7968084-Limit-State-Design-of-RCC

(1)

I, I, t:t: 'i'i

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TA

TATE

TE

M A M A A.DHIRAJLAL, A.DHIRAJLAL, B .B .E "E " M .M . C .C . F .F . E "E " G RG RA DA D C .C .E dE d L EL EE DE DS )S ) M JM J S .S .T .T .E .E . PRINCIPAL PRINCIPAL Thiagarnjar

Thiagarnjar PolytechnicPolytechnic Salem-636005

Salem-636005

Tamiln~du. Tamiln~du. and

and i>;,.i>;,.

': ': :,:,';'; .' .' .,., :~ :~ :i, :i, '" '" .-,..-,. P.RAM.'ACBANDRAN,

P.RAM.'ACBANDRAN, D.E.,D.E., M.I.S.T.E.,M.I.S.T.E.,

Salem..u3600S';

Salem..u3600S'; :c'::c': i}i} ,1,1 'il'il : -:-. '.'

Tam Tam ilnad.'ilnad.';';'

"i\;'J "i\;'J

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Preface

w a w a n en e w hw h h eh e n dn d m em en tn t n cn c n dn d " d " d IS:456~ IS:456~ 1978.1978. m a m a m em e m em e A lA l 'number'number will will b eb e e xe x r er em e lm e ly ,y , . " . " With

With ririchch expexperierienceence authorsauthors inin encenchihingng h eh e m om o h ah a 3 03 0 a ra r h e h e u du de ne n o fo f P oP o y ty te ce ch n ih n ic sc s fifi ~ . ~ .• .• . ~ • ~ • /;", /;", T h T h a ua u h oh o m em e g rg ra ta t h eh e B uB u ~ a~ a o fo f n dn d a ~a ~S :S : a na nd ad a d sd s whichwhich m i m i ISIS :4:45656-1-19797 1987. 1987. IS IS 5656-1-1 7878 ~ v~ vh ih ic hc h M a M a M a M a B h B h a la l B h B h n e h wn e h w B oB om bm b C aC a C h n dC h n d g ag a h ,h , H yH y K aK an pn p , P , P J! J! .•.• .... ~t~t ./'./'

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Preface

w a w a n en e w hw h h eh e n dn d m em en tn t n cn c n dn d " d " d IS:456~ IS:456~ 1978.1978. m a m a m em e m em e A lA l 'number'number will will b eb e e xe x r er em e lm e ly ,y , . " . " With

With ririchch expexperierienceence authorsauthors inin encenchihingng h eh e m om o h ah a 3 03 0 a ra r h e h e u du de ne n o fo f P oP o y ty te ce ch n ih n ic sc s fifi ~ . ~ .• .• . ~ • ~ • /;", /;", T h T h a ua u h oh o m em e g rg ra ta t h eh e B uB u ~ a~ a o fo f n dn d a ~a ~S :S : a na nd ad a d sd s whichwhich m i m i ISIS :4:45656-1-19797 1987. 1987. IS IS 5656-1-1 7878 ~ v~ vh ih ic hc h M a M a M a M a B h B h a la l B h B h n e h wn e h w B oB om bm b C aC a C h n dC h n d g ag a h ,h , H yH y K aK an pn p , P , P J! J! .•.• .... ~t~t ./'./'

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'.'.

10;'

14. 14. .... ~ :~: ,"," -"-" • ~ • ~ ! "! " 1'1'

C h a p t e r : : : ,

• T • T • ,• ,. ,. , i~i~ " _ " _; ,;,; .;., ~, ~ ; :;: - r-r • .•. . ,., L J L Jn tn t n .n . m i m i L i L im im i a ta t a pa p ininFlexureFlexure .. .. n g n g n fn f n sn s

~ ~

~ ~ ~ ?

~ ? ~ 1

~ 1 ~ ) ,

~ ) , ~ R

~ R r W

r W ~ r ~

~ r ~ e d

e d e , 6 ~ o

e , 6 ~ o ~ ~

~ ~ ~ ! {

~ ! {

" L :

T-Beams T-Beams Shear' Shear' ,:' ,:''':'': [4[42:2:

58

~'~ ~'~ ~7~ ~7~ «'93; «'93; r!r! .,1,;.,1,; ;,;'t,;,;'t, Ge Genenerara DeDe gngn rere enen D e D e D e D e _"_" _ :_:: i:i 1 9 1 !1 9 1 !

210'"

L i L in tn t a na n S uS un sn sh ah a Staircase Staircase Colwnns Colwnns C o C o m nm n B aB a a na n n gn g I~ I~ S~ S~

'n

eil eil ·1t'h~1 ·1t'h~1 A! A! A! A! bf bf b\ b\ B, B, ~, ~, d' d' fc fc F• F• fy fy IeIe

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;',l SYMBOLS 'f m b

in

this text .', 'i'+/J N e A e '=Ag-Asc)

Gross area of sectio

l J1 . , . .. , : st sv bf B,V fc fy Ld LL t. le lex ley A r incompression A r e a ' o f S t e n

T o t a l '

cros -sectional area irru egs

B r m , m en s n g m n f f B r e a d t h w e 1,:;,:,.::.:;, .. B a S i c Value Overa dept m e m n m en s n s o f a . ! C C t .. 1 il g· u la ; ; ; ;; ' column in h e n d considcra}r,'on, Thickness of flange' D e a : o ad " _ . -',: D ep t o f c om pr e o n r ei n o rc em en t fr~l~ h e h ig h m p a c M o ,Eccentricity C h ar ac te r s t c om p re s i v s tr er ig t o f concrete F a L oa d D e g n L oa d C ha ra c r i r en g 'f ",f D ev e o pm en t e ng t

L i v e

m p Unsupported m n

Ef ective 1lpa of beam a b n g m n

Ef ec iv lengt about a x

n g a x

.,":.

.'

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lx

ly to

.F

g t h o

Length of onger side o f a b

D i n c w e n t z e moments in beam M o d c a o n F ac to r moment M u m ) M om en t m i Section. load on c om p re s o n m e mb e ' S , X u · Dcptliof ~ : l . z'

a.p,

Ym 'tbd ever an A ng l P a Partial safety a c o r o r m a te r a l D e g n b on d e s

'1

'tC Shear

; '

m a M a h e inconerete with shear reinforcement

Nominal

a f O O I £ !

Diameter ofHYSD ba

.. '_,'

-:·,'!·~il.-jj··tn'.·r:·~j';>~~.-~i:j~~,;i~~r ~ ~ ~ ~ "-i-'-~

Ii ,! iiT·'.:)~!~":':;'~';'',-~:))~-.~J .~~~<~ ., ./

....

-; ':'_' ''",'f '. ~ " ' , ? m ~ " > ; " "_ " _ ; ~ ·=·-~~~:..:i~""ti·c h.,'o · · ., .~ < " ,· " " , :. ~ . ._;_-."··,.~~~~-,_~~:,_c·_•." '--'--': ,-,- _ ~ _ ~ ....:.-.i~.::n.--'~_l-.:-_:_~-,_"..:..~_:.;... .•,._,~,.~,.

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llJ\lTRODUCTION

~':',L ,L :I''}';,', .11:;

5 :

R~i~~rg~d' conc~et has oday' be~6~~'~id~ty'U~~il ri ~i~nJ artdis"ex ~hsi i6 J~e~'h{tOristhicti6ri()fl'

a lm o a l o f u c d in g d g w a V o i ~ ' o uh da t o n ~ 'd am s . ,

c~nci~'te

is

very compressio'n:" w ea k n te n o n

'In

prac ice, tl

the:'

snme:

~ tr uc tu ra l m em b o t o m o n a n n s ar n d a n a s h , a in ' o n a n n C> (Q e ~ s~ d

h o n i m b T o ~ e ~ m h i w h

is

str~ng,in

tensi?r

is,usedal?ng,,'rith,. c O 9 . e t~ ~ d h i c om b n a o n o f e e a n~ ' c on c e t IS n ow n a s e in fo rc e c O w re te . T h e

rn~m

r~a~p,n!J;

h in d h e h e m b n a n , is

'1;\

~ J ~ ' "

I! )~~<,,,: ,!f~

Advantages of Reinforced

Concre'te:::

:":,,,,;:_;:"Hqtnn'l :.;!';

.1 C pn c e t c om po ne n c a b e e a H ~ ~ a a ny .d e r e z ea nd :~ h ap c ~ cb ea m s ~ P '~ I 1 1~ , ~ ,~ ~ a

shell~foldedplates 1;<,:r'i';'

"2:'

; M o s t

o f h e m a te r a l r e u i e d forconcrete a r e a a n o ca l . .

3;': D i n t g r b~·made o v d e u i d eg r of strength.," 't

C on c g r g t g a n s

5" M a n t n an c low comparedt timber andsteel u c ~ re s~ '

. " ; "

" .

:i'J

6.', S t n g o f o n n c a s :,i!

. ; . " : : , F - ;; : ~ ' .~ :

--i::j";H·~;:~t~-~·:_!('~H:'l: :fO~: fJr_/i'

1. Th m em b n c n n dismantle .and

re-used,

iasinlhYi911S~:PJ; ~t~~~

. a ~ 4 , H ~ D e ~ h

2.'

'tt:i~

n a m u 3. r eq u r e c en te r n g , c ur in g etc., anddifficulttoconstruct. ,-;,:'.-:.f 4. T im e e qu ir e o r c on s u c o n

is

m o

steel.

Materials used '~ T h n c m a m en t n e g g g a n d g g g a n e w a ,:II';::I{. 1 : 1 ; 1 , ' •

Types;()'r~ement Bnd )ili i):h· :':,[i

~;.;- ; >: _- -' _; :" -l ', ;' ;. ;~ r; ~: :: /i ' ,_ ;','tl;'l ;:j:L;;,':,,_L "~~f« ' . > ' ,i ;-:,:;~J ~ - - , < 1 " < > : .~~·!L;:.;~': .-. ..•• :,;,~~,:,

m e m e will depend itsusagefor'a'

r w

1.Ord;inary Portland Cement- 18:269·

' a u ; 8 c ; h ' 6 r e t e ;

costniction

w o ~ h h e ~ e h a thesoilor in

2.,Rltpid'bardening portl~~dcen;Bent

r, .:,.

h e n in g m en t m a n l n d

rapair

~dtb i~ r id ge s a n

'roads

since

it n s h e m e n g 2 4 h o w h n a m en t

3.,porthmd

This m en t m a n c n in g w a n d m s

;.~.~-'...-.: .-,:.,~.

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h i m en t m a n c w o m a n e n d h y n s w o

5. Hig Strengfllor~inary.Portla'ndCemcnt~;IS~8112 ; }

:;••

u se d p re s r e e d o nc re t r uc tu re s u ch a b r d g a i w a e p r s c ;

6. igh' alum'ina Ce~ent,~

This

type!

of

m en t h em i a n n d n in g furnaces

a n d -

fo structures

subjectedtotheaction

i . : >- ; ~ : ~ . , • . -~ ; " ;.c,J:: .0

,The a ga i manufactured hre e grade s rJ~rrie ly33grade , grad and 53'gta:'de re~~ec ively,

T h m p n g n t g ra d m en t g i below.

n om i a l h ig h g r m en t ~ in ~ 'th~-~~{yill a v n g a l a n m en t

' o !:<; ,~

f s

at'

;Mi'Id:steclhars c dnforn1 ngtoIS:432~ 1966 Grade

m e m i 1786-1979 G r a n N o A ll r e n f m en t h a n i a l n d n t m u h e o a n g w h m a n d 43 rade M i M i C on c a n bemade1 h a n g n t n g h s n g h e n e n d h u n c , a . t s s Design M l n n

When c oncre rm... IS g ra d a n m s m p n g h , itis n ow n a s g n

ix concrete. \':1 g r g n m i n c a r g i w : . . '. 'r .' .,

_ ~ _ M _ ~ ~ ~

~

', G r m p n g 2 8 s . ID 1 0 N /m m M15 15 II ~ , ~ i , ;-

,I·i

20 .: M25 25: II JO 30 M3S, 11 M,10 40 .11

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T h e h ar a e r n g d ~ n ~ -d '

~~'-th~

~tr~~gth

'~f~~~t~ri~i

-beloWwhidl rllore'ihrii15' percerit ..', ... concretemix, lqttcrMref¢ts'·to'the a n h e n um b g i h e m in im u h a a c m p n g

I ) f , } ~ Q ,

b O b a t

r,',I;'J;,J:, : : _ :- I: ' : : / , ,. JJ ,;-, ;:.,.i:-.,~~:;<.! 1 \ 1 1 :.0" N om in a M i C on c

_Ifconcr~;t~ i~'~p~cifie aspr6pdrtion C:em~nt~·ne aggregat andc()ars aggreg~te'hYN( ur ef ~i

m i m i W Q r ~ .

g r

1 \ 1 ;

~O,or I? :er, n?minal nixn ay He used w,hich howeve isH~e

t o ; i ~ : :( )l ¥ e '

h~gh~~c~m~~.~il;:

~ol1ten,t.

Nominal

ix Y J : ; : - ; : ---T;-:-:-T -t':'---..:---'":.---;-'"-:..---.:;~ ...-.;- :~ --...;.,;.,.--~-~;.--:"~":'''"-~..;-:---:..--~-..---,_...---..;--:--;-~-:T ..~7-,.::'-;-..,.-~; •... Design MI

~---~,.--~---,.---~~---~--~-~--7-7~~---:-~--;.---~--~7~~---;---~-~,

..,.',

M a n c n g w o n da t n s ,\-.' Hs\vhichn'eed;hot

,JI"

• ' ·'-·,;-->Ii:'~.~,·~;·i._,::rjt:,.:_'-:,.,[f.~(-.<r:,': ~:qHC~)" "1 ... !:/,; :J1l!;' B ea m s S la b L .n te l S ta i R e a in in g w a & 'C Q lu mn ~ " i. ,i:"i.r.9f;.up~~r,~tQte.ys.,J,.;;..'. C o m n w e a n n g m u_ 1 ~ . Q '_ 'L " m e w a & w

Coiu111l1swhich are

he~~i1y

lo~d~d arid 'lor IO/lg\pan ;arc~e,~.;'·:}\i;

m b ~ d

F6'rpre-tensioned me~bers6 pres ress(! 1 ;

_ ,1 1 ,: _~ !. .~ , ' ,~ ;: . 1:2:4 20 1: M25 M30and 1. Strength 2. Workability 3. t':>5-:·:;";. T h c om pr e v e n g o f o nc re t d ep en d p r m a r y on ag e c e m , n t c on t n t

~vater

e me n a t o .

at2~~~ys:

daysis

about /3 of 28 a y n g a n g o n d a t ~ ~ ~ pg t li~~~y,tp be attained.

' ; f ' ~. !'

;:i~:"

W h

will

n o g n w i h i 2 8

days!

aft~r:'

ih~~

a s n g h e m em b~ r a n ~ uh da t n s a n towerc()IJn111~jn' m41tistorey

m u n g h e g i w .

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M i h h g e m em b design load/stres is.

m o factor ',:,'j ',3' 12 .1.QO 1.10 1.15 1.20 Note respect ; ~ Note W h .they h e n g m b n a a d n d n d n g n g n s n . Workability:. T h B { ~ r o m o n t w o full n p w a m e n g g r fine e me n r a o , but n g rk .Placing conditions _~"."·No"';':"'W"'''' .... _ : _ ' C on c n g h a ec ions wi vibratio C on c n g o f g h reinforce sectio wit vibratio C on c n g o f g h reinforce sections w i h o a t heavily reinforce c t o n w i v ib ra t o n .Concreting heavy n f n s withou vibration' Thesevalues of m p a r

fo

2'0rn g g n d w e m a g a V e ... Medium High Compacting factor Slump" Vee-bee m e 0.75-0.80 10-20 0.80-0.85 5-10 0.92 75-125

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,.

;;

li

h e n c m u w i n c n d the environment in

which it h e a c o r a f n g h e durabi ar quality cemen wa er and aggrega.t.e~;

w a m en t n d m p h e w a m en t h o , " o re s a n n cr ea s d ur a y , :. ': .. '. '.

---~-;;~~~~-;iQ~;;-i;c~t~~;t-;~;;[M~~-I.,

SPI~CD!IED Exposure ·]'1010 r. l~eHlljforced Minimum" .MaXini~m" , ' ~ , ;_I '. .':j ::::'1"( 1'1'(1)' f" "·(2)": 4},!, ",~(5)

"',:IIf

!,~.:" ~! (kg/m •.

::"':(kBr~~?~(:::;!~;::

---~~:~7"7~--~

..--- ...

. - -"!'"-!--~..

IIII.--...

~~M--""--~--""~-~--r-I-I--~-.~-7l:~;r:-~;T~~_~~"::rr~0.tII,':j'~;.~ Mild": Forexample, 'protected" agains a gg re s v e c on d o n e xc ep t for n or ma l W e a th e c on d o n: - . .

during cons ruction.

-~ ,~,;,:1t~"-,; . I ..': .t ~o er '; or xa pl h e n d w in d rain against

reezing, whi sa urated W i w a _'._'! _,. n c in"soil arid 'concrete

n t n u u s u n e r w a

0.6

i' ~ " ~ ;

- .- - j . .. .. .

Severe :' examp e, expose

to se water al ernat wetting an dryi~g

reezin whi st wet,

ub ec heavy condensation corrosive fumes.

310

~ W maximum water-cement ra ca be strict controlled h e m en t content h e

a bo v T ab l h em in im u m en t n t n u g g g a or 40 1 1 1 m aggregate, it should' m m it h ou l b e n cr ea se d by a bo u n t ::::,.', j ; - " , , ) ~ - ' ;-':;.~ ~"~:.'i."'lU:t.t:~~~~ ,~t~'--·- t.P_'"f.,.~~.~.,~.

~-,-,--~_...,._.,.;,,____..-Print document

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. ~ - t

' ~ r . >! '~'-l'~'1:~: 'c;

plain: concrete KN/ni

KN/m respectively.

T h m a g n by n e h e w in g m e h o

EJ~sti theoryi.e~W orkingstress ' ;

2.L im i a t and'

3. Methodbn"~periri1ental inve gat ons

',

h e c om po ne n

m a a l a c h e n g h s T h a c a f

used Se in maximum permissibl stresses fo concrete e e a r 1.78 respec ivel

~: (~

Limit state method:

Lirriit m e ~ c a l a g a d n g g h a p w i h e a f a ga in s a p a n a b <' is an part al h e e x p e r a m 1 t : f tl ll :1 H Investigations: T h a r g n a s h e a in e m en ta l n v g a n s n d m o Notes:~· , ' ,J •- ~ ' "",1 'T .rl :' \~

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~t: ..~: I· I, ~j .. f. .' .: :,:.>.~~l' 'l ~;;v·,~. :T::.; ,-;i:·'·~n;,<'·;·; ', i~':~(;'F':' '1 il'~~;{:~:)!~~~jd:·:!,~',:.i:\},;:;~·:·' <.,: ,. ~ o ; .~.':,~··.;~~.':t~·~:~:<,~,:::c<~ .:I~;i,,~,lt,~,:.K,

I>:':

·1'1

;: :i ii,l.I !l f;.irhi·

·.i!:A"PRl NC IPLESOF

:UMrr

!'";..d ;:.L:jJ '~'" c.~;:

(,;:. :,';:':".;,

ic'~U c ~ · uI/,:Jr!fi.~i:rrr,~((}1,.( j; ~~~:·T,}~\.'J::~ ~.>,\!~j~~ ]r,L

:ti~iJ;~,t~~~:fu~'t~?d:

"-"':.! :'!'~..

~·Ht<.!~fit,

ill,

;;;·':':':i:I;;:I;JI~~i);j·;~'~:;'~.~;'tf!j:;~.,.·,

.;In

limit. n c h e ~ r1 J e s h ~ d e

g I 1 c q

o'wi hs and ~~f y; aU.

,' o~ds ab ~: i'ac il it throughout.

it

life;it.;:sh~. also ,'s~tisfY '~h~"

, s e i Y k e · a b H t y . :

j,rbq~iI'e!riJ~jg,sJcl{;as'

a t o n o n_. _. a nd - a ck in g T h a c p ta b m i h e a f a n ~ l ~ - ~ ab i y J ~ qu l ~ m ~_{j:.·,11J~:j'«'.!i} _',"it":i" _""'~f~ _{;·'+t~·;::....}

~ ; L im i h e design achieve ilcceptable'prob'abilitiesthat

' : i ~ e

~ ( r 1 t q t ~ : r ~ ; \ V i n

n o m e ~ n ~ :

' \ y ~ c n

it is ,int~nded~'t l 1 8 t !;,t~l~tit '''Hl,'n~t'reaclf aJ~mit

. .

1 : ) ,

!ii,.::!tjii)l!;;·w :n .jlliji;'r:/.i,:.:<

i' ~ ~ ~ t a~lequate ,qegree

o(s¥e,~l

and:

$ e g e h e g n

'on'the

basis ;oftlwnlost'!ririticiabliiriit~statei'and,

shall bech~cice tor othe mi ates~f ':l';Y":i! i. ,dr,t'.ill! rv-: i,'! i'~'}"! 'i

'Yi;i',i'iJ!1'!,.Ji<;'::frj1/~,';l;/':~HI,

'0

·-.:·:~:!·,·;·:"f~."'<',',·· !,'~' ]T: ':,t·i<)'·<··-2_"~,'<;'::::~'V.';5);.'>~'.:r\:·:,!:,:..· · , : > 1 ' ~ ,

. F

design

'slfdutd' be ba ed .on harncrorl c: va ue o: 6r "mater al

~ t ~ n t?k~ n t a c n t h e a r a t n s m a ~ ~~ ~ ~ ng ih s h d h th e

T h h o ~ l V h

ata.ctel1~H vwue hrough he of partHU!safety factors,.one matf;ri alstr~ngth~and,~h,e.·()tqer,for:

".C' !,::'j, ,·.. ~ j' c . .,'~.

o ad s n c ~ p ~ r .should: h,ay.e'~heNal~~ls,

~y'~~ i~l~la~ jse"3S

of

I S : . : 5 ~ : t T . ? ~ J ;

according o. he~a. ~r~!~. P, ~~~ l~~~l.~¥}~d ~h~

!~~~i

~'~a~~bf~~~~

(~~~~~~~~~~2'i'jh;j

Limit

state

collapse:. ,'i

l,

",:r;,

i,

.':r'·ii;:·,'".2,·,;?i.<~,'.

_ _> ' ': ~J :::!"::::.r".;,,,,·:··!:::,!,.!;~,,,qf"'~·l-~

The tructureorpart of'the u r o u be. P t~ r n e

or m o ~ i U 9~ r g ' h ~~ f ~ i

~ wa yw h~ r a p a t~ ) ~ r n in ~ ~ e a n tobending.shear.toreion a r a x a l a d a t 'at that 'section' ptoduced'b{ithff'probableiI110st, unfavourabl combina io of loads.onthe t&cture' usin herippropfia.t~'

part~al,siifety'

facitci'rs,·'·i.~, ..,'

. ' c·:, '!.;, \'1:'.,:',-' (a

Limlt

a p i' ,.,;+';~'~_.t 'fj : ti c' ~ . . ' ~ (b) m p ~ . ,, .. '.j I••: '~ .: (c) m i h e ',1 ...(d LiPlit state co ap in or on

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(14)

ni stat ofsen'iceabiDi;ty:

ol oWing mit state o f e rv ic ea b y ar e o n d er e d e g n ; D e

;)"'e"

" " ·; · k i ·· · . .. .

!'

D e

ie deflection part. there emciency

deflection shall generally m i e d h e o l o w n g 'h·~.;di;i'·i\i!

!':~.:~..? d ~ - . h : : : ~ ' i ,( ( ~ ) ; · _ t ; > ~ " , " " : ' _ ' - _ ; : . '~ ' : _ , ; " . _ , ; ': , : _..

i:!,::r,~~(~'?;~Jjd~Q,~c~i~~~~tie,t ll lo~~smcludingtheeffects of emperature, cree and,shrin5

}P~W

~ ? m , t h ~

~a~tlevel. (I h e u pp o o f o o o o a n a l o th e h o o n a l n em b~ s~ ~O ld d ;i,1 """!:

., ::..~.,t_,.>'~:_I,.~__~_:;_".'~", .".., ': _'. _ , "" ; ' , :i;l~~,_f_ ; ~ : . / -r _:; :_;_; , '" - ~r:\,_:__::"'-:',:< ..

f'

T h l d e e C o ] n c u d n g h e ~ffects~( t~mp'eratur~ ,~,~~ep8.n~

s¥~ge

~ccunng i U f e r :eteCtio~'of

partitions application n i h e h o n o n o m a n / 20nirn

w h i c h e V e

is

! .: ;J e . s ~ ; l" ; .; :/ 1 \ ;" ,: " : '; , i, :.,:;,":;',,< ; .

o :

10r;b e a m s a n a b he vertical deflection iml may gene ru m e ' p r O V i d e d that

span todept ratios n o greater h a h e values obtained

t-Ll~L:~;'_;<~'~:

:l);.::Sasic v a u e o f p a e f e c v e d ep t r a c ~ fo spansuptqlOm

~T~ ir , " \,;.;,!

''

, .

(ii) (i) b em u l p l b y O / p an ' metres, o ~ cantilevers

" it i\ v m

a S a l n s h o m a

; " < : ~ ~ : > , ~ ~ , : ~ , ~ , ~ ~ :j, "i

D e n d n g n ' h e n d h e o f e l fo tension. reinforcement, (i a n shall

be multiplie perfig.3 o f : 45 6 1 97 8 il)'···';

(iv)D~pendihg m p m e ratio .befurther

m o d d a per.fig~4 of S: 456-1978 '!'~ ' . ,,<~ . • n g m e th (ii) modified pe n f n tm t n t g e se in ec on equa (b) Cracking C ra c n g n c h o n o a b th a p n c h e structure': (clause 34.3,2 A s g u h e w in g m a g a 8Sreasonab imi s:

surface-width o f c rn c c s should general, exceed ·O.30mrrl. For pa cu ar aggres iv

n v n m n t w a n a we ng, drying to n g w e heavy

o nd en sa t o n o r o r o s fumes, th assumed points nearest to

n f m en t n o general, m e th n om in a o v main reinforcement

Cha ll servi chee ofaJ

deri~

1\1 perce caner Until h a SJJccif ii) T I 1 € wI consid Hi) Pa: W h pnrtial chance! batchin strcngtl adOptee

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(15)

\t\i

';.:'\·....

·t.i\f"ltj\:(;·f,:"(!':j!:i'!·'-~ l'·.·.·;:···". .. !.' i' c;,.\' i\:i.! _ . ig ar ia af ct 'f~~;:·...·, design'

' d d g r ~ ' ~ "

ofsarety':and

(it

8ei{jcea~ilit.l;iith,e

"structure should be :designed onthehasis most. criticaIHJ~t.~tatel~diJh,eJ1.

tIl:

c h k ~d i o r~ d h e m i a t F o h i b e a ch i v ed . h ed e g n h ou l b e b a e d o n c ha ra c e l ~ t £ ;y a u e

t{

~ p p J i e d l o a ~ s ~

m i design v":lH~s ~re

derived from t:he characteristic value h e of partial safety

';1,

!\h;it).

fl.J '.

:,i}..

·:tj~.

i) f m

h e n ch a n g m ea n h a a l n g the in te rial below hi9h,not

mO~~J~aP5

tl\.'.··

~:::j::~e::;:I:T::::;C:::;~e:~:~~~n~J~d

~ail1es

fO;'~~\:;:

n c a r g i a u ofI8:456-1978. m a m o n g

'0

. " )ii'·-'·':('"'/"·:·

)j;~i'~,': '., j":_1 ',_:-'::'

Unti the relevant Indian a n a r a t ~ r n f n g a r modified toinclude th cQnceptof

chaiac er st cs rengh he charac er value h a b e assumedas'minimum yield/O.2 percent.proof stress.

specified

. " ~ ~ "l

ii]Design values: Is: 456,;,1978 !.

~ .~ t~ ~ m a

' '

characteristic $tr~ngth ofihcmnt~rial~(Jbk;f~r n c a n fy

Ym= Part al afet factor appropriate he nater al and he rrii state being' +.'1(1)

;i~~~:~

. a r ~ l y r a . ' t o r Y m

fo mateJa

<;,1[1

a . a l n em e limit st~te

Afcbiicip~fth~,~J'I~e

parti~lsafety

factorYrn

should betakenasLs

fo~concrete

. ,

. _ c . . :- _ll:~I - ~ : ;'j < ' ~ _C > : _ " ~ .+ l :::;,1_-'~·~-;~f'j:\:~, :,__i' ~r;.~~" "-"'J<:1

higherey'alue o f p ar t a l a fe t a c o r (Yme), ~r concre has" eenadopted ',~ecaus ;ther~.,~reigr~ater·,

I, V f ~ t n g n c due' to;improper. cp'mpacion,i.n:1ldequate

curing,irjlpr9per:

batchtng'_aJ1d'J1li~ngand The' chancesofv~riation~(irithe

r en g cifieihl'Orcementare known-t6

t i e '

m a n d h en c Y r = i Y h n :

a do p e d T hu s th m i a t m e h o the design stress n f m em . Y m s: :~ .fytl.

\. ="O.87!fY.;\~cQr~in the IS:456-1978:l~~(~~sig~p,~f1J~~e~ h~:~omp~es ~~.~~tr~n~th()(9o~ci~te, he

0,67 times the characteristic strength partial' ~ f Y m c = =

J.~ addition to '.!'. ,;,;1 . : ! /. , ; : ; \; I. i· l; ; : '· I : ;: G , " .

!.' l!' L ' ( ,; : < - 'f

\y

in

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(16)

.c 1. .. b) Imposed 1 0 1 1 . d , : ' . ' - ; j' ~ : ' ~ ' fc " ;- - :' ,} ~ : 1 :, ;~ ' :; _ _ ; _ ~ :' 2 ;> ' ·. t~ j ~ !i:i);,t~~~~~(¢ri~tii~.~OHds!:.li:': "!'

f)T~et:t~mi;Ch'ai~~eriHtic,Ioadmeans' Whlc.hhlls 95 per~entprob~~ility)fnot being

;':cxceedCd'

. d u r i n g ' the, B r structure. Since u f c ie n d a a r n o a va i expres

'statisticiUt~rms,: th loads 'given in various relevant codesshallbe assumed h e c ,h # l c ~e r~ t c lo ad s (Refer clause 3?3,.2 ofJS:456"1978)J

" ;

i,

- - <

ii) Design loads (Ffi<f.:a~redLoad or Ultimate Load

/. .' by F.yf '-~I····'\_-.'·-~_::'·>':::-·: ~:"-~-·-.,·};-'f:,:·': h e re F ' o ; g;cbaractcristictoad, and.': Y ( limit state' I'·;t)eing~·corisidered. . : ~ > : · ~ U ~ _ ' I · ; ' ~ : · ' ' ' . ' . ~ ) · !~ , _ : _ - ; ; ~. ;

iii) artial safety factlO!!' (Yf)

:Th.epartialskfety factbrs(Yf) for h a a s g i e n ofIS:456-1978.·;

faCt~rs fo Load combinations L im i '. WL' DL' (2) (3) DL+LL '.:

.l.?

1.5 1.0 ,1.0 t',5.or, 1.0 DL+WL ,. .S 0.9* 1 . 2 ' _1.2 1.0 0.8 DL+LL+WL_{"1I;:"~tltl•• IPt'".f.h} 1.2 'i\r'ote W h W . , - ._

.'::Note

, F o r

m i Y f table .are applicable

n r W h i a s n g h e n g du

likely o be p y o y be c

,+ This value 'is:to lH!icon~i~ered\Vh~n' stability agains o ~ ~ m in g stress reversal Hicritical ..

,;Factol'ed 1 ~

:: :, ,_ :" :_ »,

section moment du charac eris ic oa miltiplied .1

, a p a t a lu e p a r t i a l ~afety

.Factoredloador design o ad , F d h a a d a l

F.Y£'

Factor moment M ud

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(17)

~.!l

Ii

bv

M I

A T O F C O

E XU R

Stress-Strain relationship fO c~mc,rete

. .

T h ~ t n c h ow n fig. as accepte IS:456·1978.

L , _ - ' , ~ '

m a

m a n ~ by

, _ ,,!P~j::,

d , h ~ l u a e ng t o f c on cr e

is

less

than

cube strengt

found

laboratory.

h e

factor ~ t ~ d a c n t h e n c n d a t n g

structure

0:67

ti~~s. h e c ha ra c e ng th . T h a r a l a f a c o r (ym=Ld) a pp l e d a dd i o n to this, andhencethe maximum compressive stress in

will w- ..l_= ~-I- O.67Fc:k/l.S .. .446fc:: STRAIN i , L.~ -H .:., .... "";:"-'_L·,~~",,_~,_!__,":',,~'~".~ ::

..._;__..__...--~-«""""",' d~!:au!tt:m"'!e!lt"MS'E!9'i!!!:..,~-n~lm!tlii!i!;;;$I!' i r .l l& ' ¥ S I S : I : ! I S = = = 4 i l 1 1 ! i i4 i f l2 t = l is m l l = = S i h : ~ I ; _ ~ ~ : ~ ' . : ~ , ~ . ~~,~,~~:.?'wI:.~~.~i_~..!.iJ~

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(18)

';1

••

12

e-~,'.:'

Stress-Strahl relationship fo stee

T h ~ ~ m i h ow n g . Y ' 7 8 fy 1: f'y ---~---'--- 1.l5 I'

of

'>( 1 0 : ' 1 N/mm M i ~ r H e h e m i o m 1.15 0.87 :':'. ~;';' b) The m e :456'~i978. .i._.~", :1.~·~~[U L_: ... ..:_':I'; '200000 N/MM2, C U I ~ f 1 I1 1 1 ~1 .1 1 O A Cl ~,~. .... :, ote: n i n g m a n g h e m a g e . . __; ~_.i___..._.,,~__.,_~.,

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(19)

'1111~:~bla;llib~~~ l'highstr~ng th steel

bars

tdl/

r i 6 t

h a v ~

~:defiriiie}~~ld

~oi~l

since

tl~c'Chllnge'from'e~ea'sH~:;': plasticcp~:di io is quit gradual Here~ he design yield~stres~;"1)ecomes YY I,1 5 : f P , 8 7 z y , ' o ( ! r e } I _ 1 _ ?

i s p a r i a l _' _. _l _.

AS$Ulnptionsmade in

e m b e r s

m i a t o l a p

, . , ' ..~~."-."~~

< "~ r ~ . LC~'_' .:~-·:-"i-,~-:<' - ~

ii).

T h ~ m ,a X im u m s r a i n c on cr e at he'9.¥!err:nos compression fibres i~ taken asO.003S in bendip ..;;

iii)'

m p m a

a ng l~ , a r n y h e h a w h n g n t

a g re ~m ,e n w i h e r e u l~ s o f e s A n ac ce p a b : e r ai nc ur v g iv e } h e g l

~9H.l;p{,;

IS:.4?q, 1978 F o d e g n u r o s h e o m e s strength of concret In

0,67 m e h e h a a c n g h o n c T h a l a f

e a

iv)i!the

i~n~ile

strengt of concr~~d!i;,"i~ho:ibdJ

1;::"

.. ,'

ilDn;~i:<s'"

T h h e n f m en t a r n t n i h e

. s t T y a r g i h e f i g u r e 22 I8.:456-1978: F o ' ! ;design purpose, the·partiat,·:·;O!

a l 1. ' .

J.::'.'::.l:

vi). h e m a m u a i th n s n f m en t h e a t h e a i shall ~ncitbe less thanD,,!,!~.lh::

',',> ',

The assumptio

(vi)

h e h a is n d g o

0.87 fy

.----'"':'~;;-..

0.002 --- 0,002

where fy= characteristic strength n d

m o a s

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(20)

.;

So}~e,s.~~n.s ,s~rnindlagram

ror~in~)'!einf~r~

~tio~~will~e:~ g iv e b e o w

:"f?rL~~~

in'

steel

a n r a o n r e r ea ch e their penrusslble _{, > - '. ': : : ." ~ '} _"!.,:", _{,~~(it)'J;<t'-,o:}

1l.42 X\l(Mo.)()

,l

XU(MQ.X) SECTION, ~: . ~ r ' _ ;

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(21)

I.f. ,'

..

:SiNGLYREINFOltl(~EJlj

SECTIONS

>,; Sln'g'IY ' ,

secti~riJ if e e p ro v d e o n l e n s o n , th sectionH lcallcc{ias'singlyreinfoted ij Umiti9g L im i ~ g n eu t a l a x X u m a w h gets m e on re the"strain'in"steel a ch e h e m a x m u m i a lu e e . 0 .0 03 5 a n 0 . E s Consider n g l r e n fo rc e e c p n shown in he gu e. 0 . 4 2 F c XU(Mn)():' '. S E C T I O N R A D I G R A M '

strain diagral!;l, h e g u by,

X u m a 0.0035 . [ ~ . .;\ ""'~_...~__.L.;~~~~-- _',T ,!

... .... ..._. . ._

-(0.8 fY Es 0.002) (0 P935) 0.0035 E s X u m a _{! " . .} _ ~ _ ~ 0.bOS5Es

0.87£/;

.' :' :0.0055 (0.87 fylEs) Substituting he va ue f E

N/mm

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(22)

u ( 2 x 1 _',I . ~ . _ . - . -9.0055, x{2~1,05)t 0 . 7 _ ty , ~ 1 0 0 - , " Fe 250; X u m a 0.531 1100 (0.87 250) 0·," V a a c o r o r different g ra d o f e e ·'_:::-t· Y iC l n g o f e e

N/mm

'250 .. Xu (max)/ Dl,cUltral axis, ' ,

Total compression Average stressX a re a o f c o m p ! z on e

:= 0 .3 6 c k X u ; ~

Total tensio .. steelx i : "

:: (0 87 y) Ast

Equating abovetwo 0.36fck. Xu.b': 0.8TfY.

~,;~l,'.,t~ r.;::- D. cv:u... 6 :

______ _~______________ "..1;..i'- Co ";.l

.3 ck b.d:

N o m i g n n e X u h e th

m i n g

L e a r is dislunce between th n t w h h e m p n d h e n s a c ie., (d-0.42 Xu)

,.Tota Compression:

T o a l C o mp re s o n Tota Tension:_!:

= 'Q ,

y A s

'Resisting

Total compression lever ann 0.42 Xu) 'j

Resisting

Total ensio eVer ann 0.87

.:

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(23)

the the

Jf Jf

;L

;Limimjijititing(ng(oror BaBalalancnceded sesectctioion:n:

h e

h e e ee e p rp ro vo v d ed ed .d . ecec ioio inin C oC on cn c n dn d h eh e ~ c~ ch eh e~ }~ }~ e~ e maximum

maximum pepermirmissssibib vava ueue simultaneously,simultaneously, h eh e sectionjssectionjs calledcalled limitinglimiting sectionsection balancedbalanced ~cction.;.

~cction.;. _"','!_"','! ....j'j'

In

In thisthis typetype o fo f o no n d dd d a ia i u ru r willwill n dn d n cn c lessless

;1 ;1

/<....

/<....,~:--~~;},~:--~~;} ;(,;(, (;(; ;-:;-:,',';';'_,__,_.~.~.~.~,.,.,',':.:. l;l; -,-, neutral

neutral axisaxis m em e balbalanced.anced. sectsectionion isis calledcalled asas LiLimitmitinin NeutralNeutral

'-·l'

D I A G R A M D I A G R A M Considerthe Considerthe n g ln g l r er e n fn fo ro rc ec e e ce c o no n e ee e o no n e n se ns o no n s is id ed e o n lo n ly )y ) h o wh ow h e ph e pr ~ vr ~ v o uo u g u rg u re .e . r hr he .e . r ar a m u m u g u g u T hT h m pm p n cn c n tn t h eh e n tn t o io i o fo f h e h e c oc om pm p~ e~ e v ev e o ro rc ec e h eh e r er e ~ l~ lo co c a ~a ~ 0.420.42 X uX u m am a h eh e m em e m pm p ~ e~ e A r e A r ea "a "o fo f e ce c: ::: o no n A v eA v er ar ag ege r ~r ~s ss s ,i,i " -" -. _ . _ - "- " m am a = Q= Q X uX u m am a Leverarmdistance

Leverarmdistance b eb e w ew e h eh e c ec en tn tr or o of'compressiveof'compressive forceforce n d

n d n tn t ;l.,;l., , •, •

T h

T h m am a m u-- .. m u m om om em en tn t n cn c h eh e n gn g n ,n , Force

Force LeverLever

A rAr

~.,-~.,-.,~·.,~·-,~~-,~~d-L;d-L;,,~~,,~~;,r:·;,r:· 0.36 0.36 c kc k b .b .X uX u (m,x)(m,x) [d-0.42[d-0.42 (~ax)l,(~ax)l, . :. : !,!, .3 .3 ckck .d.d .,--.:.---.,--.:.--- 0 .0 .4 24 2 • - -• - - :

-:-d)

d)

I.~~o~ I.~~o~ '"',; '"',; ; : ; :

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(24)

,,,. ,,,. 18 18 '-, '-, '(._'(._ 'Mu(liIh) 'Mu(liIh) '!, '!,~,>J'~,>J'::-.::-. -::.-::.',', W h W h '. '. fckN/mm fckN/mm '.'.

Fy,N/mm

-,-, 250 250 415415 500500 15 15 _~~_~~2.07'.'2.07'.' 2.002.00 20 20 2.972.97 2.762.76 2.652.65 25 25 3.713.71 3.323.32 30 30 4.464.46 4.134.13 3.983.98 >=-" >=-" ~ercentagc

~ercentagc ofs'teeBofs'teeB forfor limitinglimiting sectlonsectlon (%(% stst m o

m om em e ~f~f wsistancewsistance willwill m im i g eg e n fn f m em en tn t n .n .

The

The percentagepercentage m pm p (Tu)(Tu)

m p m p

Total

Total tensile

tensile

Stress

A rA r

.:::0.87 .:::0.87 Ast.Ast. fckb. fckb. X uX u m am a 0.870.87 fyfy . b . b stst ~~~-~~~- -.-. .---~.- --~---::::::::::---~- ---~-- . - . 0.0.8787 fyfyO:O:S7S7 .. .. --- --- -.: .: 4\'.384\'.38 A s A s m )m )

--.;.~---

--.;

.~---

---

: -:- : :::: :::---

K ,K,

0.87 0.87 1;1; 41 41:3:3 fcfc

X ~ ( m a x ) . . ,

15 15 A sA s = === - .-.; .;.- ---~ . ;~.;. ~ -.~-, ;,;~ :~: '.1; '.1; 'd'd 41.38x15xO.531,ii 41.38x15xO.531,ii 1.316 1.316 250 250

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(25)

.: .: 19 19 .•.•! .•.•! i"i" =415 =415 N/mmN/mm Va Valulueses f%f% -250N/mm -250N/mm 1.757 1.757 .2.1Y7 .2.1Y7 2.637 2.637 0.955 0.955 1.194 1.194 1.433 1.433 .;.; 0.566 0.566 0.755 0.755 O~943. O~943. 1(132 1(132 .' .' d,

d,rcrcHhHh ElEl g'g' 1I1I1!1!1010meolmeolco-efficientco-efficient Qu Qu

,'·'M20

! -! - , ',' ~ :~ : : ':' Charac--teristic -teristic .strength .strength (fy)(fy)

2.23, 2.23, tHil tHil :M15 :M15 _1.318%_1.318% Con-crete crete grade grade Steel Steel grade grade (mild:

(mild: N/ffim2N/ffim2

0.531 0.531 0.479·-'bd 'bd .""-" .""-" 0.531 0.531 L757%L757% Un

Undede IlIl'e'eininfoforcrceded sesectctioion:n:

When'the

When'the percentagepercentage steelsteel h ah a h ah a inin thth

baj~~b~(~tiridlt!6~,itiMn~utr~lllbti~'

m o ~ e s

w a

w a h ah a h eh e

of

of compr

compres

essi

sion

on

force

of'tension.:of'tension.: J,J, !l·;i.~i,f}!l·;i.~i,f}

H e

tiilderii1creasing

bendingbending

m o m e n t , '

st

steel

eells

ls st

strai

rained

ned

beyond.theyieldbeyond.theyield

point

. ". " 1;:'.1,'(,'1/"-11;:'.1,'(,'1/"-1

(0.87

rYilfs);+

0.002'0.002' o ro r h eh e n cn c

attains

m a xma x m u mm um p ep er m ir m i b lb l v av a u eu e ofofstrain.strain. ie.;iOi0035ie.;iOi0035 Henc~

Henc~ thef'~lurethef'~lure

widb~

due'due't~t~

i h ~ ;

l.iliJie

~f~teel."Stict1

ifJiIUf~

i'i'dil

dilted'

ted'

c n

c n i o u

i o u { f i i

{ f i ii lill lllr e '

r e '- H c

- H c r e ;

r e ;: J:J l llle h

e h

h ehe

co

co crcretete-e- aiai s,s, thth ststrarainin ininsteelsteel willwill bebe grgreaeatete thanthan (0.87(0.87 fyfy Es)Es) 0.0020.002 thethe value.value. ofof acactutualalneutralneutral

a x

a x WilWil bebe eses erer m im i n gn g n en eu tu t a la l a xa x Xu(max).-

Xu(max).-'.;' '.;' ;:.;:. ":~":~"':l"':l -:-:':'"':'" '.'. (mild (mild N/rrun2J N/rrun2J

r=r=t==:

F e

F eA t

A t

415415 _ ( _ ( ~ t~ t N/mmN/mm 0.479 0.479 2.762.76 0.955%0.955% .... !'d:2=tC_ !'d:2=tC_ r.r. . " . " ::r:~l,!;~l~j.iwmwM-'$W=kmt*_~:m';~~~*?I¢AA.*'*.*?

::r:~l,!;~l~j.iwmwM-'$W=kmt*_~:m';~~~*?I¢AA.*'*.*? "0'""0'" 'iH:'iH:

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(26)

EcuClil"1) 0.0035 SECTION TRAI D I A G R A M fy 0.002 Es fo will b e e s h a nX u fm a x] , Tlledcsignstr6ss 0 . , ~

Moment a n o f h e o n =Tensi orce a n

= 'A s 0 . fy d -0 . X u st 0.8 7 y.

11,:,

0.42

--.l,·

d)

Substituting value

:.

---

in h e O.87fy st u.36 c k b . A s ---0.42 A s 0.87 fy 0.36 fck. b.d. _'----~-~--II

OVER REINlFonCW:O SECTION:

W h e h e n t g e m o h e n d m o

n w t h a t m p

n t h e n c n c h e m a ie ii

'1'1 ~ 0 W n m w h e called a s o ve r r e n fo rc e e c o n

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(27)

;moves'

ailure is

21

W h e h e w i h a h e fy s) O .O O a n h e

value actual neutra axis is m o h a h e ' .. '

.: SECTION 0,B7 0.002 :Es. T h v e n f o n .isundesirable n c h e m p o n a i u r is d d w i h o warning. Th m m en d h a h ' h o g n H ow e n o n c n i E in c n f m a a k a l tomoment a n h e limiting

Here

will

h a Xufrnax). ';

Mu(lim) Qu.b.d u (

(01a~~)\J';.:

, J , - - - -- - - ~ - -. ; -

ri..~---~----~--;.

..

b.d~

l_'!

,'): ~ ~ ( >

T y p e s

ngly re nfor se io . ' , , ; _ -j f ", _ . : ~ .. .

dept oflirniting neutra axis.for ~imiting,secti9n. --;---~---... ti i, Stt~ No.2 D e m in e h e ~ c a i n ~ a l 'axis -:«, .. Xu 0.87 fy.Ast fck.b.d

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(28)

i· 22

>:~;' .i. > ': - . .. .. .

under rein orced.

~u=0,87zy.ABt.d

l--:::k'

b) Xu X u m a h e h e m i M u m ) Qu.b.d 0.36 fc

-:~-~~~!---

0.4~

-~u;~,~~'{l

'c If / d Xu(max) h e h e " b n e w e M u taken equal Mu(lim)

::

0.42

~ ~ ~ = J

b,d !~..

Type To filldAst fff~ give sectio and Bendi~ Momen,t.

N o m om e M u Xu m a M u m ) 1100 0.87 fy Qu.b.d Xu(max) =0.36 fc ---

~ - ~ : : 2 1 b .

... M u M u m ) h e h e n d

E q ~ : ~ ( : ~ ,

u:::

::~:t

~ ~ ! ~ ~ ~ _ , ~ c l t i o n

fok.b.d

From the: abov equation, A s m a determined,

)0 M u m ) H en c g n '[ .. 'j

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(29)

23

Type g n D im B t n g s n g n f@ R " o r gave :H Step No.l~:~in factored m om en t M u

Ste~N().l: X u m a M u m ) ~everarm.

'.scepN~..

~ . ~ ~ ua t M u to '0 (max) m a 0.36 fc

-'d--

---.i-- :..

il Assuming Carl' m i value o f a n o un d

Step No.4: n d A s by equa in otal compres iv o ta l n s o rc e

0' ,3 ck.b X u m a 0.36 fck.b.Xu .(max) ':~ ,~.' ,_.; st ' . (max) x:..._~_;~~~~-~'~._ 0.87 fy.

~ ~ ~ ~

-m p m i m o singly n f a m m m wide'X used ..A l m i h e m i n t fe 41'S :. .. 'Mu(Iiilti·~'Q~.b.d2::.,. ';'j;;.,. ;:'.:t:~<,~:::~~.';·:::"~:~'::·!-:- ·; tl~!~'--' ,:,.i:=~.0 ! :

st

for limiting sectio 0.716% 'f

500 ·107 rrun

ProVid~4'bats'l2{Y#"

ofr~sistance ()ra singly reinforced simply supported RCbe~m;360;'

numbers n c

r" :1,' (,.'1

r'

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(30)

Since X u m a

ie,

th sectio is u nd e e in fo r e d Ast.fy 0.87 fy.Ast.d __ _M

0 . x 4 1 5 0 0 5

:~:l!

184.13kNm'"

~3bo'24'

f ;

'600

5-16

.,

b' Solution '- '- "::::i 0.37 fck.b.d 0.36 15 600

b) (max) dforM lS.andFe 415 0.479 c)

.,

c)

:Calcuiate the m om en t o f e s a n of singly' rein orce simpl supporte I)

wide effective is rein orce wit n um b o f2 0m m Fe 4 1 e e 15 concrete is 942.48 ~~

_~ ";..

,.~I:u'''''''''''': 1 _ _ =? O . 6 ~ _ g) terr..

s.s.

0.36.x '~ b) (max) fo Fe 0.479

S in c X u (max) I~ h e e c o n is ove reinforced, The over rein orce sectionisundesirable

since failure u dd e w i h ou t w ar n n g . The m m n d h a a m h ou l be redesigned,

m om e o v n f a m Mu

resistance of imi ingsectio

Qu.b.d 2 . N m

Soh. Ass

beam

m m a r n d h e m om en t n c h e m . E f a n O m. F in d h e a f w o n g

h e effective 0 mm , n d m p load, €oncrete g ra d F c4 1

re

used, b) 0,87IY,A~lf, I) .H 603.19

~---~---~- ---

=' 0:40 c) I; 0,36 fck.b.d 0.36 0 x 5 d)

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(31)

'~

I.

\:,~

Example: m e beam, 300ri1ln 5~)(Jmm

bending moment C on c e t g ra d MiS n d g ra d Fe 41

It rg

____________________

0479

1100 _I_{~ .} _{_}_, _{. "} _{; .}

xu}

Xu(max) • t section is underreinforced.

.._

='

, . ·i ·' '" ,.,i i,e',

41 =..o.~7"x415

~'rm

Fd ., ---

---

29.0 kN/m ,~ . . : . ' 29.0 W o

~l

1'9.33

KNIM

-i-. .5 .5 S u m p a d !n

kNm'

Ii ~i Solu~ion: Assuming 20n:undia ar Ef ct e; dept~"d (20/2) a ) N l u ( l \ n ~ ) ' ; j _ Q ~ 1 . b . d 2 : : : 2.07 .x

3()nx4$.:;7,,7.1~4.28

k N 1 1 1 :.,~_>:: ~-,l...: :;,.;cr.d~,,::;,;':.<;j! '. ": b) ~ en d n g m om e K N .':j :- Fac ored

c) Since'Mud <J\.fU (lim), th designed

fy.Ast.l

d) ud O.87lY.Ast.d

f c k . ~ ' j '

1 .: ' i"C:

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(32)

I\-sl:.:- 'i.'5.,

Asl.:..'2.-Soh3''4fI(_= f.k"lo·{,

'2. _'~.!32..'

I\sl!'_ '2i<~2'3'1,.bKtt:. 'J.l,£ := _'_2.b~fA.·j

'1.021

. .

f,)-

--::--3·b~ID

/"2-'2~{-e

120 10 -0.87,,415

" A s i

465

A 1 ! t

...504~.20 ::::0 (Quadratic Eqn.)

,E beam resist n d n g m om en t of 10 N1\ill.. nun

n d h e effective tensile reinforcement required. C on c g ra d MI5 steel 'grade .!

26 it'. _t _•

_{I .}

_." I" a ) F ac to r m om en t Mud 100?< 10 150.x

l0

Ntrirn .. '= b) ... _ ~ 0.479 0.87

_x u, ~~ ~2 _

.! '= Mud ---- -- b.d~ "P' .... d) Find st Xu(max) 0.87 yAst 0.36 A s A s I053mm bars. a,b\.e.(,rUJ1d _{_ l} singly reinforced beam carry

a n ) W e

load af20 e N m , T h e cletif span

h e has 300mm bearing s. beam (2/3)~.yse~15

Fe415.

.;

~I

Solution:

ltjve

load 20 N /m ; load. 20 kN/m

4, di

: 1 C L , 1

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(33)

," - : , " t., (,;i t,11 Factor~d.noment 93';.75kNm'~';',· b)~~~__. M . _ - :- _ M _ _ ; ' _ _ ' : ' ~ _ _ 0.479:

.noo

Equl1teMud Mu(1im) Assume (2/3)

Xu(max)

J '

M~ ', 0 .3 6 c k O A 2 ~ _ 93.75>:

1b6~

0.36,,)( 15 0.479 '(i' (t2 408 mrn, a y 4 1 b,·:::l ('2/3)d (2/3) mrnj ro .' ;_~':~::-L:'·{l.;'"i} Find 0.36 o.fi7

A 1 t

0;36

' K ' b , ~ 1 7 0 0

><196.39 0.87, Ast;<~~.A:tt ,''5X t~n2 .! -'... Provide3'Nos.16 bars.

'><~i:,

,l;'?i'iiifi:;!;

Exerc:isc

t"

j/ ' r > ; :

m m 0 m a s with M I o n and, mild steeLDetcrrrline th l i m i ti n g

moment n c n d h e Iimitirig

_':~'H~'~I";';"i:~~~i

An,!i~er

: M : 1 1 ;

(lim) 113. KN A B t ,

(lim)

""."c",

m m w i 4 50 m r d ee p e f e c v e is reinforced 20mm mild F in d h e m om en t n c h e ; X u n d n f M u 103.91 ~ o m m

4 : ' . :

n um b 2 5m m m i a r MIS s u Answer

:Xu

79 X u m a 0.53

f;

o v n f d , M u m ) m m an 500mm m b m m section. MIS m a 0.479; n f M u m ) 119,025 IcNM.

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(34)

5.A singly

; ~ i ~ " r " ; ~ c ~ i c e e Il t

c on cr e e a o f o ve ra l size 450mm n f w i n um b btu·s!

n i o se d o ad /r i = r= .2 0 E f a n= 6 0 m L ~ L ;

Ariswer:l'Xu(ma~)!d

• " W o

,,'I se w.cight:~f bemn=2.81 kN/M; super Imposed

.5 to .r is bending

m om e ~ OO . N m n c .Li;:~'''!',i~; :;!i

Answer:, Effective o ve r a s u m d =3 5m m ; d =S lS mm ; Mud=15 kNrn

M u n ~ = : 2 . N ~n ; M ~ d~ M u m ) n d h f A ~ = 9

• _

7 . T h c ta ng u a r r e n fo r e d o nc r e a h a r e b en d n g m o m en to f7 5 kN/m.

F in d h e e f e c v e depthaad tensilereinforcement

M u d= 1 2 . N m , c 1= 47 0m m , A s = 84 2m m e 3

8.Find o f e n e e a t singly reinforce RC r ec ta n gu la r b ea m u b e c e d

to B.M. OOkNm. Th wi th the beam Concret grade and steel

M u N m m m A s m m 'I·"

,i

. D e M I {all iriclusivej

et ee th n n e dg e of S u brick m a n r mmthick

Mud=28.8 kNm, c1==270nml,Ast=387nun Provide

2 - ' 1 6 # ,

m = m m

10 m p u p o r e d R.C., m m

udl

,.,.kN/m including its.pwti weight. If I5 concrete an 415 steel are used, calculat the effective depth

~ l a l a n m O~ l

M u d=260nun, Ast=372nUl1 Provide2,:,16 ".. ' _ • , y~

..

"".i' L·

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(35)

';.t "nh;:~;:~·i"~:·5·:· - f ' • ci Ast S E S E C TI O N- 2 ., ~-..

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(36)

a la n e d ~ c w i n c o r n g o m r e landsteel (Astj) fo resisting I-t··:- ~-w i m p

(As~

o r n g e x o m a n a dd i n a n s A s 2 ) n g n s o n Neutrnl Axis: Xu(m~) . ~ '. r , SECTION cu Strain in concre :Esc C u C om p by concret~,

=:,~ompressioT1provid~~d by compression steel. n s

Tul=: Tension provi~cdby tcnsion~teel).Astl

Tu2 Tensio prov de by tension steel, Ast2

n d neutral axiaequate r . ; : '''>.~!-' Ii.' • ' . : ; ~ . c '_ ._ ! ~ ' "; , ~ T o m p n , :=T p T en s n , T u

ie., _us Tul Tu2

A' =:O.B7fy st

O.87fy A s

0.36 fck

S t m p n s

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(37)

·31"

'-."'.

~trestii~

cOlllpressioUlsteel (fSt¥:i

C ~ n s i d e r i h ~

th

strain

diagram.

::;,t;iEscJ,!.li),I ;'''(Xu( ax)

,~ -, -,.- _. ", ". ..

0.001087

~t~elx

s,

r . ~here

N/mm2:

.! i\ '~.~

.Stress

in

compressio steel,

,f~~~p(.87,

y .

=0.8 2~0;';'2i7N/

'.

When

strain

0,001475

fs

strain

in

steel

When

strain>

0.001475aty11

Stres in st~e correspond

t9~t~~

that

I~ve ca be

found:

: ; :!, i;_:.i .:i.'·:.

strnin

Strain

steel

e _ . '

i;. :.! .'. _{Stress instee}

0.001475, 0.001

i 5 ~ ; ! ;

_•• 0.00225 ',.".335.0 • . , . ,(,.1342.0 ..:i. ,,}( 'i'Hi.'· 1t--:;".;..:;..;;...;:.;;.---1---'---__;;,3..,;.5,;;:_:0.~0---'---,---;,~1 '"I,,~ 354,8 0.0025 0.00275 0.003 0.00325 0.00j5 0.00375 0.004 , . 3 59 . 0 ;

It---t----~ .,

3-6-0 -, l--,--.!--....,.,., ".

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(38)

resistanee '::~:;:-;):'.,

U 1 m a m om en t ~ s a n M u = M j :,::;!:;!~.£; il '.:',,;:;::'

Where

U 1 n a m om en t o f a nc e o f seetion- whi h'is co pl rmed by ..:" Cue

d T t a l

U l m a t m om en t o f a n o f e d o n w h c h c ou p o rm e

Ml 0.36 (d

.0.4 Xu)

M2 (d l) A s

M2

Xu

(d- 0.42

Xu)

A s

Also I::t :,;Al:' ;:

M l u}= 0.87 fyAstl fy A s 0 , 0.g7rYA~t2(d-:d~~, ltim te en ~f e o resistance h e h a : ! . . 1."".: 1.'01:1 , .. ." " " p " " " . :. _ -'J .. i; Mu(lim)'; 0.36 X u m a X u fyAst1 ~ 0 , X u ( ma x 0.87 fy t2(d-d Mu(hm)

A 1 i ~ ' ( d - d l f " " "

Area _sc fy Astl (d 0.42 3 6 Ast] ---.::'"~.---(or)'

'0;0---S im i a r M 2 .. -. .: . ' O ~~ 2X u 0.87 fyAst:2 0.87 ',"; .'. T o a l a r n s

st

A s

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(39)

e. 33 0.87tYiAst~, (or) --fs , ' d th e re g iv e s ec t o n

"Fihd

li itin neutra xl (L.N.A ctor

'ixt

(MAx)

700 {I ."-.ri 1100 (0 87fy Fi'6d'Actunl

'?rJ,,·"t,-·.

0.87 fy A s = = . ~ ~ ~ . ~ -0.36 c ) F in d under, balanced, "':i) Xu is balanced. Xu(max) h e o n u nd e n fo r d . ijli)

I{!·:Xu

Xu(max) h e o n o ve r n fo r e d

,:Fl'he

oyeireinforced s ec t o n h o u n e · y" : ~ r _ . < :~ , . _ -~<~ .·i ,:l', A~':".;,"

•.·ci~)·~..

f'i~~,~~~i~at~moment orresistance(~fu ~.,'•

'i) MOnlCilt'ofresi'stance oflimitingse9t .~

,':,c,~,:~ :i,,:·d'.}'j~'i;;:'' ~ _ : t ~ · ! ~ ~;'~.':.~I '::;f , ' : , ~,,~,~ : . _ ;c , _ r ,~ , _ . _ . " _ - -. . ~ , ~ , - : ~ " : c . - , 1 " _ :

:'~;IF.M"u9~p,)nEtl,q.36fck

h,Xll(max)«t~

O . 4 ~ X u ( i I l a x » .

f~o,Asc(d;':d );

+- ~ , f}~::::-:~':~/':"

0.87 fy Xu(m~jY+0.87fyAst2(d;.dl)

,ii)

Morit:ent re~istance under reinforced section,

Mu;':,t:,::::;;+~·36

fc

.fC~

0.42

X U : >

fs Asc(d- d 1

a n i 1 ;

;J.87 {~d 0.42 +~,g7

Asid(d~l)

Hi)Mom~#.t n c n ' o v n , M u

; < ) ~ t j: l" ,

ru(lim)

• •

· , ," ': :: ~ u : ; , t: = ( P 6 f c k

0 .4 2 X u (M A X )

; : f s c < A s c ( 4 _ ~ l d l ) :

/'. .'.:;;."I"~l("T '.1'.<';,:':

Type ll: section for give ijeqdarig'

o m n t .

a c m om en t ' "

b) l1illd ~ 1 m i n g section fo given a n d e h .

:','..

I.

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(40)

...•. -.';;:..~;~-.il~i.:-.;;,.;;:;;;;.:,,).Il~:.;;;_··..c_c.~--""c.;"''''''_'.~''iJ~-=.;~=~-:~~~'~'!~nr

). Find area of'tcnslon ee As o r n g l reinforced s~~ionto resist 1~

.r. Astl ..~" ..= = 0.87 fy [d 0.42 Xu(max)] d) M2 e) M2

Find th excess moment f- resistance_. "M2 ..;, -"'7 Mud-Ml

Determine Asl2 to resist M2

0.87fy 'M2. :. ,Ast2 fy ' d f) 'Find tota As Ast) Ast2 'g n d m p A s M2 ,= fs M2 \' se .~ ...---_ ..

_---.1 ..Example-l !:

n d h e m om en t a n of 500mm d ee p o ve ra l is reinforced with2 Nos.

2 0 m m a r n s n e ef ec iv cove M I l F Solution: a) L.N.A

....

an ~ ~ -i, Xu(MAX) A .N . st 314.1 2 5 m m se m m X u r na x c om p e s o n e e E s 0.00 ~-. " ~-. " • ~ • •• • . ~ :: . : .~ :. ; . ~

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(41)

35

_'L-!

.\ _,I

,i.

Stressin ompression steel. fs 0_87 0.87x250 217 N/mm j;,'

fy ... , _ j' {i·\'.\; 217.0 64 2 2 = . --- .r d;,j:d': 250

J((500~140L·;

:i.-46 165.6'!fuh',", I, i, ,0 ,}",-j b) JT~nd, "- '' .,

X u m a h e section unde rein orced.

) U

',:9·,~6fck

Xu)+fscAsc(d~dl}

'I;, .;.; 'i:

""~:::Jq;~;,~t;!:1

;;,=;7' (460.';.

0.47

165,6)t, 217 226.,6(4.60-4 n',wl, '1?~t:~'16'l;'!;<n . - ~I/, -"-I\,.tl';I'·'I'~';!;·.';.';;;·~;(:·'~:'!~ :i;", J.' i,,,.,"' ;,J,.",,'1L 0..; Xu(max) 'A

b e a m ~ ~ ~ ~ i o ~ " 3 0 0 r m n

W i nOI;.'o f 2 Sm m d i n s o n n e a n 200s. 0..' , b

Determine h e m om en t of

resistance

oq~e;

s~cti~ f,I,~fM .15an

:f~t50 .are

.. ::.

...L:~~:;~:.

d " b)A.N~ '~ ,-. ,

~l ..

st X_490.87 2454.37 ..

Asc X~Ol.? 402. 2nim

i0~,"':_:'_, i ,

40'1

Strain co npre io ee

E s c '

'0.0035

' ~ L .'~ ~ ~ ~ ~ " .

-StressIn compression steel, fs 0.87 250

=2IiNlmrn2

ii;i;'t .'

0.0029 0.001087

.1,

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(42)

.. """" ":

Xu·· 087fy~SI

2i7*

'.~

f;:~:-;~~~:~'~_~M_~

---~---

.5 rn ~,': ~11'll' 1,';1,U sv i. ~.,j ~F c.

l·

Find :.':1'. Since', (max:) h e e c o n is n f i' resistance, Mu. T h n f h o g n w e h e m om en t n c h e 0 .3 6 c ld L X u m a x d -0 .4 2 X u m a x ,'t":' 4 1 5 0 ~>h·l:,~~·:tp,·;d:"<', d ; , . ~I ] , .r .:.> >;'~' m p h e m om en t n c m m n d m ~

reinforced wi

~f

2 m m d i compression

i~ne

hos~

' o r ' 2 o r r u r i

ia n s n ' o n a c e f e c v e c ov e o f 40mm. T h m a a l a r IS g ra d c on c re t Fe415 • • ~ • .. J_ ," I: ~I o' .~ ". ~> .~ "' ~' t~ .' h: " "l.,,, .. ,,,.,",,,,'.•,, Solution. " " 8 ' ) L·.1h iT · A · :'C!,' ;'. .: .. .1 .,'., .. ;,.; ,. /,., '11(11:'( ';1 , , IS andF~415,~----~~~~~--~:

~ : : " O ~ 4 7 9 ) '

X~(max)~'0.479

X'4'60 220.34~ .,:, '..~ st 1256.64 rnm

} .p .? O

=.~26.~tO:m~2,!,

Strain in compress io steel, 0 O 5 [

0 .0 0 s tr a n , sc ro previous abl 354.8 N/mm 0.003 87 (y A s 354.8 226.20:. '0 0 .3 6 c k X460 ~.. 0.60' 0.60x460 _276mm

.'

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(43)

·!I~;I~','

"ii,

.~'}?I~dun~~rTeirirorcedor over reinforced

S , c e ) C u /d . > '

over reinforced. f.;H~t~f.~mf,:k;.t',"

"i'l:lle'sc~tion~hould be .\.I·.1'd.

··::;,(~'.I~':'~i·.:( ,: ; 'I . K . ," } ' \: = : ± ' ," , ' ,: ' ''''.'',.':'- ~ : [t:i, -'~~~; L,·P·. (\ .- ~'T:. • . ..•.• ~ i, '. ·

d)lmhnateano,iDent -. , .

¥ J < O X u(max)

[~-0.42

Xu{max)] fs Asc(d-d

,i":::'f'>:~'6,3'6jX'1'5'X

250'X 220.34'X(460:0.~2X'22dj4}+354j'X'21o:20':'::

(4GQ~40)'-",,,~,,;·,,'~~·J,"~,·,,~·.

,~~~,~!~4~~·iA:O.)~

..;' .: ~;.··~,.i' \,';',-;1/. ':~if cJ h:;;:,'i;' i'.;'~ln.,t ,,1,1 i(;":':':';'_Y ,ii'~~fiH~,'I~?r

"r'-i':-'~":i"~~:":,~:i<"/f;3':'~'F'6 -d ;! .;r: ~:,,;,LYI !~ L ·> I~ ' ~ 7 ~~1~1:i!f' :/:U'~'~ "I i~~;H'-<t:')":"~~i:t~;:,t,'<t1'1h·f"~jiJ'~':

l:,~ih'~,illj:;

d'H;,;;'.~:

Ii:} '; :..: .. ·"to !,'~.·;~.~'l't'\;",i.;::·:IiE·~;;i!i;:J(f.'-l(~!tri'·~k~·";

F i n d moment resistance beam

230mni

m m dee e£fe~t~vei

~q~nfo,rE~di;'\V,~t~'ii,

Nhs, n u m p n f m en t effective

dill bars ast~.~~i~n r~i:nfo~~ement. Concret g r 2 0 n d g r ·:{~/;'A:t':;iIk~.t

,';r:

;tf"~::;·:~~~·_'::

O n : , : ~ ( ' ; ;F ~ '~';:t>"'~'f~r:,~<,.'~,,:t~,,:t-~"!' h)L'~'1

}.xr

:ltQt:.~o;,~2~

an Fe41 0.479

h:::~;X!t~·~

79.~ ~~~~)=;~03'! ml

:Asr=4Xjt4:h;~

2 5 m m 1;')· 'Asc=

2X~01.06

402.12 ::!I,;:

S C · = o :

o,...

O...

3 .

i.·.-.:.---~---

;= 0.003

For O·.003..

~ t t a i n ,

r o previous apl~,= 354, N/mm

.":;! "Q;8?<~fy:Ast fs Asc·c " •i, .• .,,',

0.8'7

i\';lri:~;\.~(J..\~ 0.41 0 .3 6 2 0 230ix .= O.4~X 6 0 m m

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(44)

38

f)

d) Ultimate ElItmU )IlUiJ: j, t:

Mu";'; 0.36 fc bXu(d-0.42 Asc(d-d , , ~;':lF);::

.3 0. 30 18 .6 (460 .4 88.6 t',3?4.~J{~0 ~12.~,(~§P?!O) Mu:= 178.85IG >l ir ; . ;.;- _. ~~" 'Ill' _: _. _{'. ..} A d n f n g ~ u ~B.Mof80~.

~ ~ ; Q ' ~ : : w i d e

anlC. ··.s00

in

of

~ ~ ~ ea m n _n in ~ h e a r ~ens,io anl ~om?re Sion rein orce ~~.nt i~ ~.~ncrete &ra~. M~l

,. !';j; '. .. .' Solution:. fact~red morrient, .Mud 2 0 N M

'·.b)

Find for/limiting, n g r e n f r c n .

,c':,

tt ~.!." 0.42

x : u < m a x J

. , . Qu.b.b ,= "0,36 O.36

.l

7 9 0.42 0.479] 240.1<460 r' n d A s :to.re;iist Ml:':

o.iri-zy

A s ! }

st ~._';_~_'_~J -;'";'M_ ;'~_

O.lnl~

[(r~"OA2 (max)]

;; t:," i'

0 4 : 9 3

: ; ---~---~-~~---:t=

791inm

O.B7x 41 460-(O 42 m om e M 2 M2 2 0 15.07 M2:: O,S? fy Ast2 (d ';. 15.07 ': ,Astz IiiI'IIII"_ ~---..--... -.~""""'---• ; ~ , . .; ~.;. (460.- 40 I , . ..!, iiI<'ol 1-" As ,"j. ~l)PiJ

;iMud

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(45)

39 "(~:' ..

;,:',i};,[t

i(.':.';; Ii

'A.B.t2'~?91.0

99.0 890!UtFI~~~~.__,.. "t

'#

bars. _,j '4 ... ',.' : ~

)fu,(,rn,',",,~),

~ " " d . 1 : 1 · !CO~p~~i~~~t~el,: Esc 0.0035

[ ~ ~ 7 9 . : : ~ ~ . l _

F o c O . 6 0 3

~traip,Jsh

354.8 N/mm j" so

---;.!.;.:.::.-

n m 0.03 [lroyide2 NOsS'#bars. ::::. Example ~'

beam is limited 'to.

n un . D e g n h e n , = 2 N/mm2 a n d y =4 .1 5 N/rrun a n

',' ':'

n d

bottom

_j,

:_

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(46)

!40

..

Qu.b.d

0.36

~ ~ (~ :

~ ~ ~ ~ < : : J

j,

,'=

0.36 20 0.479] 280 460 :d 163.20kN .' to st I . ...._.~.. ...;.,;~:.._I fI II -~ _ . . .. . . _ . " . O~~7fy [d 0.42 X u m a 0.H7 15 [4 0.479 460] h. 1 23 0 m tn ~

d}Find

he.excess RM,

ie., '.Il

M2 1 8 165.30 16.8 kN } F n d A s resist M2 ',M2 'Asti

---..:-.:..---;.--~-fy 1 6 . R O

=---

Inm 0.87 f)

'A~r';;;A'st'J'·+Ast2

~l23b'+~1

r;;T34

'~m2' ~ . j: P ro v N o 2 2 bars :'b~~L;"_-I"":~~,~ ..

---~---(d·.; 0.479 sc '0,003 ---~---

o . o m

0.479 460 F o a . 0 3 a in , 354,8 N/mm 16,80 [, sc ---;.---.;-354. (460-40) bars. ':. "~:},:~ /HI" I; :; :1 .. r: i-il_I. if i ' l..".' : ,l,~

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(47)

1;)f(Jlh#y~~~ Pl;9,frein forced

concrete

to

th centre

;fl!'/:':is:reiIlfdrced with m m m p m m

1,:,'114bUl'$of20mm moment resistance uf th section,

iin!'

grnde;M15 stee ra l';";:"i"'~\"'("''f"';'"

:; '. J: ''''~',~_,'fi ,_.fJ,-.1-~.:.t.~>~_,~!_I,:~:Cic .,.1

:rJ;~.\;.:~

AnS#~t:.

:'Wld;{) ..135:' xLJ(m~)/dFO.$:U t' un~'1~e.i!1forq~~!.'.

M u ~ . h U · 2 2

In::' .. tH'}i :::'~;1";':)~;J,~i . ] :

"j:r

, ~ ~ ' q , b : ~ R ~ t

·t~i~for~e~.

r~ct~g~lllr~~~'is'

~O?mm'wi~e"'ifh~ef~eGtiYI!::4e~J5h f!,b(!~l}1,.i~.~

? , ~ W p , ~ ,~ B

it:')' h e ': e~ ~ f v ~ c ov e to both:le~sl~n "an~_rcotnpresslOn treinforcemen ts: fl:thl~,.ou,~erf~c~,¥!.~~~:j,;~~:.

, ;: ; F i '

sodlili/fne;

mp o( reinforcements consists-of 4(~:)f,\rH'!~ 2()r11ll).~!A~]e~~'i~9~r;4r

16mm d ia m e e r r e p ec t v e y . anddt Cn11irie;:thlYi: mp Wmk of

resistu)1ce.of M15gra.de steel re , , .. ::<.

:_;-';':l',:~r~,~-·~·'·~,

Answer: xu/d=O.08; xu(max)/d=O.531 u nd e e in fo rc ed ; M u =1 38 kN m

3)'

A6;~1's~cti()l1

3001nm wide ari9.540rrirri de~p,overallis reinforced with bar~~of-75mm

tensHe:zpne 6 n x d ~ J~ .~ o p re s o n z on e E f e c o ve r b o c u 4 0m m .

Concret¢ grade)vi15 u se d F in d

m o m e n t

r e a n h e e c o n

: _ i'

Answer: Xu/d=O.61; x'u(max)/d:fO.479;,()ver reinforced; Mu=253kNm

, . ':~':J:

n f a ng u a r a m 30pmm wide m m n f

c om p e s o n z on e E f c t v e i~;it,;r:coyer;\ip:.bot cases p~wqn~n~.the. ~0 Ht?r o~ r,~~i~..tanc,~

o,f.

,~r7'

.secti.o?!

K ; ? : g r g ~ ; ~ ' J ~ ~ 7

'j',' ,;\,;1

d./l

i;;r'{':::'i'd xu/d=O.25; xu(max)/d=0.479; unde reinforced

5) D e g n beam rectangular section

30b';~m~id~\~~'d'5'6d':~i{}

depth:

eft~e8ti'vJ!:suiJj6'df~ ;itli~

u l m a t m o m n t F ac to re d g r M 1 a n g ra d A 1 are,

~ ~

_ : ~ d ~ f ~ c

= 4 m m ~ _ .

":~'~i:';;::n;;;'.

':"'~'J,'r"i,:'::r~;~r:"I'::':'

~i

<Answer:

M l= 15 4 0 kN m , M 2 =2 S 3 kN m , A s = 10 75 m Ast2.=151tll,H12,'Ast~1226rti'm2~

' ;

::'!:':"'II:~,'tt;~:,·:·~~~

300mm 450mm d e o v a l has to,resist,' a'~en~g }nomellt.

m .

concrete M 1 with an effective'

b c l v e r o f 4 0 r i U h ' a r e < t b

. '

ti~t~~:H.H

'"~

'.<{d,

A r i s w ~ E

'M1;"104.20kNm, Mi=33;80kNm,' A s = 8 m m A s 2 = 2 53 , A s 1 13 4m m

7):'\',1\

;~:~QW

y'r~intorce:d

beam 250min' wide 5QOmm deep.,Qverhl~ihasJo resi ~t!tJ~,

M~

:p(,,~QQ19i'{~J1'

"';-+The"b~ettive

cover of in inaYhe,ado!pt~d:attop and:,bottorn·Find the areh,pfcQlI1pre~~im1:llIJd

steel. ~Concrete grade 20'and

's_teer

grade)! Fe'-4t5';aI'e~used;;'i'- :;;;,;:1:: ::-,,/f:i;J,

~ ; f i ¥ ; ~ ~

m 'M ~ m , m m Ast2=73m~n2, ASt=1148mm ,'

8)

:J~~~f,b~,:.2~~;~:e:e~~~Oq~;.!~e"!1~

~~~~T~ri~%.~1~~~~Sg.~J6~1eiJ

compression reinfor2e~eritsifrb'fu "t'li~'

otit~F'stiifiicFE{iWmm:~Find-:-;theL,aieli'/6f

__ compression an te si steel, re se

,.AsQ::::1099mm Ast2=127mm2, Ast~1226mm2. Asce

---...._,.----~,~=-

_,'_;-

-,..

. .: .. .c , . • .

, . . . ~ __ . ~

.41

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(48)

'.,' ()

T-BEAM3

, . -: -, INTRODUCTION

It is economical

for'beam

a n a b c on s u c o n ~ pa n o r slab exceeds to In h i

c o n s t u c t o n , b e a m

ar placed n g h o n d is n t n u h e m s H e

alab'ond beam: ar~', cast,m o no l h ic a a n d

p o r t i o n

the, slab, helps

b.elUil,

'in,,taking, bending

~om~t;He~~e

t~e,

beam

h e h a T ' B I h en ~ .it'i~;ikn,Own

-L J. ...'o'r·,f)an·oe'd'bCl' ' ' Dt?W" :._!,~! >~(J.::.:.:~:;"~" ;:[' ',:: :I~;; j.' ~! "SLAB," C-~:.~·.r.u;~~> . ! : ~ ~ : , : o U ~ ~ ,-'. '• r • - ' ; ~ • -" ~.;I~~;')~,I'~ ;.! !: '~}~~~f'; ';:~:; f'~( .~,--~;': : _

The 'slab portiono tlieTsbeam

flange;

rind therectangular

beam

portion'ofthe-Tsbeam is

1/;-J ) .

Dimensions ortb 1f-:-llIteam.'sre indicate as given

:,:I·;::i:,L',.Ii ', i;'!:; ',C i, r.

Breadth

Effe~'tive

h , ': .- : ' . '

._ ;~I. " ' i ' i~

~ i = p~ p~ h Y '~ b ~ ~ vl Breadth of'web rib.

''-; ~ I '

i: .._;

l ,

Effective .wldth Daoge: he, absence of m o a cc u a t d e m in a o n the effective

JWidthofflangemny taken h e th 'distances to h e a d a c n t a m on either , . : : ' '. ' ~'U!'I' ' . .! £I ~1 ;,. l. '» .c 1'.' "1 •• _'Fort~berunsbf =10/6 +bw+6'Df::t> [0.5 If "I -::! I~

'j.~pt!~~'lO:f 4~starcc bC~Yfee~po~n,~sofz~ro.niome~tswit~

'I - , , - ~,

j.

, ,

' :

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(49)

I' is .: ;ctive dfthe :\"", _I; l':\Tfn<

:ll,

'i :' ',

iti::Cas :lWhen neutral Axis lies Xu:<I~~..

!~\Hcl'e h e r e d ia g a m will fig.

'1": 1L~~Fck.~Y,br

' A ) ,Hf:AHSEtT1Dt.. _'r.•

o·

liEUrp~LA xi s ITIlIN >LIIN~£

Total c om p e s v e o rc e c on cr e in compression O . bfXu' Total tensile force n f m en t n s

(ru)

Equating the above bfXu 0.87 st

st ----_ .. -- ---.,..""'-:-br Depth Factor Xu(max) 700 (0.87 fy)

Mu(lim)= Total compressiv~,forc l~ve;.Mr ,' I;.,

',:;1

<:

M u

lever~'!

0.87 ~(d~ X u

Substituting h e a l of(XuJd) ~b~~eequ~tion.'

• +

.:-. ."

.~

.1".;

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(50)

ff

Mu 0.87 st

fOD' OVCI Mu

Xu(max) h e n f w h n o n d h e

section designed as im ng section.

U n c s

2.a)

nr

O.2d)

T o C om p C om pr e o rc e in concrete in w x

n c in lange excludin portion

0.446 fc (0[.''; bw Dr T o a l e n o rc e r e n fo r e me n n s o n 1: =.0.87 Ast.·,·' Cu= ..', w X 0.446 D f O.87t)r 0.87 fy Of ", f'· .'j 0.36 fekbw Allis Factor 1100 0.87 fy m om en t a n ; , " . +-,'

ue to.C om pr e o r e s Total compressive force

Mu Xu bw)

bf d~' ",

~-,,>.

Mu 0.36

---1' ;.

O.~46 fck w ) D f

'~,'" ."

M u Total tensile a r 0.87 Ast> X u

Moanr.ni I'esistnn~~ ih section, (lim when balanced'

W h X u m a -,' th oftl De,l ,Tot! Tota Equa

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(51)

45

"MlJQim) +::,

i.'

" ~ _i':';':

Moment of resistanc h e section, Mu w h o v n f d .

m a over rein orced, de~i~~ble m om e

resistance section m e resistance the limitingsection,

~ : l' 2b)D ri Dr 0.2d) '" '"~ < C : w e 0.36 FCI(,Xubl C:rc . . D .4 4E iF CK ,W C b - 0 D N .~ Ii 1,'. .. ~.":

~. henDfis gi~aterthan 2 d h e curvilinear h e comes into the flange also. Hence str~ss~istHbuti?n. in

~ange n?tH~~fprrp'Jhie~

,ip~

~~3pt:n.~d

th~~!t,h~~~ptt~of~~,st,~~!5u_l,ar

P.~ rt,i :\

'I ofthe~tre~s.J,)l~ck IS taken "i" i' 1,;;1:,:;.;.",1 i1,..f.,'

~I ; •

actual'

Neutral is Factor

Tot~l comprrHi'. web)

n c c om p re s v e o rc e in n c t1all~

+'0.440' fc (bj-bw

'I

~i where Yf Df

Total tensile orce in ~ein orcement in en io (.Tu) 0,87 fy Ast. ,; .i .. . . NEUTRAL :: Equating the Xu 0,87 f) 0446 Ick

~---~---O~36-f~kb~

ci---*----: .. ~ . ,