal load pr nt lngth of grdr axs = Dca coffcnt for cross sctonal dformaton * WA = Dstortonal rgdt = Dmnsonlss shap coffcnt R = Rads of crvatr of grdr Hns (976), on th bass of Dabrows s formlaton obtand a fnt dffrnc solton for th bam on lastc fondaton qaton and gav a nmbr of crvs for qc dtrmnaton of th * WA for analss and dsgn of dstortonal proprt thnwalld box strctrs. Osadb and Mbajorg (6) obtand a forth ordr dffrntal qaton for dstortonal analss of thnwalld box grdr strctrs sng Vlasov s thor (96). Ths, v qd ( x) V ( x) V ''( x) V ( x) () EI whr, V ( x) fncton govrnng dstortonal dsplacmnt n th longtdnal drcton, = Nmrcal factors qd Dstortonal load EIw Dstortonal warpng rgdt As wll b sn latr, th xprssons for th valaton of th coffcnts for Dabrows s dffrntal qaton of qlbrm () ar qt complx and rqr xtnsv calclatons to dtrmn th paramtrs. Smlar problms ar ncontrd b sng Vlasov thor of thnwalld box strctrs. Thrfor thr s th nd to hav a dffrntal qaton of dstortonal qlbrm for thn walld box strctr whos coffcnts ar as to obtan and applcaton has a masr of as. Ths, howvr, was not th prpos of ths wor. Th prpos of th std was to obtan th translatonal and rotatonal componnts of a dobl smmtrc box grdr scton and to obtan a rlatonshp btwn th two dsplacmnt componnts. w 87

2 Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5) 8788 (ISSN: 76) Th dffrntal qaton for dstortonal analss of thnwalld dobl smmtrc (rctanglar) box grdr strctr was drvd sng Vlasov s thor. Also, Dabrows s qaton for dstortonal analss of crvd thn walld box grdr strctr basd on statc qlbrm was modfd to obtan th qvalnt qaton for straght grdrs. Nmrcal std was carrd ot sng th drvd qaton and th modfd Dabrows s qaton to obtan dstortonal dformatons of a smpl spportd, sngl cll rctanglar box grdr strctr. It shold b notd that Vlasov s approach dals wth lnar dsplacmnts whl that of Dabrows dals wth anglar dsplacmnts. Th rslts obtand from th two approachs wr xamnd to s f thr was an (mathmatcal) rlatonshp btwn th two dsplacmnt qantts. FORMULATION OF DIFFERENTIAL EQUATIONS OF EQUILIBRIUM FOR DOUBLY SYMMETRIC SECTIONS Gnral Exprsson for Dffrntal Eqatons of Eqlbrm Th gnral xprssons for th dffrntal qatons of dstortonal qlbrm for box grdrs drvd b Chdol () basd on Vlasov s thor ar gvn b; m m n a U ''( x) b U ( x) c V '( x) (a) j j j c h U '( x ) r h V ''( x ) shv ( x) / G q h (b) whr, qh dstortonal load pr nt wdth of grdr U ( x) dstortonal dsplacmnt n longtdnal drcton. V ( x) dstortonal dsplacmnt n transvrs drcton. a j, b j, c j, c h, r h, s h ar Vlasov s coffcnts for th dffrntal qatons of qlbrm and ar xprssd as follows; a a (a) j j j b b ' ' (b) j j j cj c j ' j (c) ch ch ' (d) rh rh h ; () sh sh (/ E) M ( s ) M h( s ) / EI ds (f) (5) whr, and = lmntar dsplacmnts of th strp fram, rspctvl ot of th plan (m dsplacmnts) and n th plan (n dsplacmnts). ', j ' ar th drvatvs of and j rspctvl. M s th transvrs dstortonal bndng momnt d to dstortonal load. A = th cross sctonal ara of box fram and E s th modls of lastct. Stran Mods Dagrams for Comptaton of Vlasov s Coffcnts Fg. (a) shows th dmnsons of a sngl cll dobl smmtrc box grdr scton sd for nmrcal std. Fgs. (b) to (h) ar th gnrald stran mod dagrams for th fram. (Fg.c) s a proprt of th cross scton obtand b plottng th dsplacmnt of th mmbrs of th cross scton whn th vrtcal () axs s rotatd throgh a nt radan. Smlarl, (Fg.) s a proprt of th cross scton obtand b plottng th dsplacmnt of th mmbrs of th cross scton whn th horontal () axs s rotatd trogh a nt radan. Th warpng fncton (Fg. g), of th bam cross scton s obtand as dtald n ltratrs: Ratch (978), Chdol and Osadb (), Osadb and Chdol (). and ar nplan dsplacmnts of th cross scton n x and x plans rspctvl whl s th dstorton dagram for th cross scton. (Fg.b), s th dsplacmnt dagram of th grdr cross scton whn th fram s rotatd on radan n sa, a clocws drcton abot th cntrodal axs. Ths, s drctl proportonal to th prpndclar dstanc (rads of rotaton) from th cntrodal axs to th mmbrs of th cross scton. In an npblshd wor, Osadb has shown that nplan dsplacmnt qantts, and ar th sam as th drctvs of thr corrspondng ot of plan dsplacmnts. Consqntl,, and ar obtand b nmrcal dffrntaton of,, and dagrams rspctvl. Vlasov s coffcnts, qn. (5) wr valatd sng th stran mods dagrams, Fgs. and, on th bass of Morh s ntgral for dsplacmnt comptaton. m 7.m mm (a) Sngl cll doblsmmtrc scton mm ψ (b) Rotaton dagram.6 875

3 Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5) 8788 (ISSN: 76) Th (s) (c) Longtdnal stran mod (d) Transvrs stran dagram (bndng abot mod n drcton axs).6. s Coffcnts h Th coffcnt s h () Longtdnal stran mod dagram ( bndng abot axs)..8ei.8ei =ω M (g) Warpng Fncton.. M (S).8 ' =ψ ' =, gvn b qn. (5f), dpnd on th bndng dformaton of th strp charactrd b M (for =,,, ). To compt th coffcnts w nd to constrct th dagram of th bndng momnts d to th stran mods,. (f) Transvrs stran mod n drcton ' = (h) Dstorton dagram Fg. Gnrald stran mods for sngl cll dobl smmtrc scton.8ei Fg. Bndng momnt dagram d to dstorton of sngl cll dobl smmtrc scton.8ei ,, and,. Incdntall,, and stran mods do not gnrat dstortonal bndng momnt as th nvolv pr bndng and pr rotaton. Onl stran mod gnrats dstortonal bndng momnt whch can b valatd sng th dstorton dagram Fg. (h). Consqntl th rlvant xprsson for th coffcnt bcoms M M s s h h (6) E s EI s whr M ( s ) s th dstortonal bndng momnt of th cross scton. Th procdr for valaton of dstortonal bndng momnt s avalabl n ltratrs [5], [6]. Th bndng momnt dagram for dstorton of sngl cll dobl smmtrc fram of Fg. (a) s shown n Fg.. Dstortonal Eqlbrm Eqaton In ths scton, spcfc xprssons for dffrntal qatons of dstortonal qlbrm ar obtand for th dobl smmtrc box grdr strctr sng qns (a) and (b). Th rlvant coffcnts for flxral dstortonal qlbrm for dobl smmtrc sctons ar a b c r b and r. All othr,,,,,, c, s ntractng coffcnts; a, b, c, r, ar ro bcas of smmtr n major and mnor axs drctons. Sbstttng ths rlvant coffcnts nto th matrx form of qns (a) and (b) w obtan '' U U '' a U b U a '' U b U ' V ' V _ c ' (7a) c V ' V and ' U V c ' V U c ' U s + V V '' V '' q r V q r '' (7b) V G q '' V q 876

4 Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5) 8788 (ISSN: 76) Expandng th abov matrx qatons w obtan a U '' b U c V ' (a) a U '' b U c V ' (b) c U ' r V '' q G (c) / c U ' r V '' s V q G (d) (8) / Smplfng b lmnaton of U, U and thr drvatvs w obtan; v / V q a E (9) v '' V V V K () whr, s / r, / s a and K q / Ea Eqns (9) and () ar not copld wth rspct to V and V sgnfng that dstortonal dformaton V s ndpndnt of mnor axs flxral dformaton,v. Hnc th two qatons can b solvd ndpndntl. Eqn. () s th dffrntal qaton of qlbrm for dstortonal analss of sngl cll dobl smmtrc box grdr profls basd on Vlasov s thor. Dabrows s Formlaton Th dstorton of a box grdr scton, as dvlopd b Dabrows [], as shown n Fg.(b), s gvn b qaton () v M x m * () WA R whr K () * WA * Ea b Av A WA 8 () A / a A v A Av () EI K v b (5) a b I I Iv / / ( I I ) / Iv 6 b / a II / Iv I I v v I (6) (7) (8) (9) / () 7o Ao bo b v I x Ix () bv R R o () 5 Ix Io I / Iv 6 b/ aioi / I v R a b I / Iv () R a b Io / Iv () A a (5) Av bv (6) A a (7) Th paramtrs n th abov xprssons ar rfrncd to Fgr and th followng. = Anglar dstorton, radans = Anglar ordnat of cross scton K = Fram stffnss,, = Dmnsonlss cross sctonal paramtrs for dformaton of a rctanglar scton Io, Iv, I = Plat stffnss vals for top flang, a vrtcal wb lmnt and th bottom flang, rspctvl Ao, Av, A = Ara of top flang, a vrtcal wb lmnt and th bottom flang, rspctvl. = posson s rato a, b,,,,, v, and ar cross sctonal dmnsons gvn n Fg.. o v o a Fg. (a) Box grdr dmnsons x v Fg. (b) Dabrows s Modl of Dstortd box scton showng rotatonal dsplacmnt b 877

5 Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5) 8788 (ISSN: 76) For a straght grdr th rads of crvatr R =, and qn. () gvs; m( x) v ( x ) ( x ) WA (8) Eqn. (8) s Dabrows s dffrntal qaton for dstorton of straght grdrs of rctanglar profl wth or wthot flang ovrhang. From qn.() w hav / WA * Sbstttng n qn.(8) gvs; v ( x ) / WA * ( x ) m ( x ) / WA * (9) A comparson of rght hand sds of Dabrows s modfd qn.(6) and Vlasov s qn.() shows that, Ea b Av A WA* EI w 8 Thrfor, WA* EI w / Usng th dmnsons n Fg.(a) w obtan th Vlasov s warpng rgdt as follows. a b a b Iw a b v A Av EI Ea b WA* w A Av 8 (Vlasov) () Ea b A A Bt v WA * (Dabrows) () 8 For a box wth no flang ovrhang =, and from qn. () w obtan; A Av () A Av For nform thcnss of top and bottom flangs, and Ao A.. Sbstttng n qn () gvs; Ea b WA* Av A 96 (Dabrows) () From qns. () and (), w obsrv that Vlasovs warpng rgdt s twc as mch as Dabrows s dstortonal rgdt for a dobl smmtrc scton wth nform flang thcnss. Dstortonal Analss: Nmrcal Std W consdr a smpl spportd m span box grdr strctr wth cross scton shown n Fg. (a). Th dstortonal componnt of th appld torsonal load comptd sng AASHTOLRFD (998) spcfcatons was 6.5KN. Vlasov s Mthod Th govrnng qaton for dstortonal analss s v '' qn. (),.., V V V K () Bondar condtons: V () V ( L) ; V ''() V ''() Th rlvant coffcnts obtand b dagram mltplcaton sng Eqn. (5) and Fgs. and ar; a 6.7; c.76 ; 9 Tang E * N / m ;.5 q 6.5KN, w obtan;.58* K.86 * 6 s.6 * 7.*, Sbstttng nto qn. () w obtan v 6 V 7.* V ''.58* V.86* (5) Solvng qn. (5) and applng th bondar condtons w obtan;.58 * Snh xcos x Cosh xcos x (6) V ( x)..56 * Cosh xsn x whr,.8radan ;.78radan Dabrows s Mthod Dabrows s dffrntal qaton for dstortonal analss of rctanglar profls s gvn b qn.(7). v ( x ) / WA * ( x ) m ( x ) / WA * (7) Th paramtrs for qn. (7) wr valatd wth rfrnc to Fgs.(a) and (a), sng qns () to (7). Ths, a = 7.m, b =.9m, =.m,.m, v.m A A.m ; I Av.58m x 6.55m, Io Iv I 7.* m.8,.59 ; ;. K.79 * E ; WA* 8.99E ;.5 (for concrt) From qn () w obtan;.8* (8) Sbstttng ths paramtrs nto qn.(7) whl tang m = 6.5KN, w obtan; v 6 ( x ).8* ( x ).59 * (9) Eqaton (9) s Dabrows s dffrntal qaton for dstorton of box grdr Fg. (a). Solvng qn. (9) b mthod of gomtrc srs wth acclratd convrgnc w obtan; ( x) 6.96* Sn( x / 5) () 878

6 Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5) 8788 (ISSN: 76) DISCUSSION OF RESULTS.58 * V ( x)..56 * Cosh xsn x whr Snh xcos x Cosh xcos x.8 radan ;.78 radan () ( x) 6.96* Sn x /5 () Eqatons () and () ar th soltons to th dstortonal qatons of qlbrm () and (7). Th xprss th dformaton of th 5m smpl spportd box grdr Fg. (a), accordng to Vlasov s thor and Dabrows s statc qlbrm approachs, rspctvl. Dstortonal Dsplacmnt (mm) Dstanc Along th Lngth of th Grdr (m) Tabl shows th comptaton of th dstortonal dsplacmnts (Vlasov s translatonal dsplacmnts and Dabrows s rotatonal dsplacmnts) whch ar graphcall prsntd n Fgs. (a) and (b) rspctvl. It was notd n scton. that Vlasov s warpng rgdt was twc as mch as Dabrows s dstortonal rgdt. Howvr, whl Vlasov s dsplacmnt fncton V, s a lnar masr, that of Dabrows s anglar masr. It thrfor follows that a box grdr strctr has dstortonaltranslatonal rgdt twc as mch as ts dstortonalrotatonal rgdt. Th maxmm (mdspan) dstortonaltranslatonal dsplacmnt for th box grdr nmrcal xampl was 6.88mm whl th maxmm dstortonalrotatonal dsplacmnt was 6.96 radans. As can b sn from Fg. (b), dstorton altrs th gomtr of th box grdr cross scton, mang t dffclt to obtan a rlatonshp btwn dstortonaltranslatonal dsplacmnt and dstortonalrotatonal dsplacmnt of th box grdr scton. Consqntl, Vlasov s dstortonaltranslatonal dsplacmnt has no rlatonshp wth Dabrows s dstortonalrotatonal dsplacmnt. Tabl : Dstortonal dsplacmnts Dstanc (x) from Lft Spport (m) Lnar dsplacmnts accordng to Vlasov s thnwalld thor V (x) x (m) Anglar dsplacmnts accordng to Dabrows s statc thor γ( x) x (radans) Dstortonal Dsplacmnt (x. radans) Fg. (a): Varaton of Vlasov s Dstortonal Dsplacmnt along th grdr Dstanc Along th Lngth of th Grdr (m) Fg. 5(a): Varaton of Dabrows s Dsplacmnts along th grdr V 6.88mm 6.96rad V 6.88mm Fg. (b) Vlasov s gnrald contor dformaton of th rctanglar cross scton showng translatonal dsplacmnt at mdspan Dstortonal 879 Fg. 5(b) Dabrows s modl of dstortd box grdr scton showng rotatonal dsplacmnt at mdspan

7 Jornal of Emrgng Trnds n Engnrng and Appld Scncs (JETEAS) (5) 8788 (ISSN: 76) CONCLUSION Eqatons () and (7) ar th drvd dffrntal qatons for dstortonaltranslatonal and dstortonalrotatonal dsplacmnt analss, rspctvl, of a dobl smmtrc box grdr strctr. Dstorton altrs th gomtr of th box grdr cross scton mang t dffclt to obtan a rlatonshp btwn dstortonaltranslatonal dsplacmnt and dstortonalrotatonal dsplacmnt. Howvr, t was ascrtand that a box grdr strctr has dstortonaltranslatonal rgdt twc as mch as ts dstortonal rotatonal rgdt. REFERENCES Amrcan Assocaton of Stat Hghwa and Transportaton Offcals (AASHTO), 998, Load and Rsstanc Factor Dsgn, LRFD, brdg dsgn Spcfcatons, Washngton, D.C. Chdol, C.A.,, Torsonaldstortonal analss of thnwalld box grdr strctrs sng Vlasov s thor, Ph.D. thss, Unvrst of Ngra Nsa Chdol, C.A., and Osadb, N.N.,, Flxraltorsonal bhavor of thnwalld mono smmtrc box grdr strctrs, Intrnatonal Jornal of Engnrng Scncs and Emrgng Tchnologs, : (9) Dabrows, R., 968, Crvd thnwalld grdrs Translatoon No., Cmnt and Concrt Assocaton, London. Hns, C.P., 976, Bndng and torson dsgn n strctral mmbrs, Lxnton Boos, Massachstts. Osadb, N.N., and Mbajorg, M.S.W., 6, Dstortonalanalss of thnwalld box grdrs, Ngran Jornal of Tchnolog, 5: (65). Osadb, N.N., and Chdol, C.A.,, Torsonaldstortonal rspons of thnwalld mono smmtrc box grdr strctrs, Intrnatonal Jornal of Engnrng and Applcaton, : (88). Rach. V.G., 978, Statc thor of thn walld spac strctrs, Mr Pblshrs, Moscow Vlasov, V.Z., 96, Thn walld lastc bams, nd Ed., Natonal Scnc Fondaton, Washngton, D.C. 88

Variational Approach in FEM Part II

Variational Approach in FEM Part II COIUUM & FIIE ELEME MEHOD aratonal Approach n FEM Part II Prof. Song Jn Par Mchancal Engnrng, POSECH Fnt Elmnt Mthod vs. Ralgh-Rtz Mthod On wants to obtan an appromat solton to mnmz a fnctonal. On of th