!"#"!$%&!$#$&#!'$%#&!$(#&!$()'$( $"*+ $&,-!$./#&$!/ $&,-!$./#&$!/&"/.$ &"/.$%/%# %/%#!!%## !!%##!+ !+
!
"#$%&'#() As
•
#"0%%$1%#", ρ =
•
"0*'$&!%$ "0*'$&!%$1%#" 1%#"2 2+3++, +3++, 0,01 ≤ ρ ≤ 0,04
•
!#"!#"!! !#"!#"!!4"$!0$ 4"$!0$%.!!$" %.!!$"%$%, %$%, 56!$!$.#$!#"!!#"! 7!$.!$"!) 8&!&$! o
o
o
Ag
'*)+,
#$9$:.$";"! *$'"(-"$
"!6%;%#"#&!$(-)&$!
*$'"(-"$.$/$'$)&*0(*"' 9!0!$#$#&!$($ "$$=!$.&$!#!"!=!6!+!6$! &%+-!!$#$2$%.$%#$$4"&!%"#+!4"&$! !$. !.- %. #"!!$! %!#$ !6! % #"+ #"!! "!= #!#3+><? #))!+ &! % 4" !$. &!%"#$! # % #$$ $$! &$!/ " #$6$%%!/4"6!"!6!% φ!&!!$.4"&!&$!+
"#$%$1$)
(C21.3.5)
(c21-1)
(c21-2)
'&!$%/'&!$"$&!)&!, Ash = 0,20
Ash = 0,30 • •
' shc f c Ag
f yh
' shc f c − 1 @ Ash = 0,06 f A ch yh
' shc f c Ag
f yh
' shc f c − 1 @ Ash = 0,09 f yh Ach
&!$!#"#$("!4"%##!$%##!%!#".!$$!" "&:!%$%%!$#$#!'$%#"+ "% #"#$( .$! &! #" % ! 1 ) &!% "!! " #&#$%%#"!6"!%#"%'$(!$%"#$
•
"#$%)&'#()*(( $*!0$!#%%&!,A5 $*!0$!#%%,A< $*2$3(**,4$'*)
#$#$(,8 $#.$%.B@>#/#%."$$=!6!$%& %.%%6!$%.B#!%##!B$!!!)%&":% 6#$%
2$*,'$3$$-$*#3$'()$"#$% %$#$$.!!.!!@/<%./53(C7 φ'!%!"+#".!$$0$@ <&!!$.&$!)%7>&!!"!=&!$#$&+
&$#%"D0$#
&$#!%"D0$#
$$D%!&
5. #"#!%'$!!/!$"#!'$)"+ "#$%$$#&!"#!'$&$#%#"'#!$#$%%EC+
•
&!$%##$(,
•
&!$!%4"$$.!$%"!=,
'
'
'
Pn = 0,85 f c ab + As f s − As f y •
#!&##!$%%##$(, '
h h a h − + As' f s' − d ' + As f y d − 2 2 2 2
M n = 0,85 f c ab •
!%$1%.#"&$!4"#%#-"%&"%*$$#"&$"*, Pu ≤ φ Pn M u ≤ φ M n
%$!%$!##$(!&!#-"%&"4"&!%"##"&! %!%'#!$#$%%+ • !'#!$#$%%#!/#!'$/#"&!#&!$(##!)&!"#$% #!#&!$( • !'#!$#$%%.-#"&!#&!$(##!#&$."#$% #!#&!$(%*#!% • ! " '#!$#$%% $!%$ %$% .#%/ $"* ##! &! #&!$("#!%#")#!&!$(! • !!%'#!$#$%%*"-$(/#!6"$!&!$(!&!% ##$(=*-%%#! • !"'#!$#$%%F#"#&!&!'$(&"!$E3)E 3 • !#!'$%$(/#"&!"#$%#!2$#
"&$#$&!.!&"%#"!6, • %&"*%!$%&!"".$##$(&#$$#%#!$%#%##$( •
∈c = 0.003
•
%!#$"$!$%##!)%#!&!&!#$%$#$-"!
•
$ f s < f y f s = E s ∈s
•
$ f s > f y f s =
•
$#"!$#$$(%##!
•
!#$(! f c y ∈c "&$"#"!6"!=%!#$(
f y
$ $"$ &" 4" %$ "! %% #&!$ * !$% &! " #!#!0$#�$#)%%#&!$%##$(#.$)"#"#$(!&! !%&"+ •
;#&!$('$,)!#!'$$+ /!$#$*'$%##!) B$!!"$%#&!$(, '
P0 = 0.85 f c ( Ag − As ) + As f y •
".#%, .) .+ ∈c = 0,003 /#!$(")% ∈s =∈ y / ∈'s &"%!")% %&%$%%!0%#"+
•
;&!'$(&"!/#"#&!#'$(&"!/E3/E 3+ ∈s ≥∈ y
•
;$('$+"*.-%%$!+##!!$$(/"!= '$$"*!%#!"$&$#%&! )+ P0t = f y As
•
#"#$&!#&$.$$%%%%!#$&!#!$(, ∈s = 0,003 f s = 0,003 E s
•
d − c c
d − c c
≤ f y
#"#$&!#&$.$$%%%%!#$&!#!#&!$(
'
' s
∈ = 0,003
c − d c '
' s
f = 0,003 E s •
c − d c
≤ f y
&!"%$%%%.4"%#&!$(%##!)!"%##!, a = β 1c ≤ h '
C = 0,85 f c ab •
;$), '
'
'
Pn = 0,85 f c ab + As f s − As f y '
h h a h − + As' f s' − d ' + As f y d − 2 2 2 2
M n = 0,85 f c ab
&$##$%6!%φ%#(%$, • $(,;#&!$('$,φE3/<&!&$!)φE3/3&!-, φ Pn = 0,75 * 0,65 * (0,85 f c' ( Ag − Ast ) + f y Ast ) ;- φ Pn = 0,80 * 0,75 * (0,85 f c' ( Ag − Ast ) + f y Ast ) &$! •
$(,D%#&!$(,φE3/<&!&$!)φE3/8<&!-
•
•
$(,$ Pb ≥ 0,10 f c Ag /#&!#!! 0,10 f c Ag )E3/ φ6!0$! 3/8<(3/<B3/3+$ Pb ≤ 0,10 f c Ag /#&!#!!. )E3/ φ6!0$ !3/8<(3/<B3/3 $(9,&!6!%!3φE3/3
-&, ! ##$( % #" &!% ##" &" % #"!6 % $!##$( # ) $ φ?+ "&$%4"##$($-#!"!=#!+ • • • •
•
G!%#!, As E<3H8E7383 E73/83'3 5
•
G!I!", Ag E3/73H3/5<E3/7< E7<+333
•
#"0%%$1%#", ρ =
As Ag
E3/38
"!=,8J> #! f c' = 28 E>+333K #! f y = 420 E53+333K E s = 200.000 E33'38K
"%#!'$*'$%#&!$( P0 , •
P0 = 0.85 f c' ( Ag − As ) + As f y
E3/>+333HL3/7<73/83'3 5MN53+333H73/83'3 5E5+5<3K •
φ Pn = 0,75φ (0,85 f c ( Ag − As ) + f y As )
E3/>3H3/8K c
0,003
".#%,./. ∈'s
∈c = .003
→c=
=
0,40
∈ y +0,003 0,003 * o,40
0,0021 + 0,003
= 0,235m
∈'s 0,003 0,003 ' ∈'s = = (c − 0,05) = 0,00236 >∈ y f s = f y c − 0,05 c c ∈s =∈ y T s = As f y = 3 * 5,10 *10 − * 420.000 = 85/83K 4
β 1c
#
'
C s = As f y = 3 * 5,10 *10
2
−4
* 420.000 = 85/83K
C c = 0,85 f c β 1cb = 0,85 * 28.000 * 0,85 * 0,235 * 0,30 = +58/73K '
Pb = C c + C s − T s = C c = +58/73K
%#!&#-"! •
T s ( d − c) + C s (c − 0,05) = T s ( d − 0,05) = C s ( d − 0,05) ;!"&!%"!=
•
M b = T s (d − 0,05) + C c c −
β 1c 2
0,85 * 0,235 = 5/85K 2
= 642,60 * (0,40 − 0,05) + 1.426,30 0,235 −
9!#φ, •
φ Pb = 0,65 *1.426,30 E/3K
•
φ M b = 0,65 * 417,64 E/5K
•
0,10 f c Ag = 0,10 * 28.000 * 0,135 = 7>/33K
"%*'$$(, Pot •
Pot = − Ast f y = −30,60 * 10 − * 420.000 = −1.286,88 K 4
9!#φ, •
φ P0t = 0,90 * −1.286,88 E+<>/3K
"#E33, '
'
'
•
Pn = 0,85 f c ab + As f s − As f y = 0
•
C c + C s = T s → Cuadrática en C
•
C c = 0.85 f c cbβ 1
•
T s = As f y
'
'
•
' s
∈ = 0,003
c − d c
'
' s
c − d
•
f = 0,003 E s
•
C s = 0,003 E s *
•
0,85 f c β 1cb + 0,003 E s
•
&-%#E3/3<337
•
C c = 0,85 f c cbβ 1 = 0,85 * 28.000 * 0,85 * 0,05 * 0,30 E737/5<K
•
T s = As f y E857/55K
•
C s = 0,003 E s *
•
' M n = 0,85 f c ab c −
c
≤ f y
c − 0,05 ' As c
'
c − 0,05 c
'
As = As f y
'
c − 0,05 ' As E7</>K c
β 1c
+ As f s (c − d ) + As f y (d − c ) E77/85K 2 '
'
9!#φ, •
φ M o = 0,90 * 233,64 E3/K
'
$!%!##$($φ
$!%!##$(#)$φO
67
λ = arctan
e x e y
= arctan
M ny M nx
M ny = Pn * e x
M nx = Pn * e y
!"$! "&!$#$ % $!##$( &! " #" %% &"% &!#! " '$( .6$ % *$$"$'$+!"6!%%%P&"%.!%$!6!%"%!#$% "!=%!#$(&! .#! "!= .!! ) !"%#&!$( ##!/ #/ "% #"#$ % 4"$$.!$ &"% %!$! / ' ) ) &"% %!$!"&"%#"!6+!%=%#&!$("&!.)! #%.!!&"%!"-%$!"!=+&!$#$&%$$#"%!%$#4"! - "! &!&%$#"! 0 % '#!$#$%%/ # 4" !" 4" &! %$!6!%&!"P%"6!%λ)%%+
5 .!%$!&!$%%6!#%+ !!%#"!6&"%!&!'$%&!#"#$(%$$, α 1
M M nx + ny M M nx 0 ny 0
α 2
= 1.0 %%,
M nx = Pn e y
M ny = Pne x
M nx 0 = M nx con M ny = 0
M ny 0 = M ny con M nx = 0
α 1 ,α 2 E '& 4" %&% % %$$ % #"/ #"0 ) %$!$."#$( %
!"!= % #! % #!#!0$# "!=%!#$( % #! ) % ##!/ !#".!$$/-&$! S!*$##α 1 = α 2 = α α
α
M nx M ny + = 1.0 M nx M ny o
o
I!!/
##/') )%*$$+ '3) )3&!%$1"$'$#%$%+$ α)$ #"#$(Q++3##"&/$#"#$(R/3#
