The following figure illustrates the cantilevers in buildings and bridges schematically.

Transcription

1 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non 8.1 Cantilvr Bams This sction covrs th following topics. Analysis Dtration of Limiting Zon Tndon Profil Introduction Prstrssd cantilvr bams ar prsnt in buildings and bridgs. Usually, th cantilvr is providd with a back span (anchor span) to rduc th torsion in th supporting mmbr. In a building, th cantilvr can b an xtnsion of a continuous bam. In a bridg, th cantilvr is a part of th balancd cantilvr girdr. Th construction can b in-situ, whr th concrt is cast in formwork that is tmporarily supportd. Els, a sgmntal construction can b adoptd, whrin slics of th girdr ar placd and tmporarily prstrssd. Th following figur illustrats th cantilvrs in buildings and bridgs schmatically. Sgmnt Back span Cantilvr span Cantilvr spans (a) Cantilvr in a building fram (b) Cantilvrs in a bridg girdr Figur Cantilvrs in buildings and bridgs Indian Institut of Tchnology adras

2 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non Figur Sgmntal construction, Bridg ovr Hooghly, Wst Bngal (Courtsy: L & T Ramboll) Indian Institut of Tchnology adras

3 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non Th analysis of a sction of a prstrssd cantilvr for flxural strsss is similar to that for a simply supportd bam. Th diffrnc is that for gravity loads, th bnding momnt in cantilvr is ngativ, that is comprssion is gnratd at th bottom. Thus, th CGS is placd abov th CGC and th ccntricity is ngativ. Th following aspcts nd to b considrd in th analysis and dsign of a prstrssd cantilvr bam. 1) Crtain portions of th back span ar subjctd to both positiv and ngativ momnts. Hnc, thr will b two dsign momnts at srvic loads. 2) Th bam may b subjctd to partial loading and point loading. 3) Th squnc of loading is important to dsign th prstrssing forc. 4) High valus of momnt and shar occur simultanously nar th support Analysis Th analysis of a cantilvr bam with a back span, is illustratd to highlight th aspcts statd arlir. Th bnding momnt diagrams for th following load cass ar shown schmatically in th following figur. 1) Dad load (DL) 2) Liv load (LL) only on th back span 3) Liv load only on th cantilvr span 4) Dad load and liv load along full lngth. Indian Institut of Tchnology adras

4 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non w DL omnt diagram for DL w LL omnt diagram for LL on back span w LL omnt diagram for LL on cantilvr span w DL + w LL omnt diagram for DL + LL along full lngth Figur omnt diagrams for dad and liv loads Indian Institut of Tchnology adras

5 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non Th nvlop momnt diagrams ar calculatd from th analysis of ach load cas and thir combinations. In th following nvlop momnt diagrams, and rprsnt th highst and lowst valus (algbraic valus with sign) of th momnts at a sction, rspctivly. Not that crtain portions of th bam ar subjctd to both positiv and ngativ momnts. Figur Envlop momnt diagrams for dad and liv loads For moving point loads as in bridgs, first th influnc lin diagram is drawn. Th influnc lin diagram shows th variation of th momnt or shar for a particular location in th girdr, du to th variation of th position of a unit point load. Th vhicl load is placd basd on th influnc lin diagram to gt th worst ffct Dtration of Limiting Zon Th limiting zon of placing th CGS of th tndons is hlpful in slcting a tndon profil. Not that for a non-prismatic sction, th sction proprtis such as A, I, k t, k b, r chang with position along th lngth. Th limiting zon was xplaind for a simply supportd bam in Sction 4.4, Dsign of Sctions for Flxur (Part III). Hr th concpt and th quations ar first rviwd for a simply supportd bam with positiv momnt. For full prstrssd mmbrs (Typ 1), tnsion is not allowd undr srvic conditions. If tnsion is also not allowd at transfr, th comprssion in concrt (C) always lis within th krn zon. Th limiting zon is dfind as th zon for placing th CGS of Indian Institut of Tchnology adras

6 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non th tndons such that C always lis within th krn zon. Also, th imum comprssiv strsss at transfr and srvic should b within th allowabl valus. For limitd prstrssd mmbrs (Typ 2 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 allowabl valus. Also, th imum comprssiv strsss at transfr and srvic should b within th allowabl valus. Th limiting zon is dtrd from th imum or imum ccntricitis of th CGS along th bam corrsponding to th xtrm positions of C. Not that, th limiting zon is rlatd with th CGS of th tndons. Individual tndons may li outsid th limiting zon. For a simply supportd bam, th imum ccntricity ( ) at any sction corrsponds to th lowst possibl location of C at transfr, that gnrats allowabl tnsil strss at th top of th sction. Th imum comprssiv strss at th bottom should b within th allowabl valu. Th imum ccntricity ( ) at any sction corrsponds to th highst possibl location of C at srvic, that gnrats allowabl tnsil strss at th bottom of th sction. Th imum comprssiv strss at th top should b within th allowabl valu. Th valus of and can b dtrd by quating th strsss at th dgs of concrt with th allowabl valus. Els, xplicit xprssions of and can b dvlopd. Th following matrial givs th xprssions of and for Typ 1 and Typ 2 sctions. Th valus of and can b dtrd at rgular intrvals along th lngth of th bam. Th zon btwn th loci of and is th limiting zon of th sction for placing th CGS. Th quations ar summarisd Typ 1 Sction Basd on th strss at th top at transfr, th following xprssion of is drivd. or, sw -k b = P 0 sw (8-1.1) Indian Institut of Tchnology adras

7 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non Also, th strss at th bottom should b lss than th allowabl valu at transfr. Basd on th strss at th bottom at srvic, th following xprssion of is drivd. or, T +k t = P T = -kt P (8-1.2) Also, th strss at th top should b lss than th allowabl valu at srvic. If for a particular sction is ngativ, it implis that th CGS can b placd abov th CGC. This happns nar th supports. Typ 2 Sction Th corrsponding quations for a Typ 2 sction ar givn blow. or, +f Ak -k b = P sw ct,all b +f Ak sw ct,all b 0 (8-1.3) -f +k t = P or, T ct,all t -f T ct,all t = -kt P Ak Ak (8-1.4) In a simply supportd bam, th xtrnal momnts ar always positiv. Th imum momnt is du to slf wight. Th imum momnt is undr srvic loads. For cantilvr bams, th imum xtrnal momnt nd not b at transfr, whn th momnt is du to slf wight ( SW ). Also, undr srvic loads thr ar two momnts and at a location, obtaind from th nvlop momnt diagrams. Th imum and imum ccntricitis and at a particular location ar first dtrd for srvic loads from and, rspctivly, at that location and th ffctiv prstrss P. Nxt, anothr st of and ar calculatd at transfr from SW and th prstrss P 0. Th final is th lowr of th two valus calculatd at srvic and at transfr. Similarly, th final is th highr of th two valus calculatd at srvic and at transfr. Indian Institut of Tchnology adras

8 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non Th xprssions of and for th simply supportd bam wr dvlopd for positiv momnts. For a cantilvr, corrsponding to a ngativ momnt, th ccntricity implis that th CGS is locatd abov CGC. Th xprssions for a cantilvr ar givn blow. Typ 1 sction At Srvic P = -kt P (8-1.5) (8-1.6) At Transfr sw sw = -kt (8-1.7) (8-1.8) Typ 2 sction At Srvic At Transfr +f Ak ct,all b P -f Ak ct,all t = +kt P +f Ak SW ct,all b -f Ak SW ct,all t = -kt (8-1.9) (8-1.10) (8-1.11) (8-1.12) Indian Institut of Tchnology adras

9 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non A typical limiting zon is shown in th following figur. k t Limiting zon k b Figur Limiting zon for a non-prismatic bam Tndon Profil Th tndon profils for a fw bams with cantilvr spans ar shown schmatically in th following figurs. Th vrtical scal is nlargd to show th location of th CGS with rspct to CGC. a) Uniform cross sction b) Varying cross sction in th cantilvr span Figur Bams with singl cantilvr span Indian Institut of Tchnology adras

10 Prstrssd Concrt Structurs Dr. Amlan K Sngupta and Prof. Dvdas non a) Uniform cross sction b) Varying lvl of soffit c) Varying lvl of top of bam Figur Bams with doubl cantilvr spans For a prismatic bam with uniform cross sction along th lngth, th tndon profil is similar to th momnt diagram undr uniform load. Thus for rgions of ngativ momnt, th CGS is locatd abov th CGC. Sinc thr cannot b a sharp kink in th tndons and, th supports ar not tru point supports, th profil is shown curvd at th right support. For a bam with varying dpth, th tndon profil can b adjustd (within and ) to b straight for convninc of layout of th tndons. Indian Institut of Tchnology adras