4.3 Design of Sections for Flexure (Part II)

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1 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon 4. Dsign of Scions for Flxur (Par II) This scion covrs h following opics Final Dsign for Typ Mmrs Th sps for Typ 1 mmrs ar xplaind in Scion 4., Dsign of Scions for Flxur (Par I) Final Dsign for Typ Mmrs For Typ mmrs, h nsil srss undr srvic loads is wihin h cracking srss of concr. Th allowal nsil srss in concr (f c,all ) as pr IS: is sam for ransfr and srvic load condiions. Th valu is. N/mm, which can incrasd o 4.5 N/mm for mporary loads. Th following marial provids h sps for scions wih small slf-wigh momn. For scions wih larg slf- wigh momn, h ccnriciy may nd o drmind asd on h covr rquirmns. 1) Calcula ccnriciy o loca h cnroid of h prsrssing sl (CGS). Undr h slf-wigh, C may li ousid h krn rgion. Th lows possil locaion of C du o slf-wigh is drmind y h allowal nsil srss a h op. Th following skch xplains h xrm locaion of C du o slf-wigh momn (M sw ) a ransfr. f c,all h C T + M SW CGC CGS k k 1 C c c f Inrnal forc in concr Srss in concr Figur 4-.1 Srss in concr du o comprssion ousid oom krn poin Indian Insiu of Tchnology Madras

2 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon In h aov skch, f f c,all h maximum comprssiv srss in concr a oom dg allowal nsil srss in concr a op dg oal high of h scion k, k disancs of uppr and lowr krn poins, rspcivly, from CGC c, c disancs of uppr and lowr dgs, rspcivly, from CGC 1 P disanc wn h oom krn poin and h locaion of comprssion disanc y which h comprssion ravls from CGS du o slf wigh prsrss a ransfr afr iniial losss. From h prvious figur, h shif of C du o slf-wigh givs an xprssion of. I is vidn ha if C is furhr shifd upwards y a disanc 1 o h oom krn poin, hr will no nsil srss a h op. M sw P (4-.1) Th valu of 1 is calculad from h xprssion of srss corrsponding o h momn du o h shif in C y 1. Pc 1 f c,all I fc,alli 1 Pc f 1 c,all P Ak (4-.) Susiuing I Ar and r /c k Th disanc of h CGS low h oom krn poin is givn as follows. M sw +fc,all Ak + 1 P (4-.) Indian Insiu of Tchnology Madras

3 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon Th ccnriciy is calculad from h following quaion. + +k 1 sw c,all +k P M +f Ak (4-.4) Th aov xprssion can compard wih h xprssion of Typ 1 mmr (M sw / P ) + k. No ha h ccnriciy has incrasd for a Typ mmr du o h allowal nsil srss f c,all. ) Rcompu h ffciv prsrss P and h ara of prsrssing sl A p. Undr h oal load, C may li ousid h krn rgion. Th highs prmissil locaion of C du o oal load is drmind y h allowal nsil srss a h oom. Th following skch xplains h highs possil locaion of C du o h oal momn (M T ). h C T + M T CGC CGS k k C c c f f c,all Inrnal forc in concr Srss in concr Figur 4-. Srss in concr du o comprssion ousid op krn poin From h prvious figur, h xprssion of is oaind y h nsil srss gnrad du o h shif of C yond h uppr krn poin. Pc f c,all I fc,all I Pc f Ak c,all (4-.5) P Susiuing I Ar and r /c k Indian Insiu of Tchnology Madras

4 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon Th shif of C du o h oal momn givs an xprssion of P. ( ) ( ) MP +k+ T P +k +f Ak c,all M-f T c,allak P +k (4-.6) Th aov xprssion can compard wih h xprssion of Typ 1 mmr P M T /( + k ). No ha h prsrssing forc has dcrasd for a Typ mmr du o h allowal nsil srss f c,all. This will lad o a dcras in h ara of prsrssing sl (A p ). Considring f p.7f pk, A p is rcompud as follows. A p P / f p (4-.7) ) Rcompu ccnriciy Firs h valu of P is updad. Th ccnriciy is rcompud wih h updad valu of P. If h variaion of from h prvious valu is larg, anohr cycl of compuaion of h prsrssing varials can undrakn. 4) Chck h comprssiv srsss in concr Th maximum comprssiv srss in concr should limid o h allowal valus. A ransfr, h srss a h oom should limid o f cc,all, whr f cc,all is h allowal comprssiv srss in concr a ransfr (availal from Figur 8 of IS:14-198). A srvic, h srss a h op should limid o f cc,all, whr f cc,all is h allowal comprssiv srss in concr undr srvic loads (availal from Figur 7 of IS:14-198). a) A Transfr Th srss a h oom can calculad from h srss diagram. ( ) C Ck+ 1 c f - - A I C kc Cc 1 f A r I (4-.8) Indian Insiu of Tchnology Madras

5 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon From f c,all C 1 c / I, susiuing C 1 / I f c,all /c C c f c,all f c A c c Ch fc,all f - - c Ac c (4-.9) To saisfy f f cc,all, h ara of h scion (A) is chckd as follows. Ch f + c,all c f Ac c cc,all f cc,all c -f c,all c A A (4-.1) f c -f c cc,all c,all If A is no adqua hn h scion has o rdsignd. ) A Srvic Th srss a h op can calculad from h srss diagram. ( ) C C k + c f - A I C kc Cc f (4-.11) A r I From f c,all C c / I, susiuing C / I f c,all /c C c fc,allc f A c c Ch fc,allc f - Ac c (4-.1) To saisfy f f cc,all, h ara of h scion (A) is chckd as follows. Ch fc,allc + Ac c f cc,all f cc,all c -f c,all c A A f c -f c (4-.1) cc,all c,all Indian Insiu of Tchnology Madras

6 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon If A is no adqua hn h scion has o rdsignd. Th following al shows a comparison of quaions for Typ 1 and Typ mmrs. Tal 4-.1 Comparison of quaions for Typ 1 and Typ mmrs Typ 1 Typ Eccnriciy M M +f Ak +k sw sw c,all +k P P Effciv prsrss Minimum ara asd on srss a oom a ransfr Minimum ara asd on srss a op a srvic M M -f Ak T T c,all P P +k +k A f c cc,all A f c -f c cc,all c,all A A f c f c -f c cc,all cc,all c,all Th following xampl shows h dsign of a Typ prsrssd mmr. Th sam scion was dsignd as a Typ 1 mmr in Scion 4., Dsign of Scions for Flxur (Par I). Th soluions of h wo xampls ar compard a h nd. Exampl 4-.1 Dsign a simply suppord Typ prsrssd am wih M T 45 knm (including an simad M SW 55 knm). Th high of h am is rsricd o 9 mm. Th prsrss a ransfr f p 15 N/mm and h prsrss a srvic f p 86 N/mm. Basd on h grad of concr, h allowal comprssiv srsss ar 1.5 N/mm a ransfr and 11. N/mm a srvic. Th allowal nsil srsss ar.1 N/mm a ransfr and 1.6 N/mm a srvic. Th propris of h prsrssing srands ar givn low. Typ of prsrssing ndon : 7-wir srand Nominal diamr 1.8 mm Nominal ara 99. mm Indian Insiu of Tchnology Madras

7 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon Soluion A) Prliminary dsign Th valus of h and M SW ar givn. 1) Esima lvr arm z. Msw 55 M % Sinc M SW <. M T, us z.5h mm T ) Esima h ffciv prsrss. Momn du o imposd loads Effciv prsrss P M IL MT -Msw kNm kN ) Esima h ara of h prsrssing sl. P A p f p mm 4) Esima h ara of h scion o hav avrag srss in concr qual o.5 f cc,all. P A.5 f cc,all mm Indian Insiu of Tchnology Madras

8 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon Th following rial scion has h rquird dph and ara. Trial cross-scion 9 9 B) Calculaion of gomric propris Valus in mm. Th scion is symmric aou h horizonal axis. Hnc, h CGC lis a mid dph. Th scion is dividd ino hr rcangls for h compuaion of h gomric propris c 46 CGC Valus in mm. Chck ara of h scion A (9 )+(7 ) 15, mm Momn of inria of h scion aou axis hrough CGC I I + I (9 ) mm 1 4 Indian Insiu of Tchnology Madras

9 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon Squar of h radius of gyraion Krn lvls of h scion r I A , 1 18,58 mm r k k c 18, mm Summary afr prliminary dsign Propris of scion A 15, mm I mm 4 c c 46 mm k k 6 mm Valus of prsrssing varials A p 96 mm P 86 kn C) Final dsign 1) Calcula ccnriciy PAf p p kn M +f Ak 1+ P sw c,all , 6 1mm Indian Insiu of Tchnology Madras

10 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon 1+ +k mm ) Rcompu h ffciv prsrss and h ara of prsrssing sl A p. M -f Ak P +k T c,all , kn Sinc P is susanially lowr han h prvious sima of 86 kn, A p, P and nd o rcalculad. P A p f p mm ) Rcompu ccnriciy PAf p p kN M +f Ak sw c,all +k P , mm Chck h covr rquirmn Assuming h our diamr of duc qual o 54 mm Clar covr for h duc mm Indian Insiu of Tchnology Madras

11 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon Th clar covr a h oom is grar han mm (Claus , IS: ), which is saisfacory. Th sid covr in h w is slighly lss han mm. Th hicknss of h w can incrasd o saisfy h rquirmn. Sinc h valu of has changd from 66 mm o 4 mm, prsrssing varials ar rcompud. M -f Ak P +k T c,all , kn P has furhr rducd from 65.6 kn. A p and P ar rcalculad. A p mm Slc (7) 7-wir srands wih A p mm Th ndons can placd in on duc. Th our diamr of h duc is 54 mm. P kN Sinc h maximum possil ccnriciy is asd on covr rquirmn, h valu of is no updad. 4) Chck h comprssiv srsss in concr. A ransfr A f c -f c cc,all c,all ,5 mm Indian Insiu of Tchnology Madras

12 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon A srvic A f c -f c cc,all c,all ,61 mm Th govrning valu of A is 18,5 mm. Th scion can rvisd. Th widh of h flang is rducd o 5 mm. Th ara of h rvisd scion is 19, mm. Anohr s of calculaions can don o calcula h gomric propris prcisly. Dsign cross-scion a mid-span CGC CGS (7) 7-wir srands wih P 719 kn Comparison of Typ 1 and Typ scions Th soluions from h xampls of Typ 1 and Typ mmrs ar placd oghr in h nx figur for comparison CGC CGS 9 9 CGC CGS 4 9 (1) 7-wir srands wih P 994 kn (7) 7-wir srands wih P 719 kn Typ 1 Typ Figur 4-. Scions dsignd as Typ 1 and Typ mmrs Indian Insiu of Tchnology Madras

13 Prsrssd Concr Srucurs Dr. Amlan K Sngupa and Prof. Dvdas Mnon Th following osrvaions can mad. 1) In Typ scion, h amoun of prsrssing sl and h prsrssing forc ar lss han hos in a Typ 1 scion. Th ara of cross-scion is lss for Typ scion. Typ scion is rlaivly conomical. ) Th ccnriciy in Typ scion is largr han in Typ 1 scion. For uni prsrssing forc, h prsrssing is mor ffciv in Typ scion. Indian Insiu of Tchnology Madras

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