PROBABILISTIC RESTORATION DESIGN FOR RC SLAB BRIDGES

Transcription

1 - Tehnl Pper - PROBABILISTIC RSTORATION DSIGN FOR RC SLAB BRIDGS Abrhm Gebre TARKGN *, Ttu TSUBAKI * ABSTRACT In e o oler brge, eng muh egn normton poble ult. Retorton egn n mportnt metho to etmte the ntl onton o truture. In th tu, probblt retorton egn, be on ltn hperube mplng (LHS) o rnom vrble, h been perorme or RC lb brge. Unertnt n entvt nl o rnom vrble re lo perorme. From the nl reult, t oberve tht meurement o totl epth n ompreve trength o onrete re the mot nluenng tor n the etmton o el trength o teel. Kewor: Retorton egn, rnom vrble, unertnt, entvt nl. INTRODUCTION Cpt perormne ement o brge one o the mportnt tk n brge mngement le. To omplh th tk, egn oument, peton n tnr re o gret mportne. In the bene o thee t, eng brge onton ult. Thu, ntl onton nee to be retore. Retorton egn metho o etmtng ntl onton o truture n t ete b urrent meurement. Sne meurement re next n ontn error o unknown mgntue, ther eet on the retorton egn hve to be nvetgte. In th tu, or uh nvetgton, probblt retorton egn ue. A probblt pproh o retorton egn o RC lb brge onerng erent ombnton o rnom vrble, ther nluene n unertnte be on the onept o entvt nl re nvetgte. There re erent rnom vrble whh et the retorton egn proe. To perorm retorton egn, erent metho n be pple. In th tu, probblt pproh o retorton egn onerng erent ombnton o rnom vrble be on the Ltn Hperube Smplng (LHS) metho ue. The probblt trbuton n the nluene o rnom vrble on the tttl vrton o the etmte el trength o teel, ro etonl re o teel n eetve epth re nvetgte. oreover, be on the tttl vrton o rnom vrble, entvt nl n the vnt o men vlue onute.. RSTORATION DSIGN Retorton egn proe o urtel erbng the ntl onton (egn vlue) o truture rom t urrent onton (tul vlue). Retorton egn ue egn vlue. On the other hn, eleton ue tonl t, t relet the preent tul vlue. Alo n nret metho or the etmton o el trength o teel neee. The requre menon o the truture re etermne through meurement []. Retorton egn, whh mportnt to etmte the ntl onton o brge, b tool or pt perormne ement. Non-etrutve tet or the etmton o urrent onrete trength, m-pn eleton o the brge rom lo tet n poton o renorng br ung n eletr mgnet eve re mn nput or the retorton egn []. Be on the low o retorton egn o RC lb brge o known eetve epth [], the low hrt or probblt retorton egn hown n Fg.. For unknown eetve epth, lo tet t let t two poton houl be one. For more urte reult, mn meurement re neee.. F Smulton For tttl nl, two e (eh onton ompre o 3 ombnton) wth totl o 64 RC lb brge re multe ung F. The meh ont o retngulr element o.5m n ze wth 4 noe n SBet mterl moel o ATNA D v 4.. [] ue. The outome o eh e, eet o the rnom vrble n the probblt trbuton o the outome re nlze n plotte. Tble how the two e onere n F multon. Tble Two e o F multon Ce Ce Norml onton xtreme onton Slp ble t br Slp llowe t br begnnng n en pont begnnng n en pont (ull nhore br) The nput prmeter ue n the F multon re brge menon n mterl properte. Thee rnom vrble re ontnuou n ontn mll tter. * Grute Shool o ngneerng, Yokohm Ntonl Unvert, JCI Stuent ember * Proeor, Fult o Urbn Innovton, Yokohm Ntonl Unvert, JCI ember

2 START Degn Retorton Inpeton t, eleton, rkng, eterorton,... Degn vlue mprl reltonhp Atul vlue Dret metho Conrete trength, L, D, B Strength evelopment urve mprl reltonhp Conrete trength, Brge menon: B, D, L Deterorton onton Deleton F multon Tet hmmer eurement n obervton Stttl rnom vrble =. 43γ. 5 =GP mprl reltonhp = α + α o =. 43γ. 5, Are o teel, A, Corroon rte = α + α o tmte A n Deleton equton Degn moment, mprl reltonhp =α +α o oment pt, Aume A n Yel trength o teel, Revew o egn tnr n peton mprl reltonhp = α + α o Yel trength o teel, ND No Compute NA epth n I e Compute eleton,,,... [Note] Superrpt egnte tul vlue I - eleton obtne rom F multon (Lo tet) =? - eleton ompute ung eleton equton Ye I e - eetve moment o nert be on ume vlue o A n Ok α, α, α n α - mgnton tor * α o, α o, α o n α o - ht o the trbuton urve * * For mplton, mmetr probblt trbuton wth no ht n wth mgnton tor o one ume. Fg. Flow o retorton egn o RC lb brge. tmton o Yel Strength o Steel mprl ormule or the etmton o el trength o teel b onerng the eet o elng moment wth trngulr eton, onrete trength, Young moulu o teel n moment pt re obtne [3]. In reerene to Fg., or oubl renore eton, the ollowng equton hol true. + = () / = A ( 3) () = A ( ) (3)

3 A = ( β + βη) B = 3. 9α ( β + 3βη) ( + α ) m C = α ( + α ) m η = ( -D)/ α = / ( = GP ) Fg. RC eton n trn grm ε = (4) ε + ε To obtn the vlue o A, tonl lo tet t erent poton houl be onute. A per AASHTO LRFD Brge Degn Speton [3], the mnmum mount o top renorement or hrnkge n temperture or RC lb brge to be prove gven ollow. ε = /, ε = / (5) A φa g, φ =. 75P (). 5 =. 43γ (6) From q. 9 n, β n be ompute. t = = D (7) A A + A = (8) A = ( ) A (9) βa A = β : benng moment rre b the onrete n prt o tenle teel (N-mm/m) : benng moment rre b A n ret o tenle teel (N-mm/m) : ultmte egn moment (N-mm/m) : tul lexurl pt o the eton A A (N-mm/m) : tul re o tenle teel (mm /m) : tul re o ompreon teel (mm /m) : tul el trength o teel (P) : urrent onrete trength (P) : Young moulu o teel (GP) : Young moulu o onrete (GP) γ : unt weght o onrete (4kN/m 3 ) β : rto o A to A m = / A (N/mm ) α = / Upon ubttuton n mplton, q. mple to the ollowng orm. A( ) + B + C = () φag β =, φ =.75P () A Subttutng q. nto q., gve the ollowng mple orm. ( ) + B + C = (3) B = 6. 9α φα ξ ( + α ) m C = α ( φαϕ ( + α ) m ) α : rto o A g to A A g : gro re o the eton=bd (mm ) ξ = ( D ) / ϕ = ( 4 3D) / For ngl renore eton, the vlue o β zero. Ue regreon nl to obtn generl emprl equton or the etmton o tul el trength o teel. Thu, the ollowng emprl reltonhp ung qurt nterpolton [] h been obtne. =. 5(+ α ) m +. 53α +. 4)( + α ) m 3. STATISTICAL ANALYSIS (. 99 (4) 3. Probblt Anl To oner the probblt trbuton o rnom vrble on eet o the retorton egn reult, the onept o norml trbuton ue. The umpton onere n the nl re rnom

4 vrble re nepenent o eh other n the ollow norml trbuton. The probblt unton, (x), ung norml trbuton [4] gven n q. 5. ( x µ ) / σ ( x) = e (5) σ π x : rnom vrble µ : men vlue σ : tnr evton The trbuton o over thkne o mn renorng br n probblt nl o rnom vrble o tnr RC lb brge, b ung LHS metho, re perorme. Quntttvel, or the RC lb brge, entvt o the prmeter n the etmton o tul el trength o teel rom q. 4 nlze. In the LHS mplng metho, the umultve trbuton unton o eh tor ve nto ntervl wth equl probblt, n then mplng one b onl rom eh ntervl [5]. The rrngement o mplng ntervl n mplng o rnom vrble hown n Fg. 3 [5]. The 3 ombnton o rnom vrble o LHS tble re uent n th multon. A tnr RC lb brge wth enter to enter length o.4m egne per AASHTO LRFD Brge Degn Speton [3] n multe ung F. To obtn the mxmum lve lo eet on the truture, the onept o nluene lne ue. The rnom vrble ue n th tu re pn length, eetve trp wth, totl epth n ompreve trength o onrete. The llowble lmt o meurement b AASHTO LRFD Brge Contruton Speton [6] re onere. In the multon, hl o the llowble lmt permtte re onere tnr evton (σ ). The tttl prmeter o rnom vrble re hown n Tble. The mterl properte ue re: =4P n =GP. Dmeter 3mm renorng br wth / png o 8mm n over thkne o 35mm re ue. A two-pont lo wth / png o.m n three lo poton (loton o rer lo re t.m, 3.m n 4.m rom the let upport) re onere. Be on the reult, nrementl ntntneou m-pn eleton ue to pple lo, the tul re o renorng br n eetve epth re ompute ung the elt eleton equton b trl n error proeure. The retorton egn heme hown n Fg. 4. Fnll, the tul el trength o teel etmte ung q. 4. For the etmton o, the ultmte egn moment,, o 75kN-m/m ue. Tble Stttl prmeter o rnom vrble No. Rnom vrble en St. vlue ev. Spn length (mm) 4 5. etve wth (mm) Totl epth (mm) Compreve trength o onrete (P) 8.4 Fg. 3 Arrngement o mplng ntervl n mplng o rnom vrble Reult o tttl nl howng men vlue, tnr evton n oeent o vrton o the et o rnom vrble hown n Tble 3. Tble 3 Reult o tttl nl tmte vlue (Ce ) tmte vlue (Ce ) (mm) A (mm /m) (mm) A (mm /m) (P) (P) en St. ev COV Preton prt en vlue o L, D, B n Lo, 3 omb. Deleton F prt or eleton (LHS, 3 ombnton o rnom vrble) Fg. 4 Retorton egn heme

5 The rto ( A o Ce ) to ( A o Ce ) ume to be equl to the rto o the tul to ultmte moment pt o the eton, /. Thu, or Ce, α o -.3 ue n the orreponng vlue o, A n re etmte orngl. The probblt trbuton o the etmte tul el trength o teel n re o teel, or both onton, ollowng the norml trbuton unton re hown n Fg. 5 n Fg. 6, repetvel. oreover the umultve perent trbuton re plotte. Probblt ent (%) Probblt_e Probblt_e Cum. perent_e Cum. perent_e Cumultve perent -3.57% rom the tul over thkne. Almot the me reult obtne or Ce. Cumultve perent Tble 4 Conene ntervl or Ce (P) Lower lmt Upper lmt Ce Ce Ce_ Ce_ Yel trength o teel, P Fg. 5 Probblt trbuton o Cover thkne, mm Fg. 7 Dtrbuton o over thkne Probblt ent (%) Probblt_e Probblt_e Cum. perent_e Cum. perent_e Cumultve perent 3. Sentvt Anl Sentvt nl the tu o how the vrton (unertnt) n the output o tttl moel n be ttrbute to erent vrton n the nput o the moel. The entvt o eh rnom vrble repreente b the qure vlue o the prtl p oeent o orrelton ( r ). The entvt tor α be on the rt-orer pproxmton eon-moment metho [7] ue. The eet o rnom vrble on the etmte tul el trength o teel nvetgte. To etermne ther eet, entvt nl perorme. Are o teel, mm /m Fg. 6 Probblt trbuton o A F x α = ( =,,3,...,n) (6) x F Be on the reult, onene lmt or the men vlue wth 95 % onene level re etmte. Thu, the onene ntervl or the men vlue o re hown n Tble 4. Ung the etmte eetve epth, the over thkne o the multe brge or both e re ompute n ther umultve perent trbuton re plotte n Fg. 7. For Ce, n verge onrete over o 33.75mm wth tnr evton o 3.mm n COV o 9.8% obtne, whh h vrton o α : entvt tor o rnom vrble x, F : men o x n F, repetvel F : unton wth tttl vrton : rnom vrble x The entvt tor α kn o nex to etmte the ontrbuton o the unertnt o x to the unertnt o F. Ung the entrl erene pproxmton equton, the unertnt o eh rnom vrble n be obtne [7].

6 . 5 F F( x + x) F( x x) = x x (7) x : nnteml prt o eh rnom vrble In th nl, / o the men vlue tken x. The ontrbuton o the unertnt o eh rnom vrble re obtne b multplng the entvt tor b t oeent o vrton. U.F = α ( COV ) (8) U.F : the ontrbuton o the unertnt o rnom vrble COV : the oeent o vrton or rnom vrble : entvt tor o rnom vrble α For entvt nl o rnom vrble, the entrl pproxmton equton gven n q. 7 ue. Th entvt meure gve the hnge n ue to hnge o one o the rnom prmeter. The entvt n ontrbuton o the unertnt o rnom vrble on re ompute ung q. 6 n 7, repetvel. The ompron o unertnt o rnom vrble o the two e hown n Fg. 8. Ce_ Ce_ Unertnt L Fg. 8 Unertnt o rnom vrble r p Ce_ Ce_ Rnom vrble D B L B D Fg. 9 Sentvt o rnom vrble A hown n Fg. 8, the lrget ontrbuton to the etmte vlue o el trength o teel re the hnge n omprehenve trength o onrete n totl epth o the brge. The ontrbuton o eetve wth ngnnt n t neglgble ompre to other prmeter. The entvt o eh rnom vrble, the qure vlue o the prtl oeent o orrelton p ( r ), lulte n t hown n Fg. 9. From the reult, t oberve tht the nluene o totl epth h omnnt tor or the etmte vlue o el trength o teel. The nluene o pn length n trp wth re mll n n be neglgble. 4. CONCLUSIONS () A probblt egn retorton heme or RC lb brge preente. () Wth the probblt egn retorton heme, not onl the men vlue but lo the onene lmt re obtne. (3) For the tpl RC lb brge e, wth 95 % onene level, onene ntervl o +.76% n +3.7% o the men vlue o or Ce n Ce re obtne repetvel. (4) From the entvt nl, t oberve tht hnge n omprehenve trength o onrete n totl epth o the brge re the mot nluenng tor n the etmton o el trength o teel. (5) A ltn hperube mplng (LHS) metho ue to mprove the omputtonl een n the ur n the etmton o. RFRNCS [] Trekegn, A.G., Tubk, T., Retorton Degn or RC Slb Brge b AASHTO LRFD, t PC mpoum, JPCI, Ot., pp [] Červenk, V., Jenele, L., n Červenk, J., ATNA Progrm Doumentton, Cervenk Conultng Lt., Prgue, rh. [3] AASHTO, Amern Aoton o Stte Hghw Ol, LRFD Brge Degn Speton, 4 th eton, Whngton, 7. [4] Shrle D., Sttt or Reerher, 3 r e, John Wle n Son, In., 4. [5] Tubk, T., Sentvt Anl, Trn. o the Jpn Conrete Inttute, Vol., 989, pp97-4. [6] AASHTO, Amern Aoton o Stte Hghw Ol, LRFD Brge Contruton Speton, Whngton,. [7] Tubk T., Ihr T. n Yoh., Stttl Vrton n oelng o Drng Shrnkge o Conrete, Trn. O the Jpn Conrete Inttute, Vol. 4, 99, pp. 3-3.