CONTENTS
Sl No
Work Sheet
Description
1
PREAMBLE
Preamble , Disclaimer and contact details.
2
PROCEDURE
List of Smbols alon! "it# a s$etc# and %roced&re follo"ed in com%&tin! t#e Moment of Resistance 'actors
(
MO'R)REC*
*ABLES Containin! Moment of Resistance 'actors for Do&bl Reinforced Rectan!&lar Concrete Sections, M &7f c$bd 2
+
)REC*
*ABLES Containin! De%t# of ne&tral a4is 'actors for Do&bl Reinforced Rectan!&lar Concrete Sections, $
-
*P)CAL
*%ical Calc&lations of Moment of Resistance 'actors
/
SOL0ED)EAMPLES
ll&strati3e E4am%les
5
RE0SO6S
Details of Re3isions
MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR CONCRETE SECTIONS 9re*0/le4 Desi!n 9re*0/le4 Desi!n of a Do&bl Reinforced Rectan!&lar Concrete Section is easier "it# t#e a3ailable of Desi!n Aids. 8o"e3er, 8o"e3er, Calc&lation of Moment Ca%acit of a !i3en Do&bl Reinforced Rectan!&lar Concrete Section b Limit State met#od is tedio&s. Str&ct&ral Desi!ners often come across %roblems of com%&tin! m oment ca%acit of a !i3en Do&bl Reinforced Rectan!&lar Concrete Section. Desi!n of Reinforced Concrete Members s&b9ected to re3ersal of bendin! moments is a 3er common %roblem in Desi!n of Reinforced Concrete %artic&larl "#en "#en lateral loads are also actin! on t#e str&ct&re. f s&itable Desi!n Aids Aids are a3ailable, it is %ossible to select a %air of reinforcement %rcenta!es on t#e t"o o%%osite faces of t #e section so as to be ade:&ate for bot# t#e factored sa!!in! and #o!!in! moments. S 1(;2< =1;;( sti%&lates, calc&lation of s#ear force d&e to formation of %lastic #in!es at bot# ends of a beam %l&s t#e factored !ra3it load on t#e s%an, "#ic# "#ic# res<s in t#e need for com%&tatin! sa!!in! and #o!!in! #o!!in! moments of resistance of t#e beam section. Considerin! t#ese re:&irements, Desi!n Aids in t#e form of Moment of Resistance 'actors #a3e been de3elo%ed. Calc&lations are made as %er S +-/ = 2<<<. Proced&re follo"ed in com%&tin! t#e 3al&es is e4%lained in t#e "or$s#eet ?PROCEDURE?. A com%&ter Pro!ram "as de3elo%ed to com%&te t#e moment of resistance factors M &7f c$bd 2 o3er a ran!e of 3al&es for p for pt7f c$ and p and pc7f c$. Res<s are com%iled and %resented in t#e form of *ABLES for 'e@+1- and 'e@-<< !rades of steel. Corres%ondin! 3al&es of k , t#e de%t# of ne&tral a4is factor are also incl&ded in se%arate *ABLES, "#ic#. "o&ld be &sef&l to 3erif t#e 3al&es of Moment of Resistance 'actors. *%ical *%ical calc&lations of Moment of Resistance 'actors and ill&strati3e E4am%les are also incl&ded in res%ecti3e >or$s#eets. Data %resented in t#is >or$boo$ s #all not be &sed for concrete of com%ressi3e stren!t# !reater t#an M--. Comments and S&!!estions are >elcome. DISCLAIMER
The +*t* cont*ine+ in this Work/ook h*s /een prep*re+ :ith #t0ost c*re The *#thor 0*kes no :*rr*nt8 o; *n8 kin+ e
R!DESAI MTech"Str#ct#res$% MTech"Str#ct#res$% MICI% MACCE"I$ D&'()% Ro*+ No G&, -ee.*n/hi0*n*1*r BANGALORE BANGALORE & ,(2 23, e0*il 4 +es*i5'6(278*hooco0
MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR CONCRETE SECTIONS 9re*0/le4 Desi!n 9re*0/le4 Desi!n of a Do&bl Reinforced Rectan!&lar Concrete Section is easier "it# t#e a3ailable of Desi!n Aids. 8o"e3er, 8o"e3er, Calc&lation of Moment Ca%acit of a !i3en Do&bl Reinforced Rectan!&lar Concrete Section b Limit State met#od is tedio&s. Str&ct&ral Desi!ners often come across %roblems of com%&tin! m oment ca%acit of a !i3en Do&bl Reinforced Rectan!&lar Concrete Section. Desi!n of Reinforced Concrete Members s&b9ected to re3ersal of bendin! moments is a 3er common %roblem in Desi!n of Reinforced Concrete %artic&larl "#en "#en lateral loads are also actin! on t#e str&ct&re. f s&itable Desi!n Aids Aids are a3ailable, it is %ossible to select a %air of reinforcement %rcenta!es on t#e t"o o%%osite faces of t #e section so as to be ade:&ate for bot# t#e factored sa!!in! and #o!!in! moments. S 1(;2< =1;;( sti%&lates, calc&lation of s#ear force d&e to formation of %lastic #in!es at bot# ends of a beam %l&s t#e factored !ra3it load on t#e s%an, "#ic# "#ic# res<s in t#e need for com%&tatin! sa!!in! and #o!!in! #o!!in! moments of resistance of t#e beam section. Considerin! t#ese re:&irements, Desi!n Aids in t#e form of Moment of Resistance 'actors #a3e been de3elo%ed. Calc&lations are made as %er S +-/ = 2<<<. Proced&re follo"ed in com%&tin! t#e 3al&es is e4%lained in t#e "or$s#eet ?PROCEDURE?. A com%&ter Pro!ram "as de3elo%ed to com%&te t#e moment of resistance factors M &7f c$bd 2 o3er a ran!e of 3al&es for p for pt7f c$ and p and pc7f c$. Res<s are com%iled and %resented in t#e form of *ABLES for 'e@+1- and 'e@-<< !rades of steel. Corres%ondin! 3al&es of k , t#e de%t# of ne&tral a4is factor are also incl&ded in se%arate *ABLES, "#ic#. "o&ld be &sef&l to 3erif t#e 3al&es of Moment of Resistance 'actors. *%ical *%ical calc&lations of Moment of Resistance 'actors and ill&strati3e E4am%les are also incl&ded in res%ecti3e >or$s#eets. Data %resented in t#is >or$boo$ s #all not be &sed for concrete of com%ressi3e stren!t# !reater t#an M--. Comments and S&!!estions are >elcome. DISCLAIMER
The +*t* cont*ine+ in this Work/ook h*s /een prep*re+ :ith #t0ost c*re The *#thor 0*kes no :*rr*nt8 o; *n8 kin+ e
R!DESAI MTech"Str#ct#res$% MTech"Str#ct#res$% MICI% MACCE"I$ D&'()% Ro*+ No G&, -ee.*n/hi0*n*1*r BANGALORE BANGALORE & ,(2 23, e0*il 4 +es*i5'6(278*hooco0
9ROCEDURE
COM9RESSION FACE b
d ' 0.0035 d '
xu
d
0.0035
1-
xu
NA
STRAIN DIAGRAM
TENSION FACE
FIG 1 DOUBLY REINFORCED RECTANGULAR SECTION
SYMBOLS
Ast Area of tension steel Asc Area of com%ression com%ressio n steel b Breadt# of beam C = Com%ressi3e force on t#e section d Effecti3e de%t# of beam d ' De%t# of com%ression reinforcement reinforcement from t#e #i!#l com%ressed com%ressed face f cc Com%ressi3e stress stress in concrete concrete at t#e le3el le3el of centroid of com%ression reinforcement reinforcement f ck C#aracteristic C#aracteristic c&be com%ressi3e stren!t# of concrete f sc Stress in com%ression reinforcement f st Stress in tension tension reinforcement reinforcement f C#aracteristic stren!t# of steel k 6e&tral 6e&t ral A4is 'actor x &7d M & 'actored moment ca%acit of t#e section M &# 'actored #o!!in! moment M &s 'actored sa!!in! moment pc
Percenta!e of com%ression reinforcement reinforcement 1<< Asc7bd
pt
Percenta!e of tension reinforcement reinforcement 1<< Ast7bd
pbot
Percenta!e bottom reinforcement reinforceme nt
pto%
Percenta!e to% reinforcement
T = *ensile *ensile force on t#e section x & = De%t# of ne&tral a4is a4is at t#e limit state of colla%se x &,ma4 = Ma4im&m de%t# of of ne&tral a4is in limit state desi!n desi!n
9roce+#re4 i3en t#e !rade of steel, d 7d, pt7f c$ and pc7f c$. 'ollo"in! Proced&re #as been follo"ed for com%&tin! t#e Moment of Resistance 'actors for Do&bl Reinforced Rectan!&lar Sections. Ste% 1= Ass&me a small 3al&e of k. Ste% 2= Com%&te t#e strains in reinforcement %ro3ided near t#e com%ression and tension faces of t#e section, considerin! linear strain distrib&tion.
Ste% (= Determine t#e stresses f sc and f st in steel corres%ondin! to t#e strains calc&lated in ste% 2, referrin! to t#e desi!n stress@strain c&r3e for steel.
Ste% += Determine t#e stress f cc in concrete at t#e le3el of centroid of com%ression reinforcement referrin! to t#e desi!n stress@strain c&r3e for concrete. Ste% -= Calc&late C 7f c$bd and T 7f c$bd &sin! t#e follo"in! e4%ressions.
C @@@@@@@@@ <.(/ k pc7f c$ Ff sc @ f ccG 7 1<< f c$ bd T @@@@@@@@@ pt7f c$ 4 f st 7 1<< f c$ bd
Ste% /= Com%are t#e 3al&es of C 7f c$bd and T 7f c$bd. *#e t"o 3al&es #a3e to be e:&al from force e:&ilibri&m condition. f t#e difference bet"een t#e 3al&e of C 7f c$bd and T 7f c$bd is more, increment t#e 3al&e of k b a small 3al&e and re%eat ste%s 2 to / Contin&e t#e abo3e iterati3e com%&tations, to determine t#e 3al&e of k "#ic# satisfies t#e force e:&ilibri&m condition. Ste% 5= 'or t#e final 3al&e of k so obtained, Com%&te t#e strains in reinforcement %ro3ided near t#e com%ression and tension faces of t#e section,considerin! linear strain distrib&tion. Determine t#e stresses f sc and f st in steel corres%ondin! to t#ese strain 3al&es. Determine t#e stress f cc in concrete at t#e le3el of centroid of com%ression reinforcement Calc&late t#e moment of resistance factor &sin! t#e e4%ression, M &7f c$bd 2 <.(/ k F1 @ <.+2 k G pc 7 f c$ Ff sc @ f ccGF1 @ d 7d G71<<
6otes = 1G f t#e ne&tral a4is is "it#in t#e co3erin! concrete Fi.e k Hd 7d G, f sc is considered ne!ati3e and f cc "ill be considered ero, since steel "ill be in tension one. 2G 0al&e of f cc Fstress in concrete at t#e le3el of centroid of com%ression reinforcementG #as been calc&late considerin! M2< rade of concrete as an a%%ro4imation to co3er all !rades of concrete normall &sed. Error in t#e 3al&e of Moment of Resistance 'actor d&e to t#is a%%ro4imation "ill be ne!li!ibl small.
d .
TABLE R&' MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&>',
d d
22,
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<(+I
<.
<.1<<1
<.12I-
<.1-(;
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<(+I
<.
<.1<2I
<.1(+2
<.1/2-
<.1I55
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<(+I
<.
<.1<((
<.1(/;
<.1/I2
<.1;/+
<.221/
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<(+I
<.
<.1<(+
<.1(5/
<.151<
<.2<2(
<.2(<+
<.2--+
@@
@@
@@
@@
@@
@@
@@
22,
<.<(+I
<.
<.1<(+
<.1(5/
<.151I
<.2<-2
<.2(/(
<.2/+(
<.2I;2
@@
@@
@@
@@
@@
@@
22(
<.<(+I
<.
<.1<(+
<.1(55
<.151;
<.2
<.2(;(
<.25<+
<.2;I2
<.(2(1
@@
@@
@@
@@
@@
223
<.<(+I
<.
<.1<(+
<.1(55
<.151;
<.2
<.2+<(
<.25(+
<.(<++
<.((22
<.(-/;
@@
@@
@@
@@
22@
<.<(+I
<.
<.1<(+
<.1(55
<.151;
<.2
<.2+<+
<.25+-
<.(<5/
<.((I+
<.(//1
<.(;<5
@@
@@
@@
226
<.<(+I
<.
<.1<(+
<.1(55
<.152<
<.2
<.2+<-
<.25+5
<.(
<.(+15
<.(52-
<.+<<<
<.+2+-
@@
@@
2'2
<.<(+I
<.
<.1<(+
<.1(55
<.152<
<.2
<.2+<-
<.25+I
<.(<;<
<.(+2;
<.(5-I
<.+
<.+((;
<.+-I(
@@
2''
<.<(+I
<.
<.1<(+
<.1(55
<.152<
<.2
<.2+<-
<.25+I
<.(<;<
<.(+(2
<.(551
<.+<;;
<.++<-
<.+/5I
<.+;21
2')
<.<(+I
<.
<.1<(+
<.1(55
<.152<
<.2
<.2+<-
<.25+I
<.(<;<
<.(+((
<.(55-
<.+11(
<.+++<
<.+5+-
<.-<15
2'?
<.<(+I
<.
<.1<(+
<.1(55
<.152<
<.2
<.2+<-
<.25+I
<.(<;1
<.(+((
<.(55/
<.+115
<.++--
<.+5I2
<.-
2'>
<.<(+I
<.
<.1<(+
<.1(55
<.152<
<.2
<.2+<-
<.25+I
<.(<;1
<.(+((
<.(55/
<.+11I
<.++/<
<.+5;5
<.-12(
2',
<.<(+I
<.
<.1<(+
<.1(55
<.152<
<.2
<.2+<-
<.25+I
<.(<;1
<.(+((
<.(55/
<.+11;
<.++/1
<.+I<2
<.-1(;
6ote=
@@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&) MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&>',
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<(+/
<.
<.<;I2
<.12//
<.1-2<
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<(+/
<.
<.<;;(
<.1(<+
<.1-I5
<.1I+<
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<(+/
<.
<.<;;-
<.1(15
<.1/2-
<.1;<5
<.21/<
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<(+/
<.
<.<;;/
<.1(1;
<.1/+<
<.1;+5
<.222I
<.2+I<
@@
@@
@@
@@
@@
@@
@@
22,
<.<(+/
<.
<.<;;/
<.1(2<
<.1/+(
<.1;/(
<.22/I
<.2-+I
<.2I<<
@@
@@
@@
@@
@@
@@
22(
<.<(+/
<.
<.<;;/
<.1(2<
<.1/++
<.1;/5
<.22I/
<.2-I;
<.2I/;
<.(12<
@@
@@
@@
@@
@@
223
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;/;
<.22;2
<.2/<;
<.2;11
<.(1;<
<.(++<
@@
@@
@@
@@
22@
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;/;
<.22;(
<.2/1/
<.2;(2
<.(2((
<.(-1<
<.(5/<
@@
@@
@@
226
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;+
<.2/15
<.2;(;
<.(2--
<.(--+
<.(I(1
<.+
@@
@@
2'2
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;+
<.2/1I
<.2;+2
<.(2/(
<.(-5I
<.(I5/
<.+1-2
<.++<<
@@
2''
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/1;
<.2;+(
<.(2//
<.(-I5
<.(;<<
<.+1;I
<.++52
<.+52<
2')
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/1;
<.2;+(
<.(2/5
<.(-;<
<.(;1<
<.+22(
<.+-1;
<.+5;(
2'?
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/1;
<.2;++
<.(2/I
<.(-;2
<.(;1-
<.+2(+
<.+-+-
<.+I+1
2'>
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/2<
<.2;++
<.(2/I
<.(-;2
<.(;1/
<.+2(;
<.+--I
<.+I/I
2',
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/2<
<.2;++
<.(2/;
<.(-;(
<.(;15
<.+2+<
<.+-/(
<.+II1
6ote =
@@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&) MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&>',
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<(+/
<.
<.<;I2
<.12//
<.1-2<
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<(+/
<.
<.<;;(
<.1(<+
<.1-I5
<.1I+<
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<(+/
<.
<.<;;-
<.1(15
<.1/2-
<.1;<5
<.21/<
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<(+/
<.
<.<;;/
<.1(1;
<.1/+<
<.1;+5
<.222I
<.2+I<
@@
@@
@@
@@
@@
@@
@@
22,
<.<(+/
<.
<.<;;/
<.1(2<
<.1/+(
<.1;/(
<.22/I
<.2-+I
<.2I<<
@@
@@
@@
@@
@@
@@
22(
<.<(+/
<.
<.<;;/
<.1(2<
<.1/++
<.1;/5
<.22I/
<.2-I;
<.2I/;
<.(12<
@@
@@
@@
@@
@@
223
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;/;
<.22;2
<.2/<;
<.2;11
<.(1;<
<.(++<
@@
@@
@@
@@
22@
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;/;
<.22;(
<.2/1/
<.2;(2
<.(2((
<.(-1<
<.(5/<
@@
@@
@@
226
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;+
<.2/15
<.2;(;
<.(2--
<.(--+
<.(I(1
<.+
@@
@@
2'2
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;+
<.2/1I
<.2;+2
<.(2/(
<.(-5I
<.(I5/
<.+1-2
<.++<<
@@
2''
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/1;
<.2;+(
<.(2//
<.(-I5
<.(;<<
<.+1;I
<.++52
<.+52<
2')
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/1;
<.2;+(
<.(2/5
<.(-;<
<.(;1<
<.+22(
<.+-1;
<.+5;(
2'?
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/1;
<.2;++
<.(2/I
<.(-;2
<.(;1-
<.+2(+
<.+-+-
<.+I+1
2'>
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/2<
<.2;++
<.(2/I
<.(-;2
<.(;1/
<.+2(;
<.+--I
<.+I/I
2',
<.<(+/
<.
<.<;;/
<.1(21
<.1/+-
<.1;5<
<.22;-
<.2/2<
<.2;++
<.(2/;
<.(-;(
<.(;15
<.+2+<
<.+-/(
<.+II1
6ote =
@@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&? MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&>',
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<(-1
<.
<.<;/+
<.12+5
<.1-<1
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<(-2
<.
<.<;/I
<.12/I
<.1-+;
<.1I<1
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<(-(
<.
<.<;/I
<.125(
<.1-51
<.1I-1
<.21<2
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<(-(
<.
<.<;/;
<.125+
<.1-5I
<.1I5-
<.21-2
<.2+<2
@@
@@
@@
@@
@@
@@
@@
22,
<.<(-(
<.
<.<;/I
<.125-
<.1-I<
<.1II(
<.215I
<.2+-+
<.25<(
@@
@@
@@
@@
@@
@@
22(
<.<(-(
<.
<.<;/I
<.125-
<.1-I1
<.1II/
<.21II
<.2+I2
<.25-/
<.(<<+
@@
@@
@@
@@
@@
223
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II5
<.21;1
<.2+;(
<.25I-
<.(<-5
<.((<-
@@
@@
@@
@@
22@
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1III
<.21;(
<.2+;5
<.25;I
<.(
<.((-;
<.(/
@@
@@
@@
226
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1III
<.21;+
<.2+;;
<.2I<2
<.(1<(
<.((;2
<.(//1
<.(;<5
@@
@@
2'2
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<<
<.2I<-
<.(1
<.(+
<.(/;-
<.(;/(
<.+2
@@
2''
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<1
<.2I
<.(111
<.(+1+
<.(512
<.(;;;
<.+2/-
@@
2')
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<1
<.2I<5
<.(112
<.(+15
<.(52<
<.+<1/
<.+(<2
<.+-/5
2'?
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<2
<.2I
<.(11(
<.(+1I
<.(52(
<.+<2-
<.+(21
<.+/<+
2'>
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<2
<.2I
<.(11+
<.(+1;
<.(52+
<.+<2;
<.+((<
<.+/2-
2',
<.<(-+
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<2
<.2I
<.(11+
<.(+2<
<.(52/
<.+<(<
<.+((-
<.+/(-
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&? MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&>',
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<(-1
<.
<.<;/+
<.12+5
<.1-<1
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<(-2
<.
<.<;/I
<.12/I
<.1-+;
<.1I<1
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<(-(
<.
<.<;/I
<.125(
<.1-51
<.1I-1
<.21<2
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<(-(
<.
<.<;/;
<.125+
<.1-5I
<.1I5-
<.21-2
<.2+<2
@@
@@
@@
@@
@@
@@
@@
22,
<.<(-(
<.
<.<;/I
<.125-
<.1-I<
<.1II(
<.215I
<.2+-+
<.25<(
@@
@@
@@
@@
@@
@@
22(
<.<(-(
<.
<.<;/I
<.125-
<.1-I1
<.1II/
<.21II
<.2+I2
<.25-/
<.(<<+
@@
@@
@@
@@
@@
223
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II5
<.21;1
<.2+;(
<.25I-
<.(<-5
<.((<-
@@
@@
@@
@@
22@
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1III
<.21;(
<.2+;5
<.25;I
<.(
<.((-;
<.(/
@@
@@
@@
226
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1III
<.21;+
<.2+;;
<.2I<2
<.(1<(
<.((;2
<.(//1
<.(;<5
@@
@@
2'2
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<<
<.2I<-
<.(1
<.(+
<.(/;-
<.(;/(
<.+2
@@
2''
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<1
<.2I
<.(111
<.(+1+
<.(512
<.(;;;
<.+2/-
@@
2')
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<1
<.2I<5
<.(112
<.(+15
<.(52<
<.+<1/
<.+(<2
<.+-/5
2'?
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<2
<.2I
<.(11(
<.(+1I
<.(52(
<.+<2-
<.+(21
<.+/<+
2'>
<.<(-(
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<2
<.2I
<.(11+
<.(+1;
<.(52+
<.+<2;
<.+((<
<.+/2-
2',
<.<(-+
<.
<.<;/I
<.125-
<.1-I2
<.1II;
<.21;-
<.2-<2
<.2I
<.(11+
<.(+2<
<.(52/
<.+<(<
<.+((-
<.+/(-
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&> MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&>',
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<(/+
<.
<.<;-1
<.122;
<.1+I2
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<(/5
<.
<.<;-2
<.12(5
<.1-12
<.15/+
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<(/;
<.
<.<;-2
<.12+<
<.1-22
<.15;/
<.2<+/
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<(5<
<.
<.<;-2
<.12+<
<.1-2/
<.1I
<.2
<.2(2I
@@
@@
@@
@@
@@
@@
@@
22,
<.<(5<
<.
<.<;-1
<.12+1
<.1-2I
<.1I1(
<.2<;+
<.2(/(
<.2/<;
@@
@@
@@
@@
@@
@@
22(
<.<(5<
<.
<.<;-1
<.12+1
<.1-2;
<.1I1-
<.21<<
<.2(I<
<.2/+5
<.2I;1
@@
@@
@@
@@
@@
223
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I15
<.21<(
<.2(I/
<.2///
<.2;(<
@@
@@
@@
@@
@@
22@
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I15
<.21<+
<.2(;<
<.2/5(
<.2;-1
<.(21+
@@
@@
@@
@@
226
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21<-
<.2(;2
<.2/55
<.2;/<
<.(2(5
<.(+;I
@@
@@
@@
2'2
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21
<.2(;(
<.2/5;
<.2;/+
<.(2+5
<.(-22
<.(5I2
@@
@@
2''
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21
<.2(;+
<.2/I1
<.2;/5
<.(2-1
<.(-(+
<.(I<5
<.+
@@
2')
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I2
<.2;/;
<.(2-+
<.(-(;
<.(I2<
<.+<;2
<.+(-<
2'?
<.<(51
<.
<.<;-<
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I(
<.2;5<
<.(2-/
<.(-+2
<.(I2/
<.+1
<.+(55
2'>
<.<(51
<.
<.<;-<
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I(
<.2;51
<.(2-I
<.(-++
<.(I2;
<.+11(
<.+(;(
2',
<.<(51
<.
<.<;-<
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I+
<.2;51
<.(2-;
<.(-+-
<.(I(2
<.+115
<.++<1
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&> MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&>',
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<(/+
<.
<.<;-1
<.122;
<.1+I2
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<(/5
<.
<.<;-2
<.12(5
<.1-12
<.15/+
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<(/;
<.
<.<;-2
<.12+<
<.1-22
<.15;/
<.2<+/
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<(5<
<.
<.<;-2
<.12+<
<.1-2/
<.1I
<.2
<.2(2I
@@
@@
@@
@@
@@
@@
@@
22,
<.<(5<
<.
<.<;-1
<.12+1
<.1-2I
<.1I1(
<.2<;+
<.2(/(
<.2/<;
@@
@@
@@
@@
@@
@@
22(
<.<(5<
<.
<.<;-1
<.12+1
<.1-2;
<.1I1-
<.21<<
<.2(I<
<.2/+5
<.2I;1
@@
@@
@@
@@
@@
223
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I15
<.21<(
<.2(I/
<.2///
<.2;(<
@@
@@
@@
@@
@@
22@
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I15
<.21<+
<.2(;<
<.2/5(
<.2;-1
<.(21+
@@
@@
@@
@@
226
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21<-
<.2(;2
<.2/55
<.2;/<
<.(2(5
<.(+;I
@@
@@
@@
2'2
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21
<.2(;(
<.2/5;
<.2;/+
<.(2+5
<.(-22
<.(5I2
@@
@@
2''
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21
<.2(;+
<.2/I1
<.2;/5
<.(2-1
<.(-(+
<.(I<5
<.+
@@
2')
<.<(51
<.
<.<;-1
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I2
<.2;/;
<.(2-+
<.(-(;
<.(I2<
<.+<;2
<.+(-<
2'?
<.<(51
<.
<.<;-<
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I(
<.2;5<
<.(2-/
<.(-+2
<.(I2/
<.+1
<.+(55
2'>
<.<(51
<.
<.<;-<
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I(
<.2;51
<.(2-I
<.(-++
<.(I2;
<.+11(
<.+(;(
2',
<.<(51
<.
<.<;-<
<.12+<
<.1-2;
<.1I1I
<.21<5
<.2(;-
<.2/I+
<.2;51
<.(2-;
<.(-+-
<.(I(2
<.+115
<.++<1
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&, MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
22,
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
22'
<.<+1I
<.
<.11;<
<.1-1(
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+1I
<.
<.12((
<.1-;I
<.1;2<
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+1I
<.
<.12+(
<.1/+(
<.2<<5
<.2(2I
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+1I
<.
<.12++
<.1/--
<.2<-2
<.2+1-
<.25(-
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+1I
<.
<.12++
<.1/-5
<.2
<.2+/2
<.2I2(
<.(1+2
@@
@@
@@
@@
@@
@@
@@
22(
<.<+1I
<.
<.12++
<.1/-5
<.2
<.2+5;
<.2I52
<.(2(2
<.(-+;
@@
@@
@@
@@
@@
@@
223
<.<+1I
<.
<.12++
<.1/-5
<.2<5<
<.2+I2
<.2I;<
<.(2I2
<.(/+<
<.(;-/
@@
@@
@@
@@
@@
22@
<.<+1I
<.
<.12++
<.1/-5
<.2<5<
<.2+I(
<.2I;+
<.((<2
<.(/;2
<.+<+;
<.+(/+
@@
@@
@@
@@
226
<.<+1I
<.
<.12++
<.1/-I
<.2<5<
<.2+I(
<.2I;-
<.((<5
<.(51+
<.+1<2
<.++-5
<.+551
@@
@@
@@
2'2
<.<+1I
<.
<.12++
<.1/-I
<.2<5<
<.2+I(
<.2I;/
<.((
<.(51;
<.+12-
<.+-12
<.+I//
<.-15I
@@
@@
2''
<.<+1I
<.
<.12++
<.1/-I
<.2<51
<.2+I(
<.2I;/
<.((<;
<.(521
<.+1(2
<.+-(5
<.+;22
<.-25+
<.--I-
@@
2')
<.<+1I
<.
<.12++
<.1/-5
<.2<51
<.2+I(
<.2I;/
<.((<;
<.(522
<.+1(+
<.+-++
<.+;+I
<.-((2
<.-/I2
<.-;;(
2'?
<.<+1I
<.
<.12+-
<.1/-I
<.2<51
<.2+I+
<.2I;/
<.((<;
<.(522
<.+1(+
<.+-+5
<.+;-/
<.-(-;
<.-5+2
<./<;1
2'>
<.<+1I
<.
<.12++
<.1/-I
<.2<51
<.2+I+
<.2I;5
<.((<;
<.(522
<.+1(-
<.+-+5
<.+;-;
<.-(/;
<.-55<
<./1-2
2',
<.<+1I
<.
<.12++
<.1/-I
<.2<51
<.2+I+
<.2I;5
<.((<;
<.(522
<.+1(-
<.+-+I
<.+;/<
<.-(52
<.-5I1
<./1I1
6ote=
@@ ndicates o3er reinforced section, Fk J x &,ma47d G
2',
TABLE R&, MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
22,
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
22'
<.<+1I
<.
<.11;<
<.1-1(
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+1I
<.
<.12((
<.1-;I
<.1;2<
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+1I
<.
<.12+(
<.1/+(
<.2<<5
<.2(2I
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+1I
<.
<.12++
<.1/--
<.2<-2
<.2+1-
<.25(-
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+1I
<.
<.12++
<.1/-5
<.2
<.2+/2
<.2I2(
<.(1+2
@@
@@
@@
@@
@@
@@
@@
22(
<.<+1I
<.
<.12++
<.1/-5
<.2
<.2+5;
<.2I52
<.(2(2
<.(-+;
@@
@@
@@
@@
@@
@@
223
<.<+1I
<.
<.12++
<.1/-5
<.2<5<
<.2+I2
<.2I;<
<.(2I2
<.(/+<
<.(;-/
@@
@@
@@
@@
@@
22@
<.<+1I
<.
<.12++
<.1/-5
<.2<5<
<.2+I(
<.2I;+
<.((<2
<.(/;2
<.+<+;
<.+(/+
@@
@@
@@
@@
226
<.<+1I
<.
<.12++
<.1/-I
<.2<5<
<.2+I(
<.2I;-
<.((<5
<.(51+
<.+1<2
<.++-5
<.+551
@@
@@
@@
2'2
<.<+1I
<.
<.12++
<.1/-I
<.2<5<
<.2+I(
<.2I;/
<.((
<.(51;
<.+12-
<.+-12
<.+I//
<.-15I
@@
@@
2''
<.<+1I
<.
<.12++
<.1/-I
<.2<51
<.2+I(
<.2I;/
<.((<;
<.(521
<.+1(2
<.+-(5
<.+;22
<.-25+
<.--I-
@@
2')
<.<+1I
<.
<.12++
<.1/-5
<.2<51
<.2+I(
<.2I;/
<.((<;
<.(522
<.+1(+
<.+-++
<.+;+I
<.-((2
<.-/I2
<.-;;(
2'?
<.<+1I
<.
<.12+-
<.1/-I
<.2<51
<.2+I+
<.2I;/
<.((<;
<.(522
<.+1(+
<.+-+5
<.+;-/
<.-(-;
<.-5+2
<./<;1
2'>
<.<+1I
<.
<.12++
<.1/-I
<.2<51
<.2+I+
<.2I;5
<.((<;
<.(522
<.+1(-
<.+-+5
<.+;-;
<.-(/;
<.-55<
<./1-2
2',
<.<+1I
<.
<.12++
<.1/-I
<.2<51
<.2+I+
<.2I;5
<.((<;
<.(522
<.+1(-
<.+-+I
<.+;/<
<.-(52
<.-5I1
<./1I1
6ote=
2',
@@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&( MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
22'
<.<+1(
<.
<.11//
<.1+I;
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+12
<.
<.11II
<.1--1
<.1I52
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+12
<.
<.11;(
<.1-55
<.1;(/
<.22--
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+12
<.
<.11;-
<.1-I(
<.1;/-
<.2(22
<.2/(;
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+12
<.
<.11;-
<.1-I-
<.1;52
<.2(-2
<.25<5
<.(<22
@@
@@
@@
@@
@@
@@
@@
22(
<.<+12
<.
<.11;-
<.1-I/
<.1;5-
<.2(/2
<.25+<
<.(<;2
<.(+
@@
@@
@@
@@
@@
@@
223
<.<+12
<.
<.11;-
<.1-I/
<.1;5/
<.2(/-
<.25-2
<.(125
<.(+55
<.(5I;
@@
@@
@@
@@
@@
22@
<.<+12
<.
<.11;-
<.1-I5
<.1;55
<.2(/5
<.25--
<.(1+1
<.(-1-
<.(I/2
<.+15(
@@
@@
@@
@@
226
<.<+12
<.
<.11;-
<.1-I5
<.1;55
<.2(/I
<.25-5
<.(1+-
<.(-(<
<.(;<2
<.+2+5
<.+--5
@@
@@
@@
2'2
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/I
<.25-I
<.(1+I
<.(-(/
<.(;1;
<.+2;<
<.+/(2
<.+;+<
@@
@@
2''
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25-;
<.(1+;
<.(-(I
<.(;2/
<.+(<;
<.+/55
<.-<15
@@
@@
2')
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-<
<.(-+<
<.(;2;
<.+(1/
<.+/;5
<.-
<.-+<2
@@
2'?
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-1
<.(-+1
<.(;(<
<.+(1;
<.+5
<.-
<.-+-1
<.-5I5
2'>
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-1
<.(-+1
<.(;(1
<.+(21
<.+51<
<.-<;/
<.-+5+
<.-I(5
2',
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-1
<.(-+2
<.(;(2
<.+(22
<.+511
<.-1<<
<.-+I/
<.-I/(
6ote =
@@ ndicates o3er reinforced section, Fk J x &,ma47d G
2'>
2',
TABLE R&( MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
22'
<.<+1(
<.
<.11//
<.1+I;
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+12
<.
<.11II
<.1--1
<.1I52
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+12
<.
<.11;(
<.1-55
<.1;(/
<.22--
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+12
<.
<.11;-
<.1-I(
<.1;/-
<.2(22
<.2/(;
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+12
<.
<.11;-
<.1-I-
<.1;52
<.2(-2
<.25<5
<.(<22
@@
@@
@@
@@
@@
@@
@@
22(
<.<+12
<.
<.11;-
<.1-I/
<.1;5-
<.2(/2
<.25+<
<.(<;2
<.(+
@@
@@
@@
@@
@@
@@
223
<.<+12
<.
<.11;-
<.1-I/
<.1;5/
<.2(/-
<.25-2
<.(125
<.(+55
<.(5I;
@@
@@
@@
@@
@@
22@
<.<+12
<.
<.11;-
<.1-I5
<.1;55
<.2(/5
<.25--
<.(1+1
<.(-1-
<.(I/2
<.+15(
@@
@@
@@
@@
226
<.<+12
<.
<.11;-
<.1-I5
<.1;55
<.2(/I
<.25-5
<.(1+-
<.(-(<
<.(;<2
<.+2+5
<.+--5
@@
@@
@@
2'2
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/I
<.25-I
<.(1+I
<.(-(/
<.(;1;
<.+2;<
<.+/(2
<.+;+<
@@
@@
2''
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25-;
<.(1+;
<.(-(I
<.(;2/
<.+(<;
<.+/55
<.-<15
@@
@@
2')
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-<
<.(-+<
<.(;2;
<.+(1/
<.+/;5
<.-
<.-+<2
@@
2'?
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-1
<.(-+1
<.(;(<
<.+(1;
<.+5
<.-
<.-+-1
<.-5I5
2'>
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-1
<.(-+1
<.(;(1
<.+(21
<.+51<
<.-<;/
<.-+5+
<.-I(5
2',
<.<+12
<.
<.11;-
<.1-I5
<.1;5I
<.2(/;
<.25/<
<.(1-1
<.(-+2
<.(;(2
<.+(22
<.+511
<.-1<<
<.-+I/
<.-I/(
6ote =
2'>
2',
@@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&3 MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<+1-
<.<5I5
<.11+(
<.1+/-
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+1/
<.<5I5
<.11-(
<.1-
<.1I2-
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+1/
<.<5I5
<.11-/
<.1-1I
<.1I/I
<.21I/
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+1/
<.<5I5
<.11-5
<.1-2(
<.1II+
<.22(1
<.2-+/
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+1/
<.<5I5
<.11-5
<.1-2-
<.1I;<
<.22-<
<.2-;-
<.2;<5
@@
@@
@@
@@
@@
@@
@@
22(
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;(
<.22-5
<.2/1/
<.2;-5
<.(2/5
@@
@@
@@
@@
@@
@@
223
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;+
<.22/<
<.2/2+
<.2;I2
<.((2<
@@
@@
@@
@@
@@
@@
22@
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;-
<.22/2
<.2/2I
<.2;;1
<.((+5
<.(/I(
@@
@@
@@
@@
@@
226
<.<+1/
<.<5I5
<.11-5
<.1-25
<.1I;/
<.22/+
<.2/(1
<.2;;-
<.((-I
<.(512
<.+<+/
@@
@@
@@
@@
2'2
<.<+1/
<.<5I5
<.11-5
<.1-25
<.1I;/
<.22/+
<.2/(2
<.2;;;
<.((/(
<.(52-
<.+<55
<.++<;
@@
@@
@@
2''
<.<+1/
<.<5I5
<.11-5
<.1-25
<.1I;/
<.22/-
<.2/((
<.(<<<
<.((//
<.(5(1
<.+<;2
<.+++2
<.+552
@@
@@
2')
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;/
<.22/-
<.2/(+
<.(<<2
<.((/;
<.(5(+
<.+<;I
<.++-;
<.+I<5
<.-1(-
@@
2'?
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;/
<.22/-
<.2/(+
<.(<<(
<.((5<
<.(5(5
<.+1<2
<.++//
<.+I2-
<.-151
<.-+;5
2'>
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;/
<.22//
<.2/(-
<.(<<(
<.((51
<.(5(;
<.+1<-
<.++5<
<.+I(+
<.-1;1
<.--(/
2',
<.<+1/
<.<5I/
<.11-5
<.1-2/
<.1I;/
<.22//
<.2/(-
<.(<<+
<.((52
<.(5+<
<.+1<5
<.++5(
<.+I(I
<.-2<2
<.---I
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&3 MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<+1-
<.<5I5
<.11+(
<.1+/-
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+1/
<.<5I5
<.11-(
<.1-
<.1I2-
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+1/
<.<5I5
<.11-/
<.1-1I
<.1I/I
<.21I/
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+1/
<.<5I5
<.11-5
<.1-2(
<.1II+
<.22(1
<.2-+/
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+1/
<.<5I5
<.11-5
<.1-2-
<.1I;<
<.22-<
<.2-;-
<.2;<5
@@
@@
@@
@@
@@
@@
@@
22(
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;(
<.22-5
<.2/1/
<.2;-5
<.(2/5
@@
@@
@@
@@
@@
@@
223
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;+
<.22/<
<.2/2+
<.2;I2
<.((2<
@@
@@
@@
@@
@@
@@
22@
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;-
<.22/2
<.2/2I
<.2;;1
<.((+5
<.(/I(
@@
@@
@@
@@
@@
226
<.<+1/
<.<5I5
<.11-5
<.1-25
<.1I;/
<.22/+
<.2/(1
<.2;;-
<.((-I
<.(512
<.+<+/
@@
@@
@@
@@
2'2
<.<+1/
<.<5I5
<.11-5
<.1-25
<.1I;/
<.22/+
<.2/(2
<.2;;;
<.((/(
<.(52-
<.+<55
<.++<;
@@
@@
@@
2''
<.<+1/
<.<5I5
<.11-5
<.1-25
<.1I;/
<.22/-
<.2/((
<.(<<<
<.((//
<.(5(1
<.+<;2
<.+++2
<.+552
@@
@@
2')
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;/
<.22/-
<.2/(+
<.(<<2
<.((/;
<.(5(+
<.+<;I
<.++-;
<.+I<5
<.-1(-
@@
2'?
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;/
<.22/-
<.2/(+
<.(<<(
<.((5<
<.(5(5
<.+1<2
<.++//
<.+I2-
<.-151
<.-+;5
2'>
<.<+1/
<.<5I5
<.11-5
<.1-2/
<.1I;/
<.22//
<.2/(-
<.(<<(
<.((51
<.(5(;
<.+1<-
<.++5<
<.+I(+
<.-1;1
<.--(/
2',
<.<+1/
<.<5I/
<.11-5
<.1-2/
<.1I;/
<.22//
<.2/(-
<.(<<+
<.((52
<.(5+<
<.+1<5
<.++5(
<.+I(I
<.-2<2
<.---I
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&@ MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<+2-
<.<5I1
<.112(
<.1++2
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+2I
<.<5I1
<.112I
<.1+/+
<.155;
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+2;
<.<5I<
<.112;
<.1+52
<.1I<+
<.2115
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+2;
<.<5I<
<.112;
<.1+5-
<.1I1-
<.21+-
<.2+--
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+(<
<.<5I<
<.112;
<.1+5/
<.1I2<
<.21-I
<.2+I5
@@
@@
@@
@@
@@
@@
@@
@@
22(
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I22
<.21/+
<.2-<1
<.2I2;
@@
@@
@@
@@
@@
@@
@@
223
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2+
<.21/I
<.2-
<.2I++
<.(15<
@@
@@
@@
@@
@@
@@
22@
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2+
<.215<
<.2-1(
<.2I-2
<.(1I5
<.(-1<
@@
@@
@@
@@
@@
226
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2-
<.2151
<.2-1/
<.2I-I
<.(1;5
<.(-(1
<.(I-1
@@
@@
@@
@@
2'2
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2-
<.2152
<.2-15
<.2I/1
<.(2<(
<.(-+1
<.(I5-
<.+1;2
@@
@@
@@
2''
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.2152
<.2-1I
<.2I/(
<.(2<5
<.(-+I
<.(II-
<.+21I
@@
@@
@@
2')
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.2152
<.2-1;
<.2I/-
<.(2<;
<.(--2
<.(I;(
<.+2(<
<.+-/<
@@
@@
2'?
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.215(
<.2-2<
<.2I//
<.(211
<.(---
<.(I;I
<.+2(I
<.+-5-
<.+;<(
@@
2'>
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.215(
<.2-2<
<.2I/5
<.(21(
<.(--5
<.(;<1
<.+2+(
<.+-I(
<.+;2<
<.-2+-
2',
<.<+(1
<.<55;
<.112I
<.1+55
<.1I2-
<.215(
<.2-2<
<.2I/5
<.(21+
<.(--;
<.(;<+
<.+2+5
<.+-II
<.+;2I
<.-2/-
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE R&@ MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS M # f ckbd ) Gr*+e o; steel
Fe&,22
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.<+2-
<.<5I1
<.112(
<.1++2
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22)
<.<+2I
<.<5I1
<.112I
<.1+/+
<.155;
@@
@@
@@
@@
@@
@@
@@
@@
@@
@@
22?
<.<+2;
<.<5I<
<.112;
<.1+52
<.1I<+
<.2115
@@
@@
@@
@@
@@
@@
@@
@@
@@
22>
<.<+2;
<.<5I<
<.112;
<.1+5-
<.1I1-
<.21+-
<.2+--
@@
@@
@@
@@
@@
@@
@@
@@
22,
<.<+(<
<.<5I<
<.112;
<.1+5/
<.1I2<
<.21-I
<.2+I5
@@
@@
@@
@@
@@
@@
@@
@@
22(
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I22
<.21/+
<.2-<1
<.2I2;
@@
@@
@@
@@
@@
@@
@@
223
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2+
<.21/I
<.2-
<.2I++
<.(15<
@@
@@
@@
@@
@@
@@
22@
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2+
<.215<
<.2-1(
<.2I-2
<.(1I5
<.(-1<
@@
@@
@@
@@
@@
226
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2-
<.2151
<.2-1/
<.2I-I
<.(1;5
<.(-(1
<.(I-1
@@
@@
@@
@@
2'2
<.<+(<
<.<5I<
<.112;
<.1+55
<.1I2-
<.2152
<.2-15
<.2I/1
<.(2<(
<.(-+1
<.(I5-
<.+1;2
@@
@@
@@
2''
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.2152
<.2-1I
<.2I/(
<.(2<5
<.(-+I
<.(II-
<.+21I
@@
@@
@@
2')
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.2152
<.2-1;
<.2I/-
<.(2<;
<.(--2
<.(I;(
<.+2(<
<.+-/<
@@
@@
2'?
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.215(
<.2-2<
<.2I//
<.(211
<.(---
<.(I;I
<.+2(I
<.+-5-
<.+;<(
@@
2'>
<.<+(1
<.<5I<
<.112I
<.1+55
<.1I2-
<.215(
<.2-2<
<.2I/5
<.(21(
<.(--5
<.(;<1
<.+2+(
<.+-I(
<.+;2<
<.-2+-
2',
<.<+(1
<.<55;
<.112I
<.1+55
<.1I2-
<.215(
<.2-2<
<.2I/5
<.(21+
<.(--;
<.(;<+
<.+2+5
<.+-II
<.+;2I
<.-2/-
6ote = 0al&es s#o"n in Red ndicates 6e&tral A4is lies "it#in t#e co3erin! concrete, Fk H d 7d G @@ ndicates o3er reinforced section, Fk J x &,ma47d G
TABLE !&' DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
22,
pt f ck
pc f ck
22'
22' 22)
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
<.
<.112-<+
<.2
<.(<-(+;
<.+<-2-/
<.-<<552
<.-55/I2
<./((552
<./5/5;5
<.5<5+++
<.52(<<1
<.5(/I+5
<.5+;2;(
<.5/<--+
<.55<5;5
<.<-//15
<.<5+;;2
<.1222(+
<.2115+-
<.(<;;
<.+<;+;1
<.-<(+(/
<.-55//;
<./(<;+-
<./52(<<
<.5<21//
<.51;<<2
<.5(2<;-
<.5+(;--
<.5-+5--
22?
<.<-++-;
<.
<.
<.1(<+/1
<.21/;++
<.(1++2;
<.+1(5<-
<.-<-;-5
<.-55/-/
<./2I(;;
<.//I(22
<./;5-+2
<.51--+(
<.525;-+
<.5(;2/;
22>
<.<-((-I
<.
<.<51I/(
<.
<.1(5//5
<.221;1<
<.(1I;12
<.+15I;;
<.-
<.-55/++
<./2/<;+
<.//+5I2
<./;(+/2
<.512-2+
<.52+(12
22,
<.<-2/;2
<.<-I;2/
<.
<.<5/(<2
<.<;1I(/
<.1++11I
<.22//51
<.(2((/1
<.+22<52
<.-1
<.-55/((
<./2(;;;
<.//1/12
<./I;I(-
<.5<;I//
22(
<.<-22+/
<.<-5(2-
<.
<.<5<5<+
<.<5;/I/
<.<;5(+I
<.1+;;I+
<.2(12-1
<.(2555I
<.+2/22I
<.-125I2
<.-55/2(
<./22
<./-I5-;
<./I/-;+
223
<.<-1;2/
<.<-/2
<.
<.
<.<5(;;(
<.
<.1<2/<1
<.1--(5;
<.2(-//5
<.((21/2
<.+(<(//
<.-1+I++
<.-55/1+
<./2<((/
<./-/15;
22@
<.<-1/I/
<.<--(I/
<.<-;/
<.
<.<5<1<1
<.<5/5
<.
<.1<5211
<.1/<(I-
<.2+<(1-
<.((/-15
<.+(++I/
<.-1/I1+
<.-55/
<./1I525
226
<.<-1+;;
<.<-+5-/
<.<-I+15
<.
<.
<.<525<-
<.<5I;I/
<.
<.111-2/
<.1/-/++
<.2++;II
<.(+
<.+(I-;<
<.-1I/;5
<.-55-;I
2'2
<.<-1(-<
<.<-+2-5
<.<-5+II
<.
<.
<.
<.<5+;+I
<.
<.<;<-2I
<.11/5/+
<.15
<.2+;/<;
<.(+-1+2
<.++2/55
<.-2<-<1
2''
<.<-1225
<.<-(I-(
<.<-/5+2
<.<-;;(/
<.
<.
<.<51II-
<.<5/;<(
<.
<.<;(2;;
<.121//-
<.15--11
<.2-+1I+
<.(+;+15
<.++/5+;
2')
<.<-112-
<.<-(-1;
<.<-/1(2
<.<-I;;+
<.
<.
<.
<.<5(5;1
<.<5I/22
<.
<.<;/5+1
<.12/2I(
<.1I<1-1
<.2-I512
<.(-(//-
2'?
<.<-1<(;
<.<-(2(I
<.<--/22
<.<-I21-
<.
<.
<.
<.<51(15
<.<5-+;2
<.
<.
<.1<<<25
<.1(
<.1I+/2+
<.2/(1;;
2'>
<.<-<;/-
<.<-2;;I
<.<--1;1
<.<-5-/<
<.
<.
<.
<.
<.<52;52
<.<55<1;
<.
<.
<.1<(<5/
<.1(+I<;
<.1II;+/
2',
<.<-<;<<
<.<-25;2
<.<-+I2<
<.<-5<<1
<.<-;(-1
<.
<.
<.
<.<5<;<1
<.<5++5(
<.<5I(;;
<.
<.<;<<<-
<.1<-;1+
<.1(I5/;
TABLE !&' DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
22,
pt f ck
pc f ck
22'
22' 22)
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
<.
<.112-<+
<.2
<.(<-(+;
<.+<-2-/
<.-<<552
<.-55/I2
<./((552
<./5/5;5
<.5<5+++
<.52(<<1
<.5(/I+5
<.5+;2;(
<.5/<--+
<.55<5;5
<.<-//15
<.<5+;;2
<.1222(+
<.2115+-
<.(<;;
<.+<;+;1
<.-<(+(/
<.-55//;
<./(<;+-
<./52(<<
<.5<21//
<.51;<<2
<.5(2<;-
<.5+(;--
<.5-+5--
22?
<.<-++-;
<.
<.
<.1(<+/1
<.21/;++
<.(1++2;
<.+1(5<-
<.-<-;-5
<.-55/-/
<./2I(;;
<.//I(22
<./;5-+2
<.51--+(
<.525;-+
<.5(;2/;
22>
<.<-((-I
<.
<.<51I/(
<.
<.1(5//5
<.221;1<
<.(1I;12
<.+15I;;
<.-
<.-55/++
<./2/<;+
<.//+5I2
<./;(+/2
<.512-2+
<.52+(12
22,
<.<-2/;2
<.<-I;2/
<.
<.<5/(<2
<.<;1I(/
<.1++11I
<.22//51
<.(2((/1
<.+22<52
<.-1
<.-55/((
<./2(;;;
<.//1/12
<./I;I(-
<.5<;I//
22(
<.<-22+/
<.<-5(2-
<.
<.<5<5<+
<.<5;/I/
<.<;5(+I
<.1+;;I+
<.2(12-1
<.(2555I
<.+2/22I
<.-125I2
<.-55/2(
<./22
<./-I5-;
<./I/-;+
223
<.<-1;2/
<.<-/2
<.
<.
<.<5(;;(
<.
<.1<2/<1
<.1--(5;
<.2(-//5
<.((21/2
<.+(<(//
<.-1+I++
<.-55/1+
<./2<((/
<./-/15;
22@
<.<-1/I/
<.<--(I/
<.<-;/
<.
<.<5<1<1
<.<5/5
<.
<.1<5211
<.1/<(I-
<.2+<(1-
<.((/-15
<.+(++I/
<.-1/I1+
<.-55/
<./1I525
226
<.<-1+;;
<.<-+5-/
<.<-I+15
<.
<.
<.<525<-
<.<5I;I/
<.
<.111-2/
<.1/-/++
<.2++;II
<.(+
<.+(I-;<
<.-1I/;5
<.-55-;I
2'2
<.<-1(-<
<.<-+2-5
<.<-5+II
<.
<.
<.
<.<5+;+I
<.
<.<;<-2I
<.11/5/+
<.15
<.2+;/<;
<.(+-1+2
<.++2/55
<.-2<-<1
2''
<.<-1225
<.<-(I-(
<.<-/5+2
<.<-;;(/
<.
<.
<.<51II-
<.<5/;<(
<.
<.<;(2;;
<.121//-
<.15--11
<.2-+1I+
<.(+;+15
<.++/5+;
2')
<.<-112-
<.<-(-1;
<.<-/1(2
<.<-I;;+
<.
<.
<.
<.<5(5;1
<.<5I/22
<.
<.<;/5+1
<.12/2I(
<.1I<1-1
<.2-I512
<.(-(//-
2'?
<.<-1<(;
<.<-(2(I
<.<--/22
<.<-I21-
<.
<.
<.
<.<51(15
<.<5-+;2
<.
<.
<.1<<<25
<.1(
<.1I+/2+
<.2/(1;;
2'>
<.<-<;/-
<.<-2;;I
<.<--1;1
<.<-5-/<
<.
<.
<.
<.
<.<52;52
<.<55<1;
<.
<.
<.1<(<5/
<.1(+I<;
<.1II;+/
2',
<.<-<;<<
<.<-25;2
<.<-+I2<
<.<-5<<1
<.<-;(-1
<.
<.
<.
<.<5<;<1
<.<5++5(
<.<5I(;;
<.
<.<;<<<-
<.1<-;1+
<.1(I5/;
TABLE !&) DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
22'
<.1<<
<.1+(+;(
<.21(;/1
<.(
<.+
<.-<1+55
<.-5I1/-
<./(+1/(
<./55
22)
<.1<<<-(
<.12-151
<.1-;51+
<.22-<<5
<.(1/112
<.+12--;
<.-<+5;I
<.-5I-I5
<./(1/;<
22?
<.1<<<(I
<.115+<<
<.1(;;12
<.1/;21;
<.2(-2<2
<.(2(12<
<.+1I<;I
<.-<5;(-
22>
<.1<<<2;
<.11(2(-
<.12;-52
<.1+;;I+
<.15I;2;
<.2+++/I
<.(2;;(I
22,
<.1<<<2+
<.11
<.12((5;
<.1(I52;
<.1-5+1+
<.1I5(+(
<.2-2;;I
22(
<.1<<<2<
<.1
<.11;2;(
<.1(1+I+
<.1+-;2I
<.1/(1/5
223
<.1<<<1I
<.1<5//+
<.11/+<5
<.12/+52
<.1(I1+2
<.1-15/-
22@
<.1<<<1/
<.1
<.11+2/-
<.122I1/
<.1(2--I
226
<.1<<<1+
<.1<-;55
<.112/1+
<.12<<(-
2'2
<.1<<<1(
<.1<-(I+
<.111(<(
<.115I-2
2''
2')
2'?
2'>
2',
<.5<5/-/
<.52(11/
<.5(/;-/
<.5+;(;/
<.5/
<.55
<./52I(;
<.5<2-/2
<.51;21/
<.5(22;I
<.5++1+I
<.5-+;+<
<.-5I;5<
<./2;+/2
<.//;
<./;I1<1
<.51-I+(
<.52I2(;
<.5(;-+1
<.+2(+;2
<.-1<;
<.-5;(2<
<./25++5
<.//-5+5
<./;+1/+
<.512I;;
<.52+/51
<.((/+;/
<.+2I5-(
<.-1(52/
<.-5;/+1
<./2-/1-
<.//25-5
<./;
<.51<(<5
<.1;/1;/
<.2/<;2+
<.(+2I22
<.+((II;
<.-1/+
<.-5;;(/
<./2(;++
<.//<
<./I5-(I
<.1/555(
<.2<+<25
<.2/I(++
<.(+I;+<
<.+(I;1<
<.-1I;/2
<.-I<2
<./22+1+
<./-5/(2
<.1+(525
<.1-//<;
<.152+-/
<.211<++
<.25-((+
<.(-+I/5
<.++(I21
<.-21+<<
<.-I<+/<
<./21<
<.12I(5(
<.1(55;2
<.1+I+I;
<.1/<5<(
<.15I(;2
<.215+<1
<.2I1;+;
<.(/
<.++I/(<
<.-2(5(<
<.-I
<.12-12I
<.1((2+/
<.1+2(+;
<.1-2/<+
<.1/+212
<.1I(/5(
<.22(5<+
<.2II2(/
<.(//21+
<.+-((++
<.-2-;-;
2''
<.1<<<11
<.1<+I;I
<.11<2(I
<.11/<;-
<.122-+<
<.12;//<
<.1(5--;
<.1+/(-I
<.1-/2<<
<.1/52-;
<.1III+-
<.2(<52;
<.2;+2(2
<.(51//<
<.+-5;//
2')
<.1<<<11
<.1<++;2
<.1<;(-/
<.11+/-<
<.12<+(1
<.12/5/(
<.1((52/
<.1+1+<5
<.1+;;1+
<.1-;(52
<.1/;;<(
<.1;+/;-
<.2(5(I-
<.2;;;/;
<.(5/;/;
2'?
<.1<<<1<
<.1<+1+I
<.1
<.11(++2
<.11I/I<
<.12+(55
<.1(<-;2
<.1(5(;+
<.1++I/(
<.1-(<;2
<.1/21;(
<.15(;;/
<.2<<1(1
<.2+(52-
<.(<-+5<
2'>
<.1<<<<;
<.1<(I-(
<.1<5;5;
<.112+1I
<.1152<+
<.122(55
<.125;I+
<.1(+<5;
<.1+<522
<.1+5;I/
<.1--;-2
<.1/+51;
<.15I1/2
<.2<-2<2
<.2+;5I-
2',
<.1<<<
<.1<(-;5
<.1<5+(2
<.111-(5
<.11-;+2
<.12
<.12-5I2
<.1(12;/
<.1(525<
<.1+(5-5
<.1-
<.1-I-+<
<.1//;;/
<.1I2<<5
<.2<;;-+
TABLE !&) DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
22'
<.1<<
<.1+(+;(
<.21(;/1
<.(
<.+
<.-<1+55
<.-5I1/-
<./(+1/(
<./55
22)
<.1<<<-(
<.12-151
<.1-;51+
<.22-<<5
<.(1/112
<.+12--;
<.-<+5;I
<.-5I-I5
<./(1/;<
22?
<.1<<<(I
<.115+<<
<.1(;;12
<.1/;21;
<.2(-2<2
<.(2(12<
<.+1I<;I
<.-<5;(-
22>
<.1<<<2;
<.11(2(-
<.12;-52
<.1+;;I+
<.15I;2;
<.2+++/I
<.(2;;(I
<.+2(+;2
22,
<.1<<<2+
<.11
<.12((5;
<.1(I52;
<.1-5+1+
<.1I5(+(
<.2-2;;I
22(
<.1<<<2<
<.1
<.11;2;(
<.1(1+I+
<.1+-;2I
<.1/(1/5
223
<.1<<<1I
<.1<5//+
<.11/+<5
<.12/+52
<.1(I1+2
<.1-15/-
22@
<.1<<<1/
<.1
<.11+2/-
<.122I1/
<.1(2--I
226
<.1<<<1+
<.1<-;55
<.112/1+
<.12<<(-
<.12I(5(
2'2
<.1<<<1(
<.1<-(I+
<.111(<(
<.115I-2
2''
2')
2'?
2'>
2',
<.5<5/-/
<.52(11/
<.5(/;-/
<.5+;(;/
<.5/
<.55
<./52I(;
<.5<2-/2
<.51;21/
<.5(22;I
<.5++1+I
<.5-+;+<
<.-5I;5<
<./2;+/2
<.//;
<./;I1<1
<.51-I+(
<.52I2(;
<.5(;-+1
<.-1<;
<.-5;(2<
<./25++5
<.//-5+5
<./;+1/+
<.512I;;
<.52+/51
<.((/+;/
<.+2I5-(
<.-1(52/
<.-5;/+1
<./2-/1-
<.//25-5
<./;
<.51<(<5
<.1;/1;/
<.2/<;2+
<.(+2I22
<.+((II;
<.-1/+
<.-5;;(/
<./2(;++
<.//<
<./I5-(I
<.1/555(
<.2<+<25
<.2/I(++
<.(+I;+<
<.+(I;1<
<.-1I;/2
<.-I<2
<./22+1+
<./-5/(2
<.1+(525
<.1-//<;
<.152+-/
<.211<++
<.25-((+
<.(-+I/5
<.++(I21
<.-21+<<
<.-I<+/<
<./21<
<.1(55;2
<.1+I+I;
<.1/<5<(
<.15I(;2
<.215+<1
<.2I1;+;
<.(/
<.++I/(<
<.-2(5(<
<.-I
<.12-12I
<.1((2+/
<.1+2(+;
<.1-2/<+
<.1/+212
<.1I(/5(
<.22(5<+
<.2II2(/
<.(//21+
<.+-((++
<.-2-;-;
2''
<.1<<<11
<.1<+I;I
<.11<2(I
<.11/<;-
<.122-+<
<.12;//<
<.1(5--;
<.1+/(-I
<.1-/2<<
<.1/52-;
<.1III+-
<.2(<52;
<.2;+2(2
<.(51//<
<.+-5;//
2')
<.1<<<11
<.1<++;2
<.1<;(-/
<.11+/-<
<.12<+(1
<.12/5/(
<.1((52/
<.1+1+<5
<.1+;;1+
<.1-;(52
<.1/;;<(
<.1;+/;-
<.2(5(I-
<.2;;;/;
<.(5/;/;
2'?
<.1<<<1<
<.1<+1+I
<.1
<.11(++2
<.11I/I<
<.12+(55
<.1(<-;2
<.1(5(;+
<.1++I/(
<.1-(<;2
<.1/21;(
<.15(;;/
<.2<<1(1
<.2+(52-
<.(<-+5<
2'>
<.1<<<<;
<.1<(I-(
<.1<5;5;
<.112+1I
<.1152<+
<.122(55
<.125;I+
<.1(+<5;
<.1+<522
<.1+5;I/
<.1--;-2
<.1/+51;
<.15I1/2
<.2<-2<2
<.2+;5I-
2',
<.1<<<
<.1<(-;5
<.1<5+(2
<.111-(5
<.11-;+2
<.12
<.12-5I2
<.1(12;/
<.1(525<
<.1+(5-5
<.1-
<.1-I-+<
<.1//;;/
<.1I2<<5
<.2<;;-+
TABLE !&? DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1(<<-;
<.15+(5;
<.2(((;;
<.(1+5/2
<.+1<+(-
<.-<((/2
<.-5;1/1
<./(+5;+
<./55++-
<.5<5I51
<.52(2(2
<.5(5
<.5+;-<<
<.5/<5-<
<.55<;I+
22)
<.1(5<(/
<.1/--22
<.2<2I5I
<.2+;;5/
<.(2//I-
<.+1;(1I
<.-
<.-I<+51
<./(2;<<
<./5(-(I
<.5<2;I<
<.51;+(1
<.5(2-<1
<.5++(+1
<.5--12+
22?
<.1+<(-1
<.1/1(1<
<.1I55-5
<.22<+;(
<.2/25-1
<.((5-11
<.+2552(
<.-1(<+(
<.-I1/-2
<./(12<-
<./5<<;/
<./;I5
<.51/1+(
<.52I-2+
<.5(;I1(
22>
<.1+2(<-
<.1-II5I
<.15;12-
<.2<(-11
<.2(2;<1
<.25+/11
<.(+5I5;
<.+(-512
<.-15+
<.-I2521
<./2;/5I
<.//5<+-
<./;+;+;
<.51(25+
<.52-<2I
22,
<.1+(-;5
<.1-5(<1
<.15(/(1
<.1;2I25
<.21-+I<
<.2+22-;
<.2I-2+;
<.(-5-5+
<.++(((+
<.-21+;5
<.-I(/;-
<./2I2;I
<.//+(22
<./;1/1/
<.51<5+I
22(
<.1++-1/
<.1-/1;I
<.1/;I-2
<.1I--I5
<.2<(I<-
<.22+;5/
<.2+;/(1
<.2;-/2/
<.(//5<2
<.+-
<.-2-(+5
<.-I+-I/
<./25<+-
<.//1I5I
<./II/+(
223
<.1+-2<+
<.1--(I(
<.1/51<2
<.1I<(;(
<.1;--(/
<.212I-/
<.2(25(;
<.2---55
<.(<+;I-
<.(5-(+<
<.+-5/2/
<.-2I;55
<.-I-+<+
<./2-;<1
<./-;/5-
22@
<.1+-5(I
<.1-+5-5
<.1/-<1-
<.15/+;;
<.1I;+
<.2<(;5-
<.22<+55
<.2(;22;
<.2/+121
<.(1(-11
<.(I(--2
<.+/+(-I
<.-(2+12
<.-I/1-I
<./2+I-(
226
<.1+/1/+
<.1-+2/1
<.1/((5I
<.15(+55
<.1I+5<(
<.1;5225
<.2112+I
<.225<<<
<.2++5-<
<.251521
<.(21(+1
<.(;1(I/
<.+5
<.-(-//-
<.-I/I-/
2'2
<.1+/-1(
<.1-(I-I
<.1/2
<.151
<.1I<;I+
<.1;1;+2
<.2<+
<.215-;(
<.2(2/-;
<.2+;-1(
<.25I-+;
<.(2I-I<
<.(;III+
<.+5511(
<.-(I5-/
2''
<.1+/I<+
<.1-(-2-
<.1/<;I<
<.1/;1<1
<.155;5-
<.1I5/;;
<.1;I(I-
<.21<1/(
<.22(1I-
<.2(5/21
<.2-(/5<
<.2I-;
<.((/<+/
<.+
<.+I2<-(
2')
<.1+5<-<
<.1-(2+-
<.1/<<5-
<.1/5+/;
<.15-+;(
<.1I+22(
<.1;(5+5
<.2<+1/+
<.21--;1
<.22I1-5
<.2+2<1+
<.2-I/I(
<.2;((((
<.(+(;5;
<.+1(<<2
2'?
<.1+52/1
<.1-(<
<.1-;(<5
<.1//<;1
<.15(+11
<.1I1(2/
<.1I;;
<.1;;22;
<.2<;(I(
<.22<+52
<.2(2/11
<.2+-;((
<.2/+5+<
<.(<<25/
<.(-1-5<
2'>
<.1+5++(
<.1-2I<<
<.1-I/+I
<.1/+;1+
<.151/+1
<.15II5/
<.1I//5-
<.1;-1<1
<.2<+222
<.21+121
<.22+III
<.2(//25
<.2+;+-+
<.25<(2I
<.(
2',
<.1+5/<(
<.1-2/2<
<.1-I<5/
<.1/(I;5
<.15<11I
<.15/55I
<.1I(;22
<.1;1/<1
<.1;;I5<
<.2
<.21I++/
<.22I;
<.2+<2/I
<.2-2/(/
<.25--
TABLE !&? DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1(<<-;
<.15+(5;
<.2(((;;
<.(1+5/2
<.+1<+(-
<.-<((/2
<.-5;1/1
<./(+5;+
<./55++-
<.5<5I51
<.52(2(2
<.5(5
<.5+;-<<
<.5/<5-<
<.55<;I+
22)
<.1(5<(/
<.1/--22
<.2<2I5I
<.2+;;5/
<.(2//I-
<.+1;(1I
<.-
<.-I<+51
<./(2;<<
<./5(-(I
<.5<2;I<
<.51;+(1
<.5(2-<1
<.5++(+1
<.5--12+
22?
<.1+<(-1
<.1/1(1<
<.1I55-5
<.22<+;(
<.2/25-1
<.((5-11
<.+2552(
<.-1(<+(
<.-I1/-2
<./(12<-
<./5<<;/
<./;I5
<.51/1+(
<.52I-2+
<.5(;I1(
22>
<.1+2(<-
<.1-II5I
<.15;12-
<.2<(-11
<.2(2;<1
<.25+/11
<.(+5I5;
<.+(-512
<.-15+
<.-I2521
<./2;/5I
<.//5<+-
<./;+;+;
<.51(25+
<.52-<2I
22,
<.1+(-;5
<.1-5(<1
<.15(/(1
<.1;2I25
<.21-+I<
<.2+22-;
<.2I-2+;
<.(-5-5+
<.++(((+
<.-21+;5
<.-I(/;-
<./2I2;I
<.//+(22
<./;1/1/
<.51<5+I
22(
<.1++-1/
<.1-/1;I
<.1/;I-2
<.1I--I5
<.2<(I<-
<.22+;5/
<.2+;/(1
<.2;-/2/
<.(//5<2
<.+-
<.-2-(+5
<.-I+-I/
<./25<+-
<.//1I5I
<./II/+(
223
<.1+-2<+
<.1--(I(
<.1/51<2
<.1I<(;(
<.1;--(/
<.212I-/
<.2(25(;
<.2---55
<.(<+;I-
<.(5-(+<
<.+-5/2/
<.-2I;55
<.-I-+<+
<./2-;<1
<./-;/5-
22@
<.1+-5(I
<.1-+5-5
<.1/-<1-
<.15/+;;
<.1I;+
<.2<(;5-
<.22<+55
<.2(;22;
<.2/+121
<.(1(-11
<.(I(--2
<.+/+(-I
<.-(2+12
<.-I/1-I
<./2+I-(
226
<.1+/1/+
<.1-+2/1
<.1/((5I
<.15(+55
<.1I+5<(
<.1;5225
<.2112+I
<.225<<<
<.2++5-<
<.251521
<.(21(+1
<.(;1(I/
<.+5
<.-(-//-
<.-I/I-/
2'2
<.1+/-1(
<.1-(I-I
<.1/2
<.151
<.1I<;I+
<.1;1;+2
<.2<+
<.215-;(
<.2(2/-;
<.2+;-1(
<.25I-+;
<.(2I-I<
<.(;III+
<.+5511(
<.-(I5-/
2''
<.1+/I<+
<.1-(-2-
<.1/<;I<
<.1/;1<1
<.155;5-
<.1I5/;;
<.1;I(I-
<.21<1/(
<.22(1I-
<.2(5/21
<.2-(/5<
<.2I-;
<.((/<+/
<.+
<.+I2<-(
2')
<.1+5<-<
<.1-(2+-
<.1/<<5-
<.1/5+/;
<.15-+;(
<.1I+22(
<.1;(5+5
<.2<+1/+
<.21--;1
<.22I1-5
<.2+2<1+
<.2-I/I(
<.2;((((
<.(+(;5;
<.+1(<<2
2'?
<.1+52/1
<.1-(<
<.1-;(<5
<.1//<;1
<.15(+11
<.1I1(2/
<.1I;;
<.1;;22;
<.2<;(I(
<.22<+52
<.2(2/11
<.2+-;((
<.2/+5+<
<.(<<25/
<.(-1-5<
2'>
<.1+5++(
<.1-2I<<
<.1-I/+I
<.1/+;1+
<.151/+1
<.15II5/
<.1I//5-
<.1;-1<1
<.2<+222
<.21+121
<.22+III
<.2(//25
<.2+;+-+
<.25<(2I
<.(
2',
<.1+5/<(
<.1-2/2<
<.1-I<5/
<.1/(I;5
<.15<11I
<.15/55I
<.1I(;22
<.1;1/<1
<.1;;I5<
<.2
<.21I++/
<.22I;
<.2+<2/I
<.2-2/(/
<.25--
TABLE !&> DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1--/-1
<.2<<2-;
<.2-I(I+
<.(25-1;
<.+1-(1(
<.-
<.-I<5;/
<./(-;-I
<./5I2I+
<.5
<.52(--I
<.5(5(/<
<.5+;5/I
<.5/<;;/
<.551212
22)
<.1/;/;/
<.2<<15-
<.2(;(<;
<.2I/;I5
<.(++1<5
<.+25;22
<.-1(-5;
<.-I(-+(
<./(-1
<./5-11(
<.5<+1(;
<.52<<+2
<.5((<-;
<.5++I-(
<.5---;-
22?
<.15/5;;
<.2<<1(2
<.22;2-/
<.2/+211
<.(<-5-5
<.(-;1;/
<.+(;21-
<.-2<+<-
<.-I-;;I
<./(+(+I
<./52(22
<.5<<(+<
<.515<
<.52;(1/
<.5+<-++
22>
<.1I11-/
<.2<<1
<.22(1;-
<.2-<(+2
<.2I22
<.(1;+2;
<.(51I(;
<.+-<<1+
<.-2/5<2
<.-II2
<./((//;
<.//;I+5
<./;/;;5
<.51+(/1
<.52/<(1
22,
<.1I+115
<.2<<
<.21;15;
<.2+12<5
<.2////-
<.2;/
<.(2;;I1
<.(I+<
<.+/<+5+
<.-(2-+<
<.-;<2<+
<./((<-I
<.//5/+1
<./;+<((
<.512<(+
22(
<.1I/2/5
<.2<<<5/
<.21/(((
<.2(+5;;
<.2--I(1
<.25;I2-
<.(<521(
<.((I+(;
<.(;-(+<
<.+5<+<-
<.-(5;5I
<.-;2<22
<./(2-<-
<.//-//<
<./;1(I;
223
<.1I5;<<
<.2<<
<.21+21-
<.2(<<5;
<.2+5;1I
<.2/I<2/
<.2;<5(+
<.(1/+<1
<.(+I<5;
<.+<-51<
<.+5;5/;
<.-+(
<.-;(++5
<./(2<<+
<.//(I5-
22@
<.1I;1I+
<.2<<<-;
<.212-I<
<.22/+/;
<.2+1;15
<.2-;1+-
<.25I+<2
<.2;;;/I
<.(2+1+/
<.(-5I-5
<.+1-25+
<.+I/+/2
<.-+5I(1
<.-;+51+
<./(1-+-
226
<.1;<22<
<.2<<<-(
<.2112I<
<.22(/22
<.2(5222
<.2-22-<
<.2/II;(
<.2I5(/5
<.(<5;1<
<.((<5I2
<.(///+5
<.+2+1-<
<.+;251(
<.--2(1I
<.-;-I/;
2'2
<.1;1<5-
<.2<<<+;
<.21<22(
<.221(21
<.2((+-5
<.2+/5-5
<.2/1(5<
<.255+-I
<.2;-21<
<.(1+I2;
<.((/-+2
<.(5+/I/
<.+(2+(+
<.+;I-5<
<.--/--2
2''
<.1;15;1
<.2<<<+-
<.2<;(+/
<.21;+2/
<.2(<(5(
<.2+22II
<.2--2I/
<.2/;+;+
<.2I-<-I
<.(<21+1
<.(2<;21
<.(+2<;(
<.(I(I2I
<.++<1;I
<.-<+
2')
<.1;2+<1
<.2<<<+1
<.2
<.215I(I
<.225I<+
<.2(I-I-
<.2-<25(
<.2/2;/I
<.25/5I5
<.2;1I-/
<.(
<.(2/((+
<.(+;;+(
<.(;2(;(
<.++I+-2
2'?
<.1;2;25
<.2<<<(I
<.2<5;5/
<.21/+II
<.22-/(1
<.2(-+5<
<.2+/<55
<.2-5-(/
<.2/;;(-
<.2I((5I
<.2;5;I<
<.(1(I/5
<.((11I<
<.(-51I/
<.+<<++;
2'>
<.1;((I+
<.2<<<(/
<.2<5+(1
<.21-(25
<.22(55<
<.2(2I1+
<.2+2-15
<.2-2;+/
<.2/+15+
<.25/2I-
<.2I;(5<
<.(<(-(1
<.(1III1
<.((--+/
<.(/(;<(
2',
<.1;(5I-
<.2<<<((
<.2
<.21+(1I
<.2221-;
<.2(<-2+
<.2(;+/<
<.2+;<22
<.2-;251
<.25<252
<.2I21<2
<.2;+I++
<.(
<.(2(+(I
<.((;-<-
TABLE !&> DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&>',
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1--/-1
<.2<<2-;
<.2-I(I+
<.(25-1;
<.+1-(1(
<.-
<.-I<5;/
<./(-;-I
<./5I2I+
<.5
<.52(--I
<.5(5(/<
<.5+;5/I
<.5/<;;/
<.551212
22)
<.1/;/;/
<.2<<15-
<.2(;(<;
<.2I/;I5
<.(++1<5
<.+25;22
<.-1(-5;
<.-I(-+(
<./(-1
<./5-11(
<.5<+1(;
<.52<<+2
<.5((<-;
<.5++I-(
<.5---;-
22?
<.15/5;;
<.2<<1(2
<.22;2-/
<.2/+211
<.(<-5-5
<.(-;1;/
<.+(;21-
<.-2<+<-
<.-I-;;I
<./(+(+I
<./52(22
<.5<<(+<
<.515<
<.52;(1/
<.5+<-++
22>
<.1I11-/
<.2<<1
<.22(1;-
<.2-<(+2
<.2I22
<.(1;+2;
<.(51I(;
<.+-<<1+
<.-2/5<2
<.-II2
<./((//;
<.//;I+5
<./;/;;5
<.51+(/1
<.52/<(1
22,
<.1I+115
<.2<<
<.21;15;
<.2+12<5
<.2////-
<.2;/
<.(2;;I1
<.(I+<
<.+/<+5+
<.-(2-+<
<.-;<2<+
<./((<-I
<.//5/+1
<./;+<((
<.512<(+
22(
<.1I/2/5
<.2<<<5/
<.21/(((
<.2(+5;;
<.2--I(1
<.25;I2-
<.(<521(
<.((I+(;
<.(;-(+<
<.+5<+<-
<.-(5;5I
<.-;2<22
<./(2-<-
<.//-//<
<./;1(I;
223
<.1I5;<<
<.2<<
<.21+21-
<.2(<<5;
<.2+5;1I
<.2/I<2/
<.2;<5(+
<.(1/+<1
<.(+I<5;
<.+<-51<
<.+5;5/;
<.-+(
<.-;(++5
<./(2<<+
<.//(I5-
22@
<.1I;1I+
<.2<<<-;
<.212-I<
<.22/+/;
<.2+1;15
<.2-;1+-
<.25I+<2
<.2;;;/I
<.(2+1+/
<.(-5I-5
<.+1-25+
<.+I/+/2
<.-+5I(1
<.-;+51+
<./(1-+-
226
<.1;<22<
<.2<<<-(
<.2112I<
<.22(/22
<.2(5222
<.2-22-<
<.2/II;(
<.2I5(/5
<.(<5;1<
<.((<5I2
<.(///+5
<.+2+1-<
<.+;251(
<.--2(1I
<.-;-I/;
2'2
<.1;1<5-
<.2<<<+;
<.21<22(
<.221(21
<.2((+-5
<.2+/5-5
<.2/1(5<
<.255+-I
<.2;-21<
<.(1+I2;
<.((/-+2
<.(5+/I/
<.+(2+(+
<.+;I-5<
<.--/--2
2''
<.1;15;1
<.2<<<+-
<.2<;(+/
<.21;+2/
<.2(<(5(
<.2+22II
<.2--2I/
<.2/;+;+
<.2I-<-I
<.(<21+1
<.(2<;21
<.(+2<;(
<.(I(I2I
<.++<1;I
<.-<+
2')
<.1;2+<1
<.2<<<+1
<.2
<.215I(I
<.225I<+
<.2(I-I-
<.2-<25(
<.2/2;/I
<.25/5I5
<.2;1I-/
<.(
<.(2/((+
<.(+;;+(
<.(;2(;(
<.++I+-2
2'?
<.1;2;25
<.2<<<(I
<.2<5;5/
<.21/+II
<.22-/(1
<.2(-+5<
<.2+/<55
<.2-5-(/
<.2/;;(-
<.2I((5I
<.2;5;I<
<.(1(I/5
<.((11I<
<.(-51I/
<.+<<++;
2'>
<.1;((I+
<.2<<<(/
<.2<5+(1
<.21-(25
<.22(55<
<.2(2I1+
<.2+2-15
<.2-2;+/
<.2/+15+
<.25/2I-
<.2I;(5<
<.(<(-(1
<.(1III1
<.((--+/
<.(/(;<(
2',
<.1;(5I-
<.2<<<((
<.2
<.21+(1I
<.2221-;
<.2(<-2+
<.2(;+/<
<.2+;<22
<.2-;251
<.25<252
<.2I21<2
<.2;+I++
<.(
<.(2(+(I
<.((;-<-
TABLE !&, DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
22,
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.
<.1(/2<;
<.2+;I((
<.(/I/-5
<.+I2---
<.-//+1/
<./2+((-
<.//+1-;
<./I-I5<
<.5<(-5(
<.51;2++
<.5((2(+
<.5+-I1+
<.5-51;I
<.5/5--/
22)
<.<-;5+;
<.
<.1+I+-(
<.2-55(;
<.(5+I+5
<.+I-;2+
<.-/-+-I
<./2<2+<
<./-I25<
<./I<525
<./;52I2
<.512
<.52-+2+
<.5(5-<;
<.5+I-2<
22?
<.<-/+I-
<.<51-/2
<.<;+-I1
<.1-;+15
<.2/-(25
<.(I<;11
<.+I;
<.-/+/2(
<./1/5(+
<./-(2/-
<./5/+<2
<./;1;(+
<.5<-;+5
<.51I//-
<.5(<2/I
22>
<.<-+I+I
<.
<.<5;;1-
<.1<155I
<.1/;25+
<.252/(I
<.(I/I-;
<.+;2<15
<.-/(I;2
<./1(/;;
<./+I;/2
<./52515
<./I5((-
<.5<
<.5125/1
22,
<.<-(I/;
<.
<.<522;(
<.
<.1<;/1/
<.15I(<-
<.25;5
<.(;2/;;
<.+;+5I-
<.-/(2+(
<./11<-<
<./+-225
<.//;-+1
<./I(((;
<./;-;/1
22(
<.<-(215
<.<-;/5(
<.
<.<55;1-
<.<;11I2
<.115
<.1I//I/
<.2I/-/2
<.(;I++<
<.+;5(;+
<.-/2///
<./
<./+1I5-
<.//-55+
<./5;I(-
223
<.<-25-(
<.<-I1--
<.
<.<52/+<
<.
<.<;-2
<.12++II
<.1;+-(;
<.2;(22(
<.+<+
<.+;;I-5
<.-/21+I
<./
<./(I<1I
<.//1/1<
22@
<.<-2+
<.<-5<+/
<.
<.
<.<5/I55
<.
<.<;I--<
<.1(11--
<.2<1;-(
<.2;;51<
<.+<;/++
<.-<21II
<.-/1/I2
<./<(/;(
<./(+-/<
226
<.<-21(/
<.<-/2<2
<.
<.
<.<52I++
<.
<.
<.1<2;;/
<.1(522+
<.2
<.(
<.+1-11;
<.-<+(;I
<.-/12-;
<./<1(+-
2'2
<.<-1;21
<.<---(I
<.<-;/--
<.
<.
<.<5/2+(
<.
<.<;2I+<
<.1<5+;-
<.1+((++
<.21-51+
<.(1222-
<.+2<-1/
<.-
<.-/
2''
<.<-15+-
<.<--<<2
<.<-I//-
<.
<.
<.<52;55
<.<5;2I+
<.
<.<;-+15
<.111-/-
<.1-<(15
<.2221-1
<.(1I255
<.+2-I(;
<.-
2')
<.<-1-;;
<.<-+-/1
<.<-5I-I
<.
<.
<.<5<+2/
<.<5-I1-
<.
<.
<.<;5/;I
<.11/1-5
<.1-/;55
<.22I(+<
<.(2+2
<.+(1<;2
2'?
<.<-1+5-
<.<-+1;1
<.<-51I;
<.
<.
<.
<.<5(<52
<.<5I+<2
<.
<.<;1-+<
<.1<<<5/
<.12<;1-
<.1/((/(
<.2(+(<-
<.((<<2/
2'>
<.<-1(/;
<.<-(I5/
<.<-//2+
<.<-;/+I
<.
<.
<.<5
<.<5--<5
<.
<.
<.<;(/2-
<.1<(+/(
<.12-+<(
<.1/;-
<.2+<1<+
2',
<.<-1255
<.<-(/<-
<.<-/1+1
<.<-I;1(
<.
<.
<.
<.<5(1+2
<.<555-5
<.
<.
<.<;--21
<.1
<.12;/-2
<.15-+(1
TABLE !&, DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
22,
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.
<.1(/2<;
<.2+;I((
<.(/I/-5
<.+I2---
<.-//+1/
<./2+((-
<.//+1-;
<./I-I5<
<.5<(-5(
<.51;2++
<.5((2(+
<.5+-I1+
<.5-51;I
<.5/5--/
22)
<.<-;5+;
<.
<.1+I+-(
<.2-55(;
<.(5+I+5
<.+I-;2+
<.-/-+-I
<./2<2+<
<./-I25<
<./I<525
<./;52I2
<.512
<.52-+2+
<.5(5-<;
<.5+I-2<
22?
<.<-/+I-
<.<51-/2
<.<;+-I1
<.1-;+15
<.2/-(25
<.(I<;11
<.+I;
<.-/+/2(
<./1/5(+
<./-(2/-
<./5/+<2
<./;1;(+
<.5<-;+5
<.51I//-
<.5(<2/I
22>
<.<-+I+I
<.
<.<5;;1-
<.1<155I
<.1/;25+
<.252/(I
<.(I/I-;
<.+;2<15
<.-/(I;2
<./1(/;;
<./+I;/2
<./52515
<./I5((-
<.5<
<.5125/1
22,
<.<-(I/;
<.
<.<522;(
<.
<.1<;/1/
<.15I(<-
<.25;5
<.(;2/;;
<.+;+5I-
<.-/(2+(
<./11<-<
<./+-225
<.//;-+1
<./I(((;
<./;-;/1
22(
<.<-(215
<.<-;/5(
<.
<.<55;1-
<.<;11I2
<.115
<.1I//I/
<.2I/-/2
<.(;I++<
<.+;5(;+
<.-/2///
<./
<./+1I5-
<.//-55+
<./5;I(-
223
<.<-25-(
<.<-I1--
<.
<.<52/+<
<.
<.<;-2
<.12++II
<.1;+-(;
<.2;(22(
<.+<+
<.+;;I-5
<.-/21+I
<./
<./(I<1I
<.//1/1<
22@
<.<-2+
<.<-5<+/
<.
<.
<.<5/I55
<.
<.<;I--<
<.1(11--
<.2<1;-(
<.2;;51<
<.+<;/++
<.-<21II
<.-/1/I2
<./<(/;(
<./(+-/<
226
<.<-21(/
<.<-/2<2
<.
<.
<.<52I++
<.
<.
<.1<2;;/
<.1(522+
<.2
<.(
<.+1-11;
<.-<+(;I
<.-/12-;
<./<1(+-
2'2
<.<-1;21
<.<---(I
<.<-;/--
<.
<.
<.<5/2+(
<.
<.<;2I+<
<.1<5+;-
<.1+((++
<.21-51+
<.(1222-
<.+2<-1/
<.-
<.-/
2''
<.<-15+-
<.<--<<2
<.<-I//-
<.
<.
<.<52;55
<.<5;2I+
<.
<.<;-+15
<.111-/-
<.1-<(15
<.2221-1
<.(1I255
<.+2-I(;
<.-
2')
<.<-1-;;
<.<-+-/1
<.<-5I-I
<.
<.
<.<5<+2/
<.<5-I1-
<.
<.
<.<;5/;I
<.11/1-5
<.1-/;55
<.22I(+<
<.(2+2
<.+(1<;2
2'?
<.<-1+5-
<.<-+1;1
<.<-51I;
<.
<.
<.
<.<5(<52
<.<5I+<2
<.
<.<;1-+<
<.1<<<5/
<.12<;1-
<.1/((/(
<.2(+(<-
<.((<<2/
2'>
<.<-1(/;
<.<-(I5/
<.<-//2+
<.<-;/+I
<.
<.
<.<5
<.<5--<5
<.
<.
<.<;(/2-
<.1<(+/(
<.12-+<(
<.1/;-
<.2+<1<+
2',
<.<-1255
<.<-(/<-
<.<-/1+1
<.<-I;1(
<.
<.
<.
<.<5(1+2
<.<555-5
<.
<.
<.<;--21
<.1
<.12;/-2
<.15-+(1
TABLE !&( DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1<5/<5
<.1//2;1
<.2-I(+I
<.(5((1;
<.+I-<25
<.-/5/I5
<./2-1/+
<.//+52/
<./I/1I-
<.5<(I-/
<.51;-<<
<.5((+/I
<.5+/<2;
<.5-5(;5
<.5/55+1
22)
<.1<+--5
<.1(I<(1
<.1I//1-
<.252+1I
<.(I(5<-
<.+;
<.-/5I22
<./215I5
<./-;(25
<./I1(
<./;5I1<
<.512-51
<.52-I/5
<.5(5;1;
<.5+I;<<
22?
<.1<(2++
<.12-;
<.1-/;;I
<.1;;I+-
<.2I+I+<
<.(;(-I<
<.+;-5/I
<.-/5;(;
<./1I;<2
<./-+5+5
<./5521(
<./;2/5/
<.5
<.51;2;+
<.5(
22>
<.1<2-15
<.11;+;I
<.1+1--;
<.15<-++
<.21(++1
<.2;/;<5
<.+<(<12
<.-<<-+5
<.-/I<+2
<./1/+12
<./-
<./5(525
<./II2/+
<.5<1+I1
<.51(--;
22,
<.1<2<--
<.11--;<
<.1(2+--
<.1-(5+;
<.1I<;<1
<.22+5-/
<.(
<.+12<-/
<.-<+;;+
<.-/I1(2
<./1+2+2
<./+5+11
<./5<52+
<./I++((
<./;/;5;
22(
<.1<15(5
<.112;51
<.12/-+<
<.1+(12-
<.1/(/15
<.1I;1/(
<.2(/2II
<.(1II(+
<.+2<5-+
<.-<;1-2
<.-/I21(
<./12((-
<./+++(1
<.//I112
<./I1<5I
223
<.1<1-<(
<.1111<<
<.122+15
<.1(-I;5
<.1-2115
<.151I2<
<.1;-;+;
<.2+/II/
<.(2I;25
<.+2;1+-
<.-1(<+;
<.-/I2I+
<./1
<./+1/52
<.//+1(5
22@
<.1<1(2-
<.1<;/;I
<.11;(II
<.1(
<.1++<1(
<.1-;I(<
<.15I55I
<.2<(--/
<.2-/-5-
<.((I-+;
<.+(52-;
<.-1/51+
<.-/I(+;
<./<;1+1
<./(I-+;
226
<.1<11I-
<.1
<.115<52
<.12/5I;
<.1(I<(1
<.1-11(5
<.1//-(I
<.1I+551
<.2115;<
<.2/--1(
<.(+55/<
<.++-121
<.-2<15<
<.-/I+
<./<55;1
2'2
<.1<1<51
<.1<55(I
<.11-2+/
<.12(5-2
<.1((+-1
<.1++-I-
<.1-5+--
<.152+(5
<.1;<<<<
<.21;2(2
<.25(I1I
<.(-//<;
<.+-25-+
<.-2(+(/
<.-/I+/1
2''
<.1<<;5I
<.1<5<25
<.11(55<
<.121(25
<.12;I(;
<.1(;+I/
<.1-<+I-
<.1/(1<+
<.155/5/
<.1;+/<;
<.22/<51
<.2I1-I+
<.(/-1(-
<.+-II<<
<.-2/-(<
2')
<.1<
<.1
<.112--+
<.11;(+/
<.12/;22
<.1(-+1-
<.1++;I-
<.1--I(1
<.1/I1;2
<.1I2(/+
<.1;I5
<.2(+1/1
<.2;<-1(
<.(5((51
<.+/((2+
2'?
<.1<
<.1<-;(-
<.111-(+
<.1155<<
<.12+-1;
<.1(2<;2
<.1+<-+(
<.1-<<2<
<.1/<5<1
<.152I<2
<.1I/-II
<.2<-122
<.2+155/
<.2;;2-2
<.(I1(++
2'>
<.1<<55-
<.1<--<5
<.11
<.11/(1<
<.122-
<.12;((2
<.1(/II/
<.1+-2I1
<.1-+/-<
<.1/-1-;
<.155<<2
<.1;<+15
<.211<(5
<.2+I;55
<.(<5/I<
2',
<.1<<52+
<.1<-1(5
<.1<;;2<
<.11-121
<.12<5;-
<.125<<-
<.1((I2/
<.1+1(+/
<.1+;/51
<.1-I;2/
<.1/;2-I
<.1I
<.1;(;<5
<.21/--I
<.2--I<;
TABLE !&( DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
2'2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1<5/<5
<.1//2;1
<.2-I(+I
<.(5((1;
<.+I-<25
<.-/5/I5
<./2-1/+
<.//+52/
<./I/1I-
<.5<(I-/
<.51;-<<
<.5((+/I
<.5+/<2;
<.5-5(;5
<.5/55+1
22)
<.1<+--5
<.1(I<(1
<.1I//1-
<.252+1I
<.(I(5<-
<.+;
<.-/5I22
<./215I5
<./-;(25
<./I1(
<./;5I1<
<.512-51
<.52-I/5
<.5(5;1;
<.5+I;<<
22?
<.1<(2++
<.12-;
<.1-/;;I
<.1;;I+-
<.2I+I+<
<.(;(-I<
<.+;-5/I
<.-/5;(;
<./1I;<2
<./-+5+5
<./5521(
<./;2/5/
<.5
<.51;2;+
<.5(
22>
<.1<2-15
<.11;+;I
<.1+1--;
<.15<-++
<.21(++1
<.2;/;<5
<.+<(<12
<.-<<-+5
<.-/I<+2
<./1/+12
<./-
<./5(525
<./II2/+
<.5<1+I1
<.51(--;
22,
<.1<2<--
<.11--;<
<.1(2+--
<.1-(5+;
<.1I<;<1
<.22+5-/
<.(
<.+12<-/
<.-<+;;+
<.-/I1(2
<./1+2+2
<./+5+11
<./5<52+
<./I++((
<./;/;5;
22(
<.1<15(5
<.112;51
<.12/-+<
<.1+(12-
<.1/(/15
<.1I;1/(
<.2(/2II
<.(1II(+
<.+2<5-+
<.-<;1-2
<.-/I21(
<./12((-
<./+++(1
<.//I112
<./I1<5I
223
<.1<1-<(
<.1111<<
<.122+15
<.1(-I;5
<.1-2115
<.151I2<
<.1;-;+;
<.2+/II/
<.(2I;25
<.+2;1+-
<.-1(<+;
<.-/I2I+
<./1
<./+1/52
<.//+1(5
22@
<.1<1(2-
<.1<;/;I
<.11;(II
<.1(
<.1++<1(
<.1-;I(<
<.15I55I
<.2<(--/
<.2-/-5-
<.((I-+;
<.+(52-;
<.-1/51+
<.-/I(+;
<./<;1+1
<./(I-+;
226
<.1<11I-
<.1
<.115<52
<.12/5I;
<.1(I<(1
<.1-11(5
<.1//-(I
<.1I+551
<.2115;<
<.2/--1(
<.(+55/<
<.++-121
<.-2<15<
<.-/I+
<./<55;1
2'2
<.1<1<51
<.1<55(I
<.11-2+/
<.12(5-2
<.1((+-1
<.1++-I-
<.1-5+--
<.152+(5
<.1;<<<<
<.21;2(2
<.25(I1I
<.(-//<;
<.+-25-+
<.-2(+(/
<.-/I+/1
2''
<.1<<;5I
<.1<5<25
<.11(55<
<.121(25
<.12;I(;
<.1(;+I/
<.1-<+I-
<.1/(1<+
<.155/5/
<.1;+/<;
<.22/<51
<.2I1-I+
<.(/-1(-
<.+-II<<
<.-2/-(<
2')
<.1<
<.1
<.112--+
<.11;(+/
<.12/;22
<.1(-+1-
<.1++;I-
<.1--I(1
<.1/I1;2
<.1I2(/+
<.1;I5
<.2(+1/1
<.2;<-1(
<.(5((51
<.+/((2+
2'?
<.1<
<.1<-;(-
<.111-(+
<.1155<<
<.12+-1;
<.1(2<;2
<.1+<-+(
<.1-<<2<
<.1/<5<1
<.152I<2
<.1I/-II
<.2<-122
<.2+155/
<.2;;2-2
<.(I1(++
2'>
<.1<<55-
<.1<--<5
<.11
<.11/(1<
<.122-
<.12;((2
<.1(/II/
<.1+-2I1
<.1-+/-<
<.1/-1-;
<.155<<2
<.1;<+15
<.211<(5
<.2+I;55
<.(<5/I<
2',
<.1<<52+
<.1<-1(5
<.1<;;2<
<.11-121
<.12<5;-
<.125<<-
<.1((I2/
<.1+1(+/
<.1+;/51
<.1-I;2/
<.1/;2-I
<.1I
<.1;(;<5
<.21/--I
<.2--I<;
TABLE !&3 DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1(5I5-
<.1;/II-
<.25-;-I
<.(5;/5<
<.+II++5
<.-/;-2+
<./2/+
<.//--;-
<./I//5+
<.5<+(<2
<.51;;1<
<.5((I-+
<.5+/(;-
<.5-55++
<.5/I<52
22)
<.1+215/
<.15;512
<.2(
<.2;/<;+
<.(;+;(<
<.+;5<2+
<.-51222
<./2+
<.//<;(5
<./I2211
<./;I/(I
<.51(((-
<.52/-I-
<.5(I/<1
<.5+;--<
22?
<.1++2<5
<.151+-/
<.2<5<+5
<.2-25/+
<.(1(I5/
<.+
<.-<+I2/
<.-52/I+
<./221<;
<./-/;;-
<./5I+//
<./;(I((
<.5<55<+
<.52<(<<
<.5(1I12
22>
<.1+-(;5
<.1//521
<.1;(-/2
<.225152
<.2/;
<.(2;1
<.+211<1
<.-11;55
<.-5(;-I
<./2<+
<./-(/1;
<./5-2I1
<./I;5
<.5<2I(<
<.51+I2(
22,
<.1+/1I<
<.1/(/5/
<.1I-<+5
<.211<;/
<.2+2;-(
<.2I1I+/
<.(+252;
<.+((+(I
<.-1I-51
<.-5-<55
<./1I;2;
<./-
<./52-+<
<./I/1(1
<./;I-51
22(
<.1+/5(-
<.1/1-/1
<.15;2+5
<.2<<(
<.22--21
<.2--I15
<.2;22<5
<.(-/2
<.++-(/1
<.-2+/I+
<.-5/<5<
<./15/(1
<./+I1+(
<./5<1-I
<./I2;;5
223
<.1+51+I
<.1/<<
<.15-
<.1;2/+-
<.21(2;I
<.2(5/II
<.2//-I(
<.(<2(1I
<.(/I-I<
<.+-//(I
<.-2;I<;
<.-5/;--
<./1/+I+
<./+-I;+
<.//I
22@
<.1+5+/;
<.1-II21
<.151;12
<.1I/;-5
<.2<+(+-
<.22+--(
<.2+I1++
<.25-55/
<.(1(5//
<.(I<<+(
<.+/+(
<.-((2+;
<.-555-1
<./1-+/(
<./+(I;;
226
<.1+552-
<.1-5II-
<.1/;+-5
<.1I2-I1
<.1;5-+(
<.21+/I5
<.2(++(<
<.2-52/<
<.2I(5++
<.(2+11(
<.(;<5(5
<.+51+((
<.-(/(/;
<.-5I+/;
<./1+-+5
2'2
<.1+5;((
<.1-5125
<.1/5+;2
<.15;115
<.1;221/
<.2<5<+5
<.22(;15
<.2+(1;2
<.2/-2;;
<.2;<5(-
<.(((-+I
<.+<<552
<.+5I
<.-(;21/
<.-5;122
2''
<.1+I1<5
<.1-/-<2
<.1/-II-
<.15/(1<
<.1I5;+2
<.2<<;5/
<.21-/++
<.2(2222
<.2-1<((
<.252+-/
<.2;/;(<
<.(+(<-2
<.+1<2(2
<.+I+(22
<.-+1I22
2')
<.1+I2-+
<.1--;5I
<.1/+-+/
<.15(;;1
<.1I+++1
<.1;/<+/
<.2
<.22(+/;
<.2(;5+-
<.2-I1<(
<.25III1
<.(<-((+
<.(-(1<1
<.+1;1I5
<.+;<1I+
2'?
<.1+I(5;
<.1---(1
<.1/(+1+
<.152<++
<.1I1-2(
<.1;1;/;
<.2<(-21
<.21/(++
<.2(
<.2+//<<
<.2/+-1I
<.2I+/I/
<.(1(512
<.(/2/
<.+2I--2
2'>
<.1+I+II
<.1--1+5
<.1/2++-
<.15<(II
<.15;<--
<.1II-++
<.1;I;/(
<.21<+++
<.22(1(-
<.2(5212
<.2-2I5;
<.25<(51
<.2I;;/(
<.(21-2I
<.(51/(5
2',
<.1+I-I(
<.1-+I12
<.1/1/<5
<.1/I;/1
<.15/;+2
<.1I-/2;
<.1;-1<;
<.2<-+I-
<.21/I5I
<.22;+25
<.2+(2;1
<.2-I/-/
<.25-5(I
<.2;+5I-
<.(2II+;
TABLE !&3 DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
2',
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1(5I5-
<.1;/II-
<.25-;-I
<.(5;/5<
<.+II++5
<.-/;-2+
<./2/+
<.//--;-
<./I//5+
<.5<+(<2
<.51;;1<
<.5((I-+
<.5+/(;-
<.5-55++
<.5/I<52
22)
<.1+215/
<.15;512
<.2(
<.2;/<;+
<.(;+;(<
<.+;5<2+
<.-51222
<./2+
<.//<;(5
<./I2211
<./;I/(I
<.51(((-
<.52/-I-
<.5(I/<1
<.5+;--<
22?
<.1++2<5
<.151+-/
<.2<5<+5
<.2-25/+
<.(1(I5/
<.+
<.-<+I2/
<.-52/I+
<./221<;
<./-/;;-
<./5I+//
<./;(I((
<.5<55<+
<.52<(<<
<.5(1I12
22>
<.1+-(;5
<.1//521
<.1;(-/2
<.225152
<.2/;
<.(2;1
<.+211<1
<.-11;55
<.-5(;-I
<./2<+
<./-(/1;
<./5-2I1
<./I;5
<.5<2I(<
<.51+I2(
22,
<.1+/1I<
<.1/(/5/
<.1I-<+5
<.211<;/
<.2+2;-(
<.2I1I+/
<.(+252;
<.+((+(I
<.-1I-51
<.-5-<55
<./1I;2;
<./-
<./52-+<
<./I/1(1
<./;I-51
22(
<.1+/5(-
<.1/1-/1
<.15;2+5
<.2<<(
<.22--21
<.2--I15
<.2;22<5
<.(-/2
<.++-(/1
<.-2+/I+
<.-5/<5<
<./15/(1
<./+I1+(
<./5<1-I
<./I2;;5
223
<.1+51+I
<.1/<<
<.15-
<.1;2/+-
<.21(2;I
<.2(5/II
<.2//-I(
<.(<2(1I
<.(/I-I<
<.+-//(I
<.-2;I<;
<.-5/;--
<./1/+I+
<./+-I;+
<.//I
22@
<.1+5+/;
<.1-II21
<.151;12
<.1I/;-5
<.2<+(+-
<.22+--(
<.2+I1++
<.25-55/
<.(1(5//
<.(I<<+(
<.+/+(
<.-((2+;
<.-555-1
<./1-+/(
<./+(I;;
226
<.1+552-
<.1-5II-
<.1/;+-5
<.1I2-I1
<.1;5-+(
<.21+/I5
<.2(++(<
<.2-52/<
<.2I(5++
<.(2+11(
<.(;<5(5
<.+51+((
<.-(/(/;
<.-5I+/;
<./1+-+5
2'2
<.1+5;((
<.1-5125
<.1/5+;2
<.15;115
<.1;221/
<.2<5<+5
<.22(;15
<.2+(1;2
<.2/-2;;
<.2;<5(-
<.(((-+I
<.+<<552
<.+5I
<.-(;21/
<.-5;122
2''
<.1+I1<5
<.1-/-<2
<.1/-II-
<.15/(1<
<.1I5;+2
<.2<<;5/
<.21-/++
<.2(2222
<.2-1<((
<.252+-/
<.2;/;(<
<.(+(<-2
<.+1<2(2
<.+I+(22
<.-+1I22
2')
<.1+I2-+
<.1--;5I
<.1/+-+/
<.15(;;1
<.1I+++1
<.1;/<+/
<.2
<.22(+/;
<.2(;5+-
<.2-I1<(
<.25III1
<.(<-((+
<.(-(1<1
<.+1;1I5
<.+;<1I+
2'?
<.1+I(5;
<.1---(1
<.1/(+1+
<.152<++
<.1I1-2(
<.1;1;/;
<.2<(-21
<.21/(++
<.2(
<.2+//<<
<.2/+-1I
<.2I+/I/
<.(1(512
<.(/2/
<.+2I--2
2'>
<.1+I+II
<.1--1+5
<.1/2++-
<.15<(II
<.15;<--
<.1II-++
<.1;I;/(
<.21<+++
<.22(1(-
<.2(5212
<.2-2I5;
<.25<(51
<.2I;;/(
<.(21-2I
<.(51/(5
2',
<.1+I-I(
<.1-+I12
<.1/1/<5
<.1/I;/1
<.15/;+2
<.1I-/2;
<.1;-1<;
<.2<-+I-
<.21/I5I
<.22;+25
<.2+(2;1
<.2-I/-/
<.25-5(I
<.2;+5I-
<.(2II+;
TABLE !&@ DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1/(55;
<.222/<;
<.2;;/;-
<.(;
<.+;2;--
<.-51--/
<./25II+
<.////I;
<./I5(12
<.5<+I;5
<.52<+/I
<.5(+(5(
<.5+/I5;
<.5-I1;I
<.5/I+;;
22)
<.15-(55
<.21-21(
<.2/5--+
<.((2-I+
<.+12<+/
<.-<-<;(
<.-5+;5<
<./2/I<5
<.//2;+I
<./I((5-
<./;;5(2
<.51+(/;
<.525--/
<.5(;-12
<.5-<+<;
22?
<.1I1222
<.211+<;
<.2+;;I-
<.2;552-
<.(--/((
<.+(
<.-1-5I1
<.-55;
<./2-I;2
<./-;5I-
<./I<<5(
<./;-(-(
<.5<;1+-
<.521//;
<.5((1<1
22>
<.1I+5;1
<.2<;112
<.2(;(+I
<.25/
<.(2<(I/
<.(5(2(1
<.++//-;
<.-2-(1+
<.-I<+/1
<./2-1<(
<./-5<5-
<./552//
<./;1-;2
<.5<+/2(
<.51/-(/
22,
<.1I52
<.2<5-I<
<.2(2((2
<.2/15I1
<.2;/I(+
<.((I+12
<.(I5((<
<.+/
<.-(2(I<
<.-I25<+
<./2++1I
<./-+5(1
<./5+I+;
<./II((<
<.5<
22(
<.1II;-5
<.2
<.225(;(
<.2-1I1I
<.2I<+(/
<.(1(;;5
<.(-(252
<.(;;/;1
<.+51/I5
<.-(55--
<.-I+/;(
<./2(I15
<./-2/I2
<./525+I
<./I-+5+
223
<.1;<2I(
<.2<-//I
<.22(5+2
<.2++-(1
<.2/I-(/
<.2;/(25
<.(2I-(I
<.(/-I(1
<.+1(;/;
<.+I1//<
<.-+2-2+
<.-I/+/5
<./2(2I-
<./-
<./5<;<-
22@
<.1;1(2(
<.2<-<((
<.22<;(I
<.2(I;;/
<.2-;-5;
<.2I(11I
<.(1<1<5
<.(+1
<.(5//+2
<.+2/II(
<.+;<5+;
<.-+/5I5
<.-II
<./22I12
<./+;25/
226
<.1;21/1
<.2<+-2-
<.21I521
<.2(+//1
<.2-2/25
<.252;+/
<.2;/<<1
<.(2221;
<.(-2<5-
<.(I/<5;
<.+(I/51
<.+;;
<.--
<.-I;-<(
<./22(II
2'2
<.1;2I-2
<.2<+111
<.21/;2-
<.2(115;
<.2+5<;<
<.2/+;11
<.2I+;(-
<.(<5-<<
<.((2;I2
<.(/1I<2
<.(;++1(
<.++;-12
<.-
<.--+1<-
<.-;
2''
<.1;(+(<
<.2<(5/+
<.21-++<
<.22I(2+
<.2+2-I-
<.2-I+22
<.25/
<.2;-55(
<.(15I-<
<.(+2/((
<.(5<+;/
<.+<+(-+
<.+-;I/-
<.-1(;15
<.--5252
2')
<.1;(;21
<.2<(+5(
<.21+1;+
<.22-;+2
<.2(II-(
<.2-(
<.2/II12
<.2I/2-<
<.(<-/(+
<.(252(+
<.(-1(-2
<.(5I(2-
<.+1+;-;
<.+/I<;5
<.-2<--;
2'?
<.1;+(++
<.2<(22(
<.21(1((
<.22(;25
<.2(-51(
<.2+I/1;
<.2/25I5
<.25I(I/
<.2;-/<-
<.(1+/-;
<.((-5;2
<.(-;2I<
<.(I-+2+
<.+2+I-(
<.+5-5/+
2'>
<.1;+512
<.2<(<<5
<.21221;
<.2222<<
<.2((<(5
<.2++I(-
<.2-551<
<.2515;-
<.2I52+/
<.(<+2(+
<.(22;-;
<.(+(/+<
<.(//-2I
<.(;1I;5
<.+(+122
2',
<.1;-<(+
<.2<2I15
<.211+2+
<.22<5<(
<.2(<5(<
<.2+1-I;
<.2-((5/
<.2//2<<
<.2I<1I/
<.2;-+5-
<.(1222I
<.((
<.(-
<.(5(1I5
<.(;I1I1
TABLE !&@ DE9T OF NEUTRAL AIS FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTIONS% k
Gr*+e o; steel
Fe&,22
d d
2)2
pt f ck
pc f ck
22'
22)
22?
22>
22,
22(
223
22@
226
2'2
2''
2')
2'?
2'>
2',
22'
<.1/(55;
<.222/<;
<.2;;/;-
<.(;
<.+;2;--
<.-51--/
<./25II+
<.////I;
<./I5(12
<.5<+I;5
<.52<+/I
<.5(+(5(
<.5+/I5;
<.5-I1;I
<.5/I+;;
22)
<.15-(55
<.21-21(
<.2/5--+
<.((2-I+
<.+12<+/
<.-<-<;(
<.-5+;5<
<./2/I<5
<.//2;+I
<./I((5-
<./;;5(2
<.51+(/;
<.525--/
<.5(;-12
<.5-<+<;
22?
<.1I1222
<.211+<;
<.2+;;I-
<.2;552-
<.(--/((
<.+(
<.-1-5I1
<.-55;
<./2-I;2
<./-;5I-
<./I<<5(
<./;-(-(
<.5<;1+-
<.521//;
<.5((1<1
22>
<.1I+5;1
<.2<;112
<.2(;(+I
<.25/
<.(2<(I/
<.(5(2(1
<.++//-;
<.-2-(1+
<.-I<+/1
<./2-1<(
<./-5<5-
<./552//
<./;1-;2
<.5<+/2(
<.51/-(/
22,
<.1I52
<.2<5-I<
<.2(2((2
<.2/15I1
<.2;/I(+
<.((I+12
<.(I5((<
<.+/
<.-(2(I<
<.-I25<+
<./2++1I
<./-+5(1
<./5+I+;
<./II((<
<.5<
22(
<.1II;-5
<.2
<.225(;(
<.2-1I1I
<.2I<+(/
<.(1(;;5
<.(-(252
<.(;;/;1
<.+51/I5
<.-(55--
<.-I+/;(
<./2(I15
<./-2/I2
<./525+I
<./I-+5+
223
<.1;<2I(
<.2<-//I
<.22(5+2
<.2++-(1
<.2/I-(/
<.2;/(25
<.(2I-(I
<.(/-I(1
<.+1(;/;
<.+I1//<
<.-+2-2+
<.-I/+/5
<./2(2I-
<./-
<./5<;<-
22@
<.1;1(2(
<.2<-<((
<.22<;(I
<.2(I;;/
<.2-;-5;
<.2I(11I
<.(1<1<5
<.(+1
<.(5//+2
<.+2/II(
<.+;<5+;
<.-+/5I5
<.-II
<./22I12
<./+;25/
226
<.1;21/1
<.2<+-2-
<.21I521
<.2(+//1
<.2-2/25
<.252;+/
<.2;/<<1
<.(2221;
<.(-2<5-
<.(I/<5;
<.+(I/51
<.+;;
<.--
<.-I;-<(
<./22(II
2'2
<.1;2I-2
<.2<+111
<.21/;2-
<.2(115;
<.2+5<;<
<.2/+;11
<.2I+;(-
<.(<5-<<
<.((2;I2
<.(/1I<2
<.(;++1(
<.++;-12
<.-
<.--+1<-
<.-;
2''
<.1;(+(<
<.2<(5/+
<.21-++<
<.22I(2+
<.2+2-I-
<.2-I+22
<.25/
<.2;-55(
<.(15I-<
<.(+2/((
<.(5<+;/
<.+<+(-+
<.+-;I/-
<.-1(;15
<.--5252
2')
<.1;(;21
<.2<(+5(
<.21+1;+
<.22-;+2
<.2(II-(
<.2-(
<.2/II12
<.2I/2-<
<.(<-/(+
<.(252(+
<.(-1(-2
<.(5I(2-
<.+1+;-;
<.+/I<;5
<.-2<--;
2'?
<.1;+(++
<.2<(22(
<.21(1((
<.22(;25
<.2(-51(
<.2+I/1;
<.2/25I5
<.25I(I/
<.2;-/<-
<.(1+/-;
<.((-5;2
<.(-;2I<
<.(I-+2+
<.+2+I-(
<.+5-5/+
2'>
<.1;+512
<.2<(<<5
<.21221;
<.2222<<
<.2((<(5
<.2++I(-
<.2-551<
<.2515;-
<.2I52+/
<.(<+2(+
<.(22;-;
<.(+(/+<
<.(//-2I
<.(;1I;5
<.+(+122
2',
<.1;-<(+
<.2<2I15
<.211+2+
<.22<5<(
<.2(<5(<
<.2+1-I;
<.2-((5/
<.2//2<<
<.2I<1I/
<.2;-+5-
<.(1222I
<.((
<.(-
<.(5(1I5
<.(;I1I1
TY9ICAL CALCULATION OF MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTION E<*0ple '4 Calc&lation of moment of resistance factor for a do&bl reinforced rectan!&lar section DATA4 f +1- 67S: mm, pt7f c$ <.<-, pc7f c$ <.
Sol#tion4 k x &7d <.1(2--I, obtained b iterati3e com%&tations FRefer *ABLE @2G
Strain in Com%ression steel <.<<
TY9ICAL CALCULATION OF MOMENT OF RESISTANCE FACTORS FOR DOUBLY REINFORCED RECTANGULAR SECTION E<*0ple '4 Calc&lation of moment of resistance factor for a do&bl reinforced rectan!&lar section DATA4 f +1- 67S: mm, pt7f c$ <.<-, pc7f c$ <.
Sol#tion4 k x &7d <.1(2--I, obtained b iterati3e com%&tations FRefer *ABLE @2G
Strain in Com%ression steel <.<<
<.(/ 4 <.1(2--I <.
<.<- 4 (/<.I;;;;+ 7 1<<
<.1I<+-<
0al&e of C 7f c$bd and T 7f c$bd are almost e:&al and #ence force e:&ilibri&m is ac#ie3ed M &7f c$bd 2 <.(/ K k F1 @ <.+2 K k G pc 7 f c$ Ff sc @ f ccGF1 @ d 7d G71<<
<.(/ K <.1(2--IF1 @ <.+2 K <.1(2--IG <.
Sol#tion4 k x &7d <.1+/-1(, obtained b iterati3e com%&tations FRefer *ABLE @(G
6e&tral A4is lies "it#in t#e co3erin! concrete F k Hd 7d G Strain in Com%ression steel F*o% reinforcementG <.<<<
C @@@@@@@@@ <.(/ 4 k pc7f c$ Ff sc @ f ccG 7 1<< f c$ bd
<.(/ 4 <.1+/-1( <.1< 4 F@1/./-;<(- @ <.<<<<<
<.<1< 4 (/<.I;;;;+ 7 1<< <.<(/<;< 0al&e of C 7f c$bd and T 7f c$bd are almost e:&al and #ence force e:&ilibri&m is ac#ie3ed
M &7f c$bd 2 <.(/ K k F1 @ <.+2 K k G pc 7 f c$ Ff sc @ f ccGF1 @ d 7d G71<<
<.(/ K <.1+/-1(F1 @ <.+2 K <.1+/-1(G <.1
ILLUSTRATIE EAM9LES I$ An*l8sis o; * 1i.en +o#/l8 rein;orce+ concrete section 4 E<*0ple ' 4 & Calc&late t#e moment ca%acit of a do&bl reinforced rectan!&lar beam section (<< mm "ide 4 /<< mm dee% b Limit State Met#od. *#e beam section is %ro3ided "it# 2 6os 1/ mm bars on t#e com%ression side and - 6os 1/ mm bars on t#e tension side. Ado%t M2< concrete and 'e +1- Steel. Concrete co3er to t#e centre of reinforcement d +< mm 300 mm 2 Nos. 16mm Bars
600 mm
5 Nos. 16mm Bars
Sol&tion = Effecti3e de%t# d /<< @ +< -/< mm, b (<< mm d 7d +< 7 -/< <.<5, t#e ne4t #i!#er 3al&e of <.1< "ill be ado%ted. pt - 4 2<1 41<< 7 F(<< 4 -/
pt7f c$ <.-;I 7 2< <.<2;;, Sa <.<(
pc 2 4 2<1 41<< 7 F(<< 4 -/
'or t#e !i3en !rade of steel, considerin! d 7d <.1<, *ABLE R@2 to be reffered. Referrin! to col&mn corres%ondin! to pt7f c$ <.<(, B Linear inter%olation, M &7f c$bd 2 <.<;I2 F<.<;;( @ <.<;I2G 7 F <.<2 @ <.<1 G 4 F <.<11;- @ <.<1G
<.<;I+ Moment Ca%acit of t#e section M & <.<;I+ 4 2< 4 (<< 4 -/< 2 7 1
E<*0ple ) 4 Rectan!&lar section of a Recreinforced concrete beam is s&b9ected to a factored #o!!in! moment of 1<< $6m and a factored sa!!in! moment of -< $6m &nder t"o different critical load combinations. *#e beam is 2(< mm "ide 4 +-< mm dee%. Ass&min! a concrete co3er of +< mm to t#e centre of reinforcement, determine t#e main reinforcement re:&ired for t#e beam. Sol#tion4 b 2(< mm, d +-< @ +< +1< mm, d +< mm d 7 d +< 7 +1< <.<;I, 6e4t #i!#er 3al&e of <.1< "ill be considered.
Step ' 4 'irst consider t#e smaller 3al&e of t #e t"o !i3en moment 3al&es, i.e -< $6m FSa!!in!G
Pa!e 2;
'actored sa!!in! moment , M &s -< $6m, M &s 7 f c$bd 2 -< 4 1< /7 F2< 4 2(< 4 +1< 2G <.
'or 'e@+1- steel and d 7d <.1<, *ABLE R@2 is rele3ant to be referred. Referrin! to *ABLE R@2, moment of resistance factors &nder t#e col&mns corres%ondin! to pt7f c$ <.<2 are close to t#e re:&ired 3al&e of M &7f c$bd 2 of <.
Step ) 4 Consider t#e #i!#er 3al&e of t#e t"o !i3en moment 3al&es, i.e 1<< $6m F8o!!in!G M &# 7 f c$bd 2 1<< 4 1< /7 F2< 4 2(< 4 +1< 2G <.12;(
Under #o!!in! moment, bottom reinforcement "ill be in com%ression, 8ence pc7f c$ pbot7f c$ <.<2 as obtained from Ste% 1. Referrin! to t#e rele3ant *ABLE R@2, from t#e ro" corres%ondin! to pc7f c$ <.<2, M &7f c$bd 2 3al&e corres%ondin! to pt7f c$ <.<+ is <.1(<+ "#ic# is more t#an t#e re:&ired 3al&e of <.12;(. 8ence O. Under #o!!in! moment, to% reinforcement "ill be in tension, 8ence pto%7f c$ pt7f c$ <.<+.
Step ?4 Calc&lation of area of steel. Percenta!e reinforcement at to%, pto% <.<+ 4 2< <.I Area of steel at to% <.I 4 2(< 4 +1< 7 1<< 5-+.+< mm 2 Pro3ide + 6os 1/ mm Bars Percenta!e reinforcement at bottom, pbot <.<2 4 2< <.+ Area of steel at bottom <.+ 4 2(< 4 +1< 7 1<< (55.2
450 mm
2 Nos. 16mm Bars
RE6'ORCEME6* DE*ALS
Pa!e (<
*#is >or$boo$ "as firs
Sl No
D*te o; Re.ision
Re.ision No
