DIAGRAMA MOMEN MOMENTO TO CURVATURA CURVATURA SECCIONES SIMPLEMENTE REFORZADAS SECCION: Peralte F'c= F!= Es= #=
2 10 . 0 0 "200.00 20 00 00 0. 0 0 $0. 0 0
%$=
0 . &( &( 2
c 0. 00 01 0. 00 02 0. 00 0$ 0. 00 0" 0. 00 05 0. 00 0 0. 00 0( 0. 00 0 0. 00 0& 0. 00 10 0.0011 0. 00 12 0. 00 1$ 0. 00 1" 0. 00 15 0. 00 1 0. 00 1( 0. 00 1 0. 00 1& 0. 00 20 0. 00 21 0. 00 22 0. 00 2$ 0. 00 2" 0. 00 25 0. 00 2 0. 00 2( 0. 00 2 0. 00 2& 0. 00 $0 0. 00 $1 0. 00 $2 0. 00 $$ 0. 00 $" 0. 00 $5 0. 00 $ 0. 00 $( 0. 00 $ 0. 00 $& 0. 00 "0
%1 0. 0 " &1 ( 0. 0 & ( 0 0. 1 " 250 0 0. 1 ( 0. 2 2 &20 0 0. 2 ( 000 0 0. $ 0 &20 0 0. $ " ( 0. $ 250 0 0. " 1 ( 0. " " &20 0 0. " 000 0 0. 5 0 &1 ( 0. 5 $ ( 0. 5 250 0 0. 5 ( 0. 0 && ( 0. $ 000 0 0. " &1 ( 0. ( 0. 2$" 0 0. & 21 2 0. ( 0 50 0 0. ( 1 &"" " 0. ( 2 &20 0 0. ( $ (( 2 0. ( " 50 0 0. ( 5 2$ 1 0. ( 5 (0 0 0. ( $ & 0. ( 50 0 0. ( ( 2&1 ( 0. ( ( 0 0 0. ( ( &&0 2 0. ( 20 0 0. ( 51 5 0. ( ("0 0 0. ( &0$ 5 0. ( & 0&0 0 0. ( & 1 (
kg/cm2 kg k g/cm2 kg/cm2 cm
%2 0 . $$" ("5 0 . $$ 20( 0 . $$( (00 0 . $$& 2 0 . $"0 &00 0 . $"2 5&$ 0 . $"" $00 0 . $" 15" 0 . $" 000 0 . $50 000 0 . $52 000 0 . $5" 1( 0 . $5 $$ 0 . $5 & 0.$1110 0 . $$ $ 0 . $ 20 0 . $& 0" 0 . $(1 &51 0 . $(5 000 0 . $( 2&0 0 . $1 05 0 . $5 "00 0 . $& 10$ 0 . $&2 00 0 . $& $( 0 . "00 000 0 . "0$ "0 0 . "0 (00 0 . "10 100 0 . "1$ $00 0 . "1 ""2 0 . "1& 500 0 . "22 "$( 0 . "25 $00 0."2110 0 . "$0 00 0 . "$$ " 0 . "$ 100 0 . "$ 5&
d1=
55. 0 0 c m
C 20 . "5 20 . 5& 20 . (2 20 . ( 21 . 01 21 . 1 21 . $1 21 . " 21 . 2 21 . ( 21 . &5 22 . 12 22 . 2& 22 . "( 22 . 22 . 5 2$ . 0$ 2$ . 2" 2$ . "5 2$ . 2$ . 2".11 2" . $" 2" . 5 2" . 2 25 . 0 25 . $0 25 . 5" 25 . ( 2 . 0$ 2 . 2( 2 . 50 2 . (" 2 . &( 2( . 20 2( . "$ 2( . 2( . 2 . 10 2 . $2
Fs1 $ $ & & &2 1 $0 & 1 1 1 &1 & 2 21 " 2 50 0 2 (( & $ 05 0 $ $1 $ $ 5 ( $ 1 " " 05 2 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0 " 20 0
As1=
C 2 0 . "5 2 0 . 5& 2 0 . (2 20 . ( 21 .0 1 21 .1 21 .$ 1 21 ." 21 . 2 21 .( 21 .& 5 22 .1 2 22 .2 & 22 ." ( 22 .2 2 21 .$ 1 20 ." & 1& . " 1& .2 1 .( 5 1 .$ 2 1( .& 5 1( . " 1( .$ 1( .1 " 1 .& " 1 .( ( 1 . 1 1 ." 1 .$ 1 .2 ( 1 .1 ( 1 .1 0 1 .0 $ 15 .& ( 15 .& 2 15 . 15 . " 15 . 1 15 .( &
Area acero 1 .1 5 c m 2
) *to+,m2. & 5 5. $ . " 11.$ 1" . 0 5 1 . " 1& . 1 5 21 . 5 2$ . & " 2 . 2 2 2 . " 2 $0 . 5 " $2 . 5 ( $" . 5 2 $5 . 1 $ . 0 2 $ . 2 0 $ . $ " $ . " ( $ . 5 ( $ . " $ . ( 0 $ . ( " $ . ( ( $ . ( & $ . 1 $ . 1 $ . 2 $ . 2 $ . 1 $ . 0 $ . ( & $ . ( $ . ( ( $ . ( 5 $ . ( $ $ . ( 1 $ . & $ . ( $ . 5
DIAGRAMA MOMENTO-CURVATURA "0.00 $5.00 $0.00 25.00 O T N E M O M
20.00 15.00 10.00 5.00 0.00 0.0
5.0
10.0
15.0
CURVATURAx10-5
20.0
25.0
$0.0
φ10,5
0.5 1.0 1." 1.& 2." 2. $.$ $.( ".2 ". 5.0 5." 5. .2 .( (.5 .$ &.1 &.& 10.( 11.5 12.$ 1$.0 1$. 1". 15.$ 1.1 1.& 1(. 1.$ 1&.1 1&. 20.5 21.2 21.& 22. 2$.$ 2".0 2".( 25.$
DIAGRAMA MOMENTO CURVATURA EECA: C34,a AOS: P= 0.00 F'c= 210.00 #= $0.00 F!= "200 Es= 270007000 8= 50.00 9t= 2.(1 %$= 0.&(&(2 c 0.0001 0.0002 0.000$ 0.000" 0.0005 0.000 0.000( 0.000 0.000& 0.0010 0.0011 0.0012 0.001$ 0.001" 0.0015 0.001 0.001( 0.001 0.001& 0.0020 0.0021 0.0022 0.002$ 0.002" 0.0025 0.002 0.002( 0.002 0.002& 0.00$0 0.00$1 0.00$2 0.00$$ 0.00$" 0.00$5 0.00$ 0.00$( 0.00$ 0.00$& 0.00"0
%1 0.0"&1( 0.0&(0 0.1"2500 0.1( 0.22&200 0.2(0000 0.$0&200 0.$"( 0.$2500 0."1( 0.""&200 0."0000 0.50&1( 0.5$( 0.52500 0.5( 0.0&&( 0.$0000 0."&1( 0.( 0.2$"0 0.&212 0.(0500 0.(1&""" 0.(2&200 0.($((2 0.("500 0.(52$1 0.(5(00 0.($& 0.(500 0.((2&1( 0.((00 0.((&&02 0.(200 0.(515 0.(("00 0.(&0$5 0.(&0&00 0.(&1(
d1 kg kg/cm2 cm kg/cm2 kg/cm2 cm cm
%2 0.$$"("5 0.$$20( 0.$$((00 0.$$&2 0.$"0&00 0.$"25&$ 0.$""$00 0.$"15" 0.$"000 0.$50000 0.$52000 0.$5"1( 0.$5$$ 0.$5& 0.$1110 0.$$$ 0.$20 0.$&0" 0.$(1&51 0.$(5000 0.$(2&0 0.$105 0.$5"00 0.$&10$ 0.$&200 0.$&$( 0."00000 0."0$"0 0."0(00 0."10100 0."1$$00 0."1""2 0."1&500 0."22"$( 0."25$00 0."2110 0."$000 0."$$" 0."$100 0."$5&
d1= d2= Po= P#= )#= o= C 15.$1 15."0 15."& 15.5 15. 15.(( 15.( 15.&( 1.0 1.1 1.2& 1."0 15.$" 1"."0 1$.5& 12.&1 12.$1 11.1 11.$( 10.&& 10.5 10.$ 10.10 &. &.( &."& &.$2 &.1( &.0$ .&1 .(& . .5 ."& ."0 .$2 .2" .1( .10 .0$
Peralte Area acero 5.00 cm As1= .&( cm2 "5.00 cm As2= 1".25 cm2 21.22 $5$.0& to+ &.&1 to+ $5.&$ to+,m A)= ,&.12 to+ AP= Fs1 1$5 2(0 "0 5"$ 1 20 &5& 10&& 12"0 1$2 1525 1 1(5$ 12( 1&( 1&1 201& 20( 212& 21(& 222 22( 2$2" 2$(0 2"15 2"0 250" 25" 25&0 2$$ 2(5 2(15 2(55 2(&5 2$" 2(2 2&10 2&" 2&" $022
Fs2 E;I4I<3IO ,$ 0.000 ,(& 0.000 ,11"$ 0.000 ,1510 0.000 ,1(1 0.000 ,222$ 0.000 ,25& 0.000 ,2&0( 0.000 ,$2$ 0.000 ,$52 0.000 ,$(( 0.000 ,"15 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000 ,"200 0.000
$0.00 25.00 20.00 O T N E M O M
15.00 10.00 5.00 0.00 0.00
10.00
20.00
$0.00
CURVATURA X10-5
"0.00
50.00
0.00
Ec= I= 6cr= )cr= φ cr10,5=
d2 As2
1.52 0.25
) *to+,m2.21 ".$( ."& .5 10.5& 12.5( 1".51 1.$& 1.2$ 20.0$ 21.(( 2$."( 2$.(2 2$. 2$.& 2".0 2".1( 2".2$ 2".2& 2".$$ 2".$( 2"."0 2"."2 2"."" 2"." 2"."( 2"." 2"."& 2"."& 2".50 2".50 2".51 2".51 2".51 2".51 2".51 2".51 2".51 2".51 2".51
DIAGRAMA MOMENTO-CURVATURA
As1
)!= )= = φ 10,5 0.5 1.$0 1.&" 2.5( $.1& $.0 "."1 5.01 5.0 .1 .(5 (.$2 ."( &.($ 11.0$ 12.$& 1$.1 15.2" 1.(1 1.21 1&.(2 21.2" 22.( 2".$0 25.5 2(."0 2.& $0.52 $2.10 $$.( $5.25 $.5 $."5 "0.05 "1. "$.2 "".&0 ".52 ".1 "&.(
2$.(2 2".51 5.(
220000 $12500 2$.1& 2.&0 0."2
kg/cm2 cm" kg/cm2 to+,m
DIAGRAMA MOMENTO CURVATURA EECA: C34,# AOS: P= 0.00 F'c= 210.00 #= "5.00 F!= "200.00 Es= 270007000 8= 0.00 9t= $0.00 %$= 0.&(&(2 c 0.0001 0.0002 0.000$ 0.000" 0.0005 0.000 0.000( 0.000 0.000& 0.0010 0.0011 0.0012 0.001$ 0.001" 0.0015 0.001 0.001( 0.001 0.001& 0.0020 0.0021 0.0022 0.002$ 0.002" 0.0025 0.002 0.002( 0.002 0.002& 0.00$0 0.00$1 0.00$2 0.00$$ 0.00$" 0.00$5 0.00$ 0.00$( 0.00$ 0.00$& 0.00"0
%1 0.0"&1( 0.0&(0 0.1"2500 0.1( 0.22&200 0.2(0000 0.$0&200 0.$"( 0.$2500 0."1( 0.""&200 0."0000 0.50&1( 0.5$( 0.52500 0.5( 0.0&&( 0.$0000 0."&1( 0.( 0.2$"0 0.&212 0.(0500 0.(1&""" 0.(2&200 0.($((2 0.("500 0.(52$1 0.(5(00 0.($& 0.(500 0.((2&1( 0.((00 0.((&&02 0.(200 0.(515 0.(("00 0.(&0$5 0.(&0&00 0.(&1(
d1 kg kg/cm2 cm kg/cm2 kg/cm2 cm cm
%2 0.$$"("5 0.$$20( 0.$$((00 0.$$&2 0.$"0&00 0.$"25&$ 0.$""$00 0.$"15" 0.$"000 0.$50000 0.$52000 0.$5"1( 0.$5$$ 0.$5& 0.$1110 0.$$$ 0.$20 0.$&0" 0.$(1&51 0.$(5000 0.$(2&0 0.$105 0.$5"00 0.$&10$ 0.$&200 0.$&$( 0."00000 0."0$"0 0."0(00 0."10100 0."1$$00 0."1""2 0."1&500 0."22"$( 0."25$00 0."2110 0."$000 0."$$" 0."$100 0."$5&
d1= d2= d$= Po= P#= )#= o=
Peralte 5.00 $0.00 55.00 5&.5( 222.2 5&." ,112.$&
C 1$. 1$.&" 1".0$ 1".11 1".20 1".2& 1".$ 1$.( 12.$ 12.12 11.5( 11.1" 10.51 &.&2 &."2 &.00 ." .$" .0 (. (.( (.50 (.$ (.2$ (.12 (.0$ .&" . .(& .(2 .( .1 .5 .52 ."( ."$ ."0 .$ .$$ .$0
Fs1 12 25( $ 51( " (0 &1$ 1020 10&& 11(5 12"& 1$2$ 1$" 1$ 1"0 1"22 1"$2 1""2 1""& 1"55 1"1 1" 1"(5 1"$ 1"&1 1500 150 151 152( 15$ 15"& 155& 15(0 152 15&" 105 11( 12& 1"1 155
Area acero cm As1= 11."0 cm As2= $.& cm As$= 11."0 to+ 2.( to+ 0.&& to+,m A)= to+ AP= Fs2 ,2$$ ,"1 ,$ ,&01 ,111$ ,1$1& ,1520 ,12 ,2"0& ,2&52 ,$50 ,"05 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200
Fs$ ,5&" ,11( ,1(5$ ,2$1 ,2($ ,$"1 ,$&5" ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200 ,"200
cm2 cm2 cm2 cm2
$0.00 25.00 20.00 15.00 10.00 5.00 0.00 10.00
20.00
$0.00
"0.00
CURVATURA X10-5
50.00
0.00
As2 As$
E;I4I<3IO 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000
$5.00
0.00
d$
2.$1 0.$
DIAGRAMA MOMENTO-CURVATURA
O T N E M O M
As1
(0.00
)!= )= = ) *to+,m$." (.21 10.(1 1".15 1(.5$ 20.$ 2".0 25.&0 2.5 2(.21 2(.5 2." 2.(1 2.0 2.( 2.&$ 2.& 2&.02 2&.05 2&.0 2&.10 2&.12 2&.1$ 2&.1" 2&.1" 2&.15 2&.15 2&.15 2&.15 2&.15 2&.15 2&.15 2&.15 2&.1" 2&.1" 2&.1" 2&.1$ 2&.1$ 2&.12 2&.12
25.&0 2&.12 10.&$ φ 10,5 0.(2 1."$ 2.1" 2.$ $.52 ".20 ".( 5.0 (.01 .25 &.51 10.(( 12.$ 1".12 15.&2 1(.( 1&. 21.5 2$.51 25."5 2(.$& 2&.$2 $1.25 $$.1( $5.0& $(.00 $.&2 "0.2 "2.($ "".2 ".51 "."1 50.$0 52.1 5".0 55.&5 5(.$ 5&.(1 1.5& $."5
CURVA ESFUERZO - DEFORMACION DEL CONCRETO Y DEL ACERO SECCION: 6'c= %$= 6>c= εo= ?= εc 0.0000 0.0001 0.0002 0.000$ 0.000" 0.0005 0.000 0.000( 0.000 0.000& 0.0010 0.0011 0.0012 0.001$ 0.001" 0.0015 0.001 0.001( 0.001 0.001& 0.0020 0.0021 0.0022 0.002$ 0.002" 0.0025 0.002 0.002( 0.002 0.002& 0.00$0 0.00$1 0.00$2 0.00$$ 0.00$" 0.00$5 0.00$ 0.00$( 0.00$ 0.00$& 0.00"0
210.00 kg/cm2 0.&(&(2 20$.2( kg/cm2 0.0020 0.001 (5.00 %1 0.000000 0.0"&1( 0.0&(0 0.1"2500 0.1( 0.22&200 0.2(0000 0.$0&200 0.$"( 0.$2500 0."1( 0.""&200 0."0000 0.50&1( 0.5$( 0.52500 0.5( 0.0&&( 0.$0000 0."&1( 0.( 0.2$"0 0.&212 0.(0500 0.(1&""" 0.(2&200 0.($((2 0.("500 0.(52$1 0.(5(00 0.($& 0.(500 0.((2&1( 0.((00 0.((&&02 0.(200 0.(515 0.(("00 0.(&0$5 0.(&0&00 0.(&1(
F!= Es= ε!=
%2
%1 0.000000 0.0"&1( 0.0&( 0.1"2500 0.1( 0.22&1( 0.2(0000 0.$0&1( 0.$"( 0.$2500 0."1( 0.""&1( 0."0000 0.50&1( 0.5$( 0.52500 0.5( 0.0&1( 0.$0000 0."&1( 0.(
0.$$"("5 0.$$20( 0.$$((00 0.$$&2 0.$"0&00 0.$"25&$ 0.$""$00 0.$"15" 0.$"000 0.$50000 0.$52000 0.$5"1( 0.$5$$ 0.$5& 0.$1110 0.$$$ 0.$20 0.$&0" 0.$(1&51 0.$(5000 0.$(2&0 0.$105 0.$5"00 0.$&10$ 0.$&200 0.$&$( 0."00000 0."0$"0 0."0(00 0."10100 0."1$$00 0."1""2 0."1&500 0."22"$( 0."25$00 0."2110 0."$000 0."$$" 0."$100 0."$5&
"200.00 kg/cm2 2000000.00 kg/cm2 0.0021
%2 0.$$$$$$ 0.$$"(" 0.$$20( 0.$$((1& 0.$$&2 0.$"0&0& 0.$"25&$ 0.$""$"0 0.$"15" 0.$"0$& 0.$50000 0.$520"1 0.$5"1( 0.$5$$ 0.$5& 0.$1111 0.$$$ 0.$2(& 0.$&0" 0.$(1&51 0.$(5000
6c 0.00 1&.2 $.2 5."1 ($.1 .&$ 10$.( 11(.$& 1$0.10 1"1.( 152." 12.11 1(0.(5 1(.$( 1".& 1&0.5( 1&5.1" 1&.(0 201.2" 202.(( 20$.2( 201.2" 1&&.21 1&(.1 1&5.1" 1&$.11 1&1.0 1&.0" 1(.01 1".& 12.&5 10.&1 1(. 1(.5 1(".2 1(2.( 1(0.(5 1.(2 1. 1".5 12.2
0.001(5
εs 0.0000 0.0011 0.0021 0.00$2 0.00"2 0.005$ 0.00$ 0.00(" 0.00" 0.00&5 0.0105 0.011 0.012 0.01$( 0.01"( 0.015 0.01 0.01(& 0.01& 0.0200 0.0210 0.0221 0.02$1 0.02"2 0.0252 0.02$ 0.02($ 0.02" 0.02&" 0.0$05 0.0$15 0.0$2 0.0$$ 0.0$"( 0.0$5( 0.0$ 0.0$( 0.0$& 0.0$&& 0.0"10 0.0"20
}}` left[{ {2ε rSub{size8{C}}} over{εrSub{size8{O}}}} -left({{εrSub{size8{C}}} over{εrSub{size8{O}}} }right)rSup{size8{2}}right]}{} ¿ ¿
f C=f C
}} ` left [1 - 1` left(εrSub { size 8{C}} - ε rSub { size 8{O} } right ) right ]} {} ¿
0.0 2100.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0 "200.0
6! C
¿
f C =f C
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s f
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} } } {} ¿ ¿
c f
6s
f C ¿
C
O Z R E U F S E
O Z R E U F S E
εs!
A 0.0000
o=
0.0020
DEFORMACION eC
0.00"0
0.001 4a#orator@o 6! es! Es
"(&.0 0.020 2$&7&&
0.0021
DEFORMACION eS
BARRA DE CONSTRUCCION DIMENSIONES Y PESOS ACEROS AREUIPA IA)E3O A3EA PESO cm2 kg/m
No
4ISO 4ISO
B mm B 1/"> B mm B $/> B 12 mm B 1/2> B 5/> B $/"> B 1> B 1 $/>
$ " 5 11
0.2 0.$2 0.50 0.(1 1.1$ 1.2& 2.00 2." 5.10 10.0
0.22 0.25 0."0 0.5 0.& 0.&& 1.55 2.2" $.&( (.&1
SIDERPERU A3EA PESO cm2 kg/m 0.2 0.$2 0.50 0.(1 1.1$ 1.2( 1.& 2.5 5.0( &.5
EUIVALENCIAS ENTRE UNIDADES 1 4#/lg2= 0.00& N/mm2 1 4#/lg2= 0.0(0$1 kg/cm2 1 N/mm2= 10.20"1 kg/cm2 1 )Pa= 1 N/mm2 )AE3IA4 )%S S.I. IN94ES kg/cm2 N/mm2 %@s/lg2 6'c= 210 20. $ Ec= 2007000 1&700 27"5 6!= "7200 "12 0 Es= 270007000 1&7000 27""5
6'c= =
210 kg/cm2 0.20
Ec
=
1!
Gc = Peso = )asa =
c √ f 'tomar: =
Ec
(
2 1 + μ
)
=
21(7$(1 kg/cm2 2207000 kg/cm2 &17( kg/cm2
2."00 to+/m$ 0.2""5 to+,seg2/m
$7000 $07000
0.22 0.25 0.$& 0.5 0.& 0.&& 1.55 2.2" $.& (.&1
A3EA lg2
0.05 0.11 0.20 0.$1 0."" 0.(& 1."
SELECCI!N DEL REFUERZO Acero Nega=
(.&2 1> $/"> 5/> 1/2>
Acero Pos@t=
2 1
$.(2 1> $/"> 5/> 1/2>
1.5 0.00 5.(0 1.& 0.00 (.
0.($
2
0.00 0.00 $.& 0.00 $.&
11."
