4.2 Design of Sections for Flexure

Transcription

1 4. Dsign of Sctions for Flxur This sction covrs th following topics Prliminary Dsign Final Dsign for Typ 1 Mmbrs Spcial Cas Calculation of Momnt Dmand For simply supportd prstrssd bams, th maximum momnt at th span is givn by th bam thory. For continuous prstrssd bams, th analysis can b don by momnt distribution mthod. Th momnt cofficints in Tabl 1 of IS: can b usd undr conditions of uniform cross-sction of th bams, uniform loads and similar lngths of span. Th dsign is don for th critical sction. For a simply supportd bam undr uniform loads, th critical sction is at th mid span. For a continuous bam, thr ar critical sctions at th supports and at th spans. For dsign undr srvic loads, th following quantitis ar known. M DL M LL = momnt du to dad load (xcluding slf-wight) = momnt du to liv load. Th matrial proprtis ar slctd bfor th dsign. Th following quantitis ar unknown. Th mmbr cross-sction and its gomtric proprtis, M SW A p P = momnt du to slf-wight, = amount of prstrssing stl, = th ffctiv prstrss, = th ccntricity. Thr ar two stags of dsign. 1) Prliminary: In this stag th cross-sction is dfind and P and A p ar stimatd.

2 ) Final: Th valus of (at th critical sction), P, A p and th strsss in concrt at transfr and undr srvic loads ar calculatd. Th strsss ar chckd with th allowabl valus. Th sction is modifid if rquird Prliminary Dsign Th stps of prliminary dsign ar as follows. 1) Slct th matrial proprtis f ck and f pk. ) Dtrmin th total dpth of bam (h). Th total dpth can b basd on architctural rquirmnt or, th following mpirical quation can b usd. h = 0.0 M to 0.04 M (4-.1) Hr, h is in mtrs and M is in knm. M is th total momnt xcluding slf-wight. ) Slct th typ of sction. For a rctangular sction, assum th bradth b = h/. 4) Calculat th slf-wight or, stimat th slf-wight to b 10% to 0% of th load carrid. 5) Calculat th total momnt M T including slf-wight. Th momnt du to slf-wight is dnotd as M sw. 6) Estimat th lvr arm (z). z 0.65h, if M sw is larg (M sw > 0.M T ). z 0.5h, if M sw is small. 7) Estimat th ffctiv prstrss (P ) P = M T / z, if M sw is larg. P = M I L / z, if M sw is small. If M sw is small, th dsign is govrnd by th momnt du to imposd load (M I L = M T M SW ). 8) Considring f p = 0.7f pk, calculat ara of prstrssing stl A p = P / f p. 9) Chck th ara of th cross-sction (A). Th avrag strss in concrt at srvic C/A (= P /A) should not b too high as compard to 50% of th allowabl comprssiv strss f cc,all. If it is so, incras th ara of th sction to A = P /(0.5f cc,all ).

3 4.. Final Dsign for Typ 1 Mmbrs Th cod IS: dfins thr typs of prstrssd mmbrs. Typ 1: In this typ of mmbrs, no tnsil strss is allowd in concrt at transfr or undr srvic loads. Typ : In ths mmbrs, tnsil strss is within th cracking strss of concrt. Typ : Hr, th tnsil strss is such that th crack width is within th allowabl limit. Th final dsign involvs th chcking of th strsss in concrt at transfr and undr srvic loads with rspct to th allowabl strsss. Sinc th allowabl strsss dpnd on th typ of mmbr (Typ 1, Typ or Typ ), th quations vary for th diffrnt typs. Hr, th stps of final dsign ar xplaind for Typ 1 mmbrs. Th stps for Typ mmbrs ar xplaind in Sction 4., Dsign of Sctions for Flxur (Part II). Th stps for Typ mmbrs ar similar to Typ, th only diffrnc bing th valu of th allowabl tnsil strss in concrt. For small momnt du to slf-wight (M sw 0.M T ), th stps ar as follows. 1) Calculat ccntricity to locat th cntroid of th prstrssing stl (CGS). With incrasing load, th comprssion (C) movs upward from th location of th tnsion (T) at CGS. At transfr, undr th slf-wight, C should li within th krn zon to avoid tnsil strss at th top. Th krn points and krn zon ar xplaind in Sction., Analysis of Mmbr undr Flxur (Part II). Th lowst prmissibl location of C du to slf-wight is at th bottom krn point (at a dpth k b blow CGC) to avoid tnsil strss at th top. Th dsign procdur basd on th xtrm location of C givs an conomical sction. Th following sktch xplains th lowst prmissibl location of C du to slf-wight momnt (M sw ) at transfr.

4 0 h + C T M sw CGC CGS k t k b C c t c b C/A = P 0 /A f b Intrnal forc in concrt Strss in concrt Figur 4-.1 Strss in concrt du to comprssion at bottom krn point In th abov sktch, A = gross ara of cross sction f b h = maximum comprssiv strss in concrt at bottom dg = total hight of th sction k t, k b = distancs of uppr and lowr krn points, rspctivly, from CGC c t, c b = distancs of uppr and lowr dgs, rspctivly, from CGC P 0 = prstrss at transfr aftr initial losss. Th shift of C du to slf-wight givs an xprssion of. = (M sw / P 0 ) + k b (4-.) Hr, th magnitud of C or T is qual to P 0. Th valu of P 0 can b stimatd as follows. a) 90% of th initial applid prstrss (P i ) for pr-tnsiond mmbrs. b) Equal to P i for post-tnsiond mmbrs. Th valu of P i can b stimatd from th amount of prstrssing stl dtrmind in th prliminary dsign. P i = A p (0.8f pk ) (4-.) Hr, th prmissibl prstrss in th stl is 0.8f pk, whr f pk is th charactristic tnsil strngth. ) Rcomput th ffctiv prstrss P and th ara of prstrssing stl A p. With incrasing load, C furthr movs up. Undr th srvic loads, C should li within th krn zon to avoid tnsil strss at th bottom. Th highst prmissibl location of

5 C du to total load is at th top krn point (at a hight k t abov CGC) to avoid tnsil strss at th bottom. Th following sktch xplains th highst possibl location of C du to th total momnt (M T ). f t h M T k t k b + CGC C c CGS b C/A = T P /A C c t 0 Intrnal forc Strss in in concrt concrt Figur 4-. Strss in concrt du to comprssion at top krn point In th abov sktch, f t = maximum comprssiv strss in concrt at top dg. Th shift of C du to th total momnt givs an xprssion of P. P = M T /( + k t ) (4-.4) Considring f p = 0.7f pk, th ara of prstrssing stl is rcomputd as follows. A p = P / f p (4-.5) ) Rcomput ccntricity First th valu of P 0 is updatd. Th ccntricity is rcomputd with th updatd valu of P 0. If th variation of from th prvious valu is larg, anothr cycl of computation of th prstrssing variabls can b undrtakn. 4) Chck th comprssiv strsss in concrt. Th maximum comprssiv strss in concrt should b limitd to th allowabl valus. At transfr, th strss at th bottom should b limitd to f cc,all, whr f cc,all is th allowabl comprssiv strss in concrt at transfr (availabl from Figur 8 of IS:14

6 - 1980). At srvic, th strss at th top should b limitd to f cc,all, whr f cc,all is th allowabl comprssiv strss in concrt undr srvic loads (availabl from Figur 7 of IS: ). a) At Transfr Th strss at th bottom can b calculatd from th avrag strss P 0 /A. f b =- Ac t P0 h (4-.6) To satisfy f b f cc,all, th ara of th sction (A) is chckd as follows. Ph 0 A f c cc,all t (4-.7) If A is not adquat thn th sction has to b rdsignd. b) At Srvic Th strss at th top can b calculatd from th avrag strss P /A. f =- t P h Ac b (4-.8) To satisfy f t f cc,all, th ara of th sction (A) is chckd as follows. Ph A (4-.9) f c cc,all b If A is not adquat thn th sction has to b rdsignd. 4.. Spcial Cas For larg momnt du to slf-wight (M sw > 0. M T ), th ccntricity according to = (M w / P 0 ) + k b may violat th covr rquirmnts or, may vn li outsid th bam. In such cass, locat as pr covr rquirmnts. Th location of C at transfr will b within th krn zon without zro strss at th top. Th xprssion of strss at th bottom is diffrnt from that givn arlir. Th othr stps ar sam as bfor.

7 At transfr, th strss at th bottom is calculatd using th following strss profil. CGC M SW /P 0 C C/A =P 0 /A f b Figur 4-. Strss in concrt du to comprssion abov bottom krn point Msw P0 - cb P P 0 0 f b =- - (4-.10) A I Substituting I = Ar and r /c b = k t M - P P f b =- 1+ A kt sw 0 0 (4-.11) To satisfy f b f cc,all, th ara of th sction (A) is chckd as follows. Msw - P 0 P0 A 1+ fcc,all kt (4-.1) Th following xampl shows th dsign of a Typ 1 prstrssd mmbr. Exampl 4-.1 Dsign a simply supportd Typ 1 prstrssd bam with M T = 45 knm (including an stimatd M SW = 55 knm). Th hight of th bam is rstrictd to 90 mm. Th prstrss at transfr f p0 = 105 N/mm and th prstrss at srvic f p = 860 N/mm. Basd on th grad of concrt, th allowabl comprssiv strsss ar 1.5 N/mm at transfr and 11.0 N/mm at srvic.

8 Th proprtis of th prstrssing strands ar givn blow. Typ of prstrssing tndon : 7-wir strand Nominal diamtr = 1.8 mm Nominal ara = 99. mm Solution A) Prliminary dsign Th valus of h and M SW ar givn. 1) Estimat lvr arm z. Msw M T 55 = 45 =1.5% Sinc M SW < 0. M T, Us z = 0.5h = = 460 mm ) Estimat th ffctiv prstrss. Momnt du to imposd loads Effctiv prstrss M IL =MT -Msw = P = 80 knm = 460 =86kN ) Estimat th ara of th prstrssing stl. P A p = f p = 860 = 960 mm

9 4) Estimat th ara of th sction to hav avrag strss in concrt qual to 0.5 f cc,all. P A = 0.5 f cc,all = = mm Th following trial sction has th rquird dpth and ara. Trial cross-sction B) Calculation of gomtric proprtis Valus in mm. Th sction is symmtric about th horizontal axis. Hnc, th CGC lis at mid dpth. Th sction is dividd into thr rctangls for th computation of th gomtric proprtis c t = 460 CGC Valus in mm. Chck ara of th sction A = A + A 1 = (90 100) +(70 100) =150,000 mm

10 Momnt of inrtia of th sction about axis through CGC I= I + I = (90 100) = mm 10 4 Squar of th radius of gyration r = I A = 150, =108,580 mm Krn lvls of th sction r kt = kb = c 108,580 = 460 = 6 mm t Summary aftr prliminary dsign Proprtis of sction A = 150,000 mm I = mm 4 c t = c b = 460 mm k t = k b = 6 mm Proprtis of prstrssing stl A p = 960 mm P = 86 kn

11 C) Final dsign 1) Calculat ccntricity P = A f 0 p p0 = =99.6kN M sw = +kb P = mm ) Rcomput th ffctiv prstrss and associatd variabls. M T P= +k t = (90 + 6) =87kN Sinc P is vry clos to th prvious stimat of 86 kn, A p, P 0 and rmain sam. Th tndons ar placd in two ducts. Th outr diamtr of ach duct is 54 mm. Slct (10) 7-wir strands with A p = = 99.0 mm ) Chck th comprssiv strsss in concrt. a) At transfr Ph 0 A f c cc,all = t b) At srvic = 158,976 mm Ph A f c cc,all b = =150,64 mm

12 Th govrning valu of A is 158,976 mm. Th sction nds to b rvisd. Th width of th flang is incrasd to 45 mm. Th ara of th rvisd sction is 159,000 mm. Anothr st of calculations can b don to calculat th gomtric proprtis prcisly. Dsignd cross-sction at mid-span CGC CGS (10) 7-wir strands with P 0 = 994 kn