4.4 Design of Sections for Flexure (Part III)

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4.4 Dsign of Sctions for Flxur (Part ) This sction covrs th following topics. Choic of Sctions Dtrmination of Limiting Zon Post-tnsioning in Stags 4.4.1 Choic of Sctions Th typ of sction is slctd asd on th us of th structur, architctural rquirmnts, casting and farication options, availal tchnology and skilld work forc.

1 4.4 Dsign of Sctions for Flxur (Part ) This sction covrs th following topics. Choic of Sctions Dtrmination of Limiting Zon Post-tnsioning in Stags Choic of Sctions Th typ of sction is slctd asd on th us of th structur, architctural rquirmnts, casting and farication options, availal tchnology and skilld work forc. Hr, a fw commnts ar givn for th availal typs of sctions. 1) Th sction should hav larg dpth low th CGC, so as to hav th provision of larg ccntricity. Th prstrssing forc can thn rducd. ) Thr should adquat concrt at th top and ottom to satisfy th allowal strsss. 3) Th nd sction is usually solid to incras th shar capacity and prvnt anchorag zon failur. Hr, th sctions ar roadly groupd undr rctangular sction, T-sction, -sction and invrtd T-sction. Som variations of ach typ ar shown undr th corrsponding road groups. Th sctions in ach group hav similar analysis procdur. Th sctions shown ar not xclusiv.

2 Tal Typs of sctions Broad groups of sctions Rctangular T-sction -sction nvrtd T- sction Variations (a) () Rmarks on th sctions (a) or room for tndons in lowr Torsionally stiff flang. and strong. () Bttr staility during rction. Farication Easy Easy Expnsiv form work Difficult Spac for rinforcmnt Adquat Lss than adquat Good Good 1) Vry Effcincy for non-composit sctions Poor, z 0.4h Good, z 0.5h Vry good, z 0.7h infficint. ) Small ultimat momnt capacity 1) Espcially 1) Good for Application of 1) Light load and good for long long span and non-composit short span. span roofs, whn havy loads. sw / T larg sctions ) sw / T larg. LL «DL. ) sw / T ) sw / T larg. small. Effcincy for Vry good ncrass load Vry good composit particularly whn capacity only Vry good with cast-in- sctions sction is shord. slightly. plac flang.

3 1) Topping srvs to ti all Application of composit sctions Good for uilding construction sctions togthr. ) No form Long span uildings and ridgs. Bridgs rquird for composit pour. Th diffrnt typs of sctions can compard y a masur of flxural fficincy η. Th flxural fficincy is dfind in trms of th radius of gyration r as follows. r η cc t r c+c t cc t h r r + c ct h k+k t (4-4.1) η h Thus for a givn valu of th dpth h, if th krn zon (k t +k ) is larg thn th sction is fficint. For a rctangular sction, η For an -sction, η > Dtrmination of Limiting Zon For full prstrssd mmrs (Typ 1), tnsion is not allowd undr srvic conditions. f tnsion is also not allowd at transfr, C always lis within th krn zon. Th limiting zon is dfind as th zon for placing th CGS of th tndons such that C always lis within th krn zon. Also, th maximum comprssiv strsss at transfr and srvic should within th allowal valus. For limitd prstrssd mmrs (Typ and Typ 3), tnsion is allowd at transfr and undr srvic conditions. Th limiting zon is dfind as th zon for placing th CGS such that th tnsil strsss in th xtrm dgs ar within th allowal valus. Also, th maximum comprssiv strsss at transfr and srvic should within th

4 allowal valus. Not that th limiting zon is a rstriction for th CGS. Th individual tndons may li outsid th limiting zon. Th following figur shows th limiting zon (as th shadd rgion) for a simply supportd am sujctd to uniformly distriutd load. CL CGC Locus of min Locus of max Figur Limiting zon for a simply supportd am Th limiting zon is dtrmind from th maximum or minimum ccntricitis of th CGS along th am corrsponding to th xtrm positions of C. Th maximum ccntricity ( max ) at any sction corrsponds to th lowst possil location of C at transfr, that gnrats allowal tnsil strss at th top of th sction. Also, th maximum comprssiv strss at th ottom should within th allowal valu. Th minimum ccntricity ( min ) at any sction corrsponds to th highst possil location of C at srvic, that gnrats allowal tnsil strss at th ottom of th sction. Also, th maximum comprssiv strss at th top should within th allowal valu. Th following matrial givs th xprssions of max and min for Typ 1 and Typ sctions. Th zon twn th loci of max and min is th limiting zon of th sction for placing th CGS. Th valus of max and min can dtrmind y quating th strsss at th dgs of concrt with th allowal valus. Els, xplicit xprssions of max and min can usd. Hr, th xprssions of max and min asd on allowal tnsil strss ar givn. Typ 1 Sction At Transfr

5 Th following sktch shows th strss profil in concrt whn C is at th lowst prmissil location du to slf-wight momnt ( sw ) at transfr. 0 k t CGC k CGS C T ntrnal forcs c t max c f Strss in concrt Figur 4-4. Strss in concrt du to comprssion at ottom krn point From th shift of C du to slf-wight, th following xprssion can drivd. sw max -k P 0 or, (4-4.) sw max +k P0 Not that sinc SW varis, max varis along th lngth of th am. Also, th strss at th ottom nds to chckd to satisfy th condition f f cc,all. At Srvic Th following sktch shows th strss profil in concrt whn C is at th highst possil location du to th total momnt ( T ). f t k t CGC k CGS ntrnal forcs C T min c t c 0 Strss in concrt Figur Strss in concrt du to comprssion at top krn point From th shift of C du to total momnt, th following xprssion can drivd. or, T min +k t P T min -kt P

6 (4-4.3) Not that sinc T varis, min varis along th lngth of th am. Also, th strss at th top nds to chckd to satisfy th condition f t f cc,all. f for a particular sction min is ngativ, it implis that th CGS can placd aov CGC. This happns nar th supports. Typ Sction At Transfr Th following sktch shows th strss profil in concrt whn C is at th lowst prmissil location du to slf-wight momnt ( sw ) at transfr. f ct,all k CGC t k 1 CGS C T max ntrnal forcs c t c f Strss in concrt Figur Strss in concrt du to comprssion outsid ottom krn point +f Ak max -k P sw ct,all +f Ak or, (4-4.4) sw ct,all max +k P0 0 Not that max for a Typ sction is largr than that for a Typ 1 sction du to th trm f ct,all Ak in th numrator. Th strss at th ottom nds to chckd to satisfy th condition f f cc,all. At Srvic Th following sktch shows th strss profil in concrt whn C is at th highst possil location du to th total momnt ( T ).

7 f t CGC CGS 3 k t k C T min c t c f ct,all ntrnal forcs Strss in concrt Figur Strss in concrt du to comprssion outsid top krn point -f min +k t P T ct,all t T ct,all t min -kt P Ak -f Ak or, (4-4.5) Not that min for a Typ sction is smallr than that for a Typ 1 sction du to th trm f ct,all Ak t in th numrator. Th strss at th ottom nds chckd to satisfy th condition f t f cc,all. Th zon twn max and min is th limiting zon of th sction for placing th CGS for a givn loading condition. Th valus of max and min for svral sctions can dtrmind at rgular intrvals along th lngth of th am to gt thir loci. Not that th limiting zon for a Typ mmr is largr than th limiting zon for a Typ 1 mmr. Th following tal shows a comparison of quations for Typ 1 and Typ mmrs. Tal 4-4. Comparison of quations for Typ 1 and Typ mmrs Typ 1 Typ aximum ccntricity +f Ak +k sw sw ct,all +k P 0 P 0 inimum ccntricity T -kt P -f Ak T ct,all t -kt P Th following xampl shows th calculation of limiting zon asd on quating th strsss at th dgs of concrt with th allowal valus.

8 Exampl For th Typ post-tnsiond am with a flangd sction as shown, th span is 18 m. For uniform loads, th profil of th CGS is paraolic. Th liv load momnt at mid-span ( LL ) is 648 knm. Th prstrss aftr transfr (P 0 ) is 1600 kn. Assum 15% loss at srvic. Evaluat th limiting zon of CGS, if th allowal strsss at transfr and at srvic ar as follows. For comprssion, f cc,all 18.0 N/mm For tnsion, f ct,all 1.5 N/mm CGS Valus in mm. Solution A) Calculation of gomtric proprtis Th sction is dividd into thr rctangls for th computation of th gomtric proprtis. Th cntroid of ach rctangl is locatd from th soffit.

9 1 c t y c CGC Ara of th sction Valus in mm. Ara of 1 A ,000 mm Ara of A ,000 mm Ara of 3 A ,000 mm A A 1 + A + A 3 40,000 mm Distanc of CGC from th soffit A A A3 0 y A mm Thrfor, c mm ct mm omnt of inrtia of 1 aout axis through CGC A ( ) mm omnt of inrtia of A ( ) mm 3

10 omnt of inrtia of A ( ) mm omnt of inrtia of th sction ( ).55 mm 4 Calculation of momnt du to slf wight. w SW 3 1 m 4 kn/m 40,000 mm mm 5.76 kn/m 3 SW wsw L kNm B) Dtrmination of limiting zon Th valus of max and min ar dtrmind y quating th strsss at th dgs of concrt with th allowal valus. Th xprssion of strss is givn low. CGC P P Py y f- ± ± A + + P/A ±Py/ ±y/ Rsultant strss profil Limiting position at mid-span For max, considr th load stag at transfr. i) Calculat asd on f 18.0 N/mm.

11 P 1600 A N/mm Pc c SW N/mm f N/mm Solving, mm ii) Calculat asd on f t 1.5 N/mm. Pc 0 t SWct N/mm ft N/mm Solving, mm Out of th two valus of, th lowr valu 455. mm govrns. max 455.mm

12 For min, considr th load stag at srvic. i) Calculat asd on f t 18.0 N/mm. P A P A N/mm Pc t LLct N/mm ft N/mm Solving, mm ii) Calculat asd on f 1.5 N/mm. Pc LLc N/mm f N/mm Solving, mm Out of th two valus of, th highr valu mm govrns. min mm

13 Limiting position at nd At transfr For f 18.0 N/mm 0.0 f SW LL N/mm For f t 1.5 N/mm Solving, mm ft N/mm Solving, mm Slcting th lowr valu max mm At srvic For f 1.5 N/mm Solving f N/mm mm For f t 18.0 N/mm Solving ft N/mm mm Sinc th valus of ar ngativ th CGS lis aov CGC. Th position of CGS closr to th CGC is slctd. min mm Similarly, th valus of max and min can dtrmind at rgular intrvals along th span. Th limiting zon is availal y joining th points y straight lins.

14 n th following sktch th limiting zon is shown shadd. CL CGC Valus in mm From th sktch of th limiting zon, it is vidnt that th tndons can sprad out at th nds. This is ncssary to anchor th tndons and rduc th strss concntration at th nds Post-tnsioning in Stags n th prvious xprssions of max and min, th valus of P 0 and P can for diffrnt lvls of prstrssing for post-tnsiond mmrs. At transfr th mmr can partially prstrssd in th casting yard, from which P 0 is calculatd. Aftr th mmr is placd in its prmannt location, it can furthr prstrssd for it is put into srvic. Th application of prstrss in diffrnt stags is trmd as post-tnsioning in stags. Th valu of P is calculatd from th rvisd prstrssing forc. With rducd P 0 at transfr, max is incrasd. Thus, th limiting zon for placing th CGS and th availal zon for th shift in C undr srvic loads ar also incrasd.

4.2 Design of Sections for Flexure

4.2 Design of Sections for Flexure 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