! "#! $ % !&'$ !"#$#%%% !""%&##%!! %&&' •
(&&$ (&&$! !)&# )&#% % ρ < ρ bal %$ %!& %!&
•
% ρ > ρ bal % %*! *!& &
• •
!%&*#%% !%&*#%%! !' ' !&%+ !&%+&, &,$" $"%' %'
•
-)+-%$" -)+-%$""% "%"%+ "%+&&. &&. M u ≤ φ M n = φ ( M n1 + M n 2 )
•
/& /&& &&0&" &0&"% % M u . M n1 '
'
M n1 = As f y ( d − d ) •
&*&&& &*&&& " " % % ) ) . % ρ max = 0,75 ρ bal
M n 2 #&**"
a ' M n 2 = ( As − As ) f y ( d − ) %%. 2 '
•
a=
( As − As ) f y '
0,85 f c b
'
=
( ρ − ρ ) f y d '
0,85 f c
. ρ =
As bd
) ρ = '
As' bd
φ M n = φ ( M n1 + M n 2 ) a
φ M n = φ As' f y (d − d ' ) + φ ( As − As' ) f y (d − ) 2
•
" As ) As' * f y )""" )""" .
•
ρ b = ρ bal + ρ '
•
ρ max = ρ max + ρ ' = ρ 0 , 004 + ρ ' ' $ && & % %% " $
&&!% "! "!& &1% 1%. .
• • •
&∈'s =∈ y c 0,003
=
c − d '
c=
∈ y
%&%2&#&+%%
0,003 0,003− ∈ y
'
d
"+. As f y = 0,85 f c β 1cb + As' f y , %$%%+%3) '
•
ρ = 0,85 β 1
f c c f y d
+ ρ ' #&!%
'
f c d '
•
ρ cy = 0,85 β 1
•
ρ < ρ cy f s' <
•
(
0,003
f y d 0,003− ∈ y
'
) + ρ
ρ > ρ cy f s' =
f y
' d (0,003+ ∈ y ) ≤ f y f = Ε s ∈ = Ε s 0,003 − d ' s
' s
'
'
f s
•
ρ max = ρ 0,004 + ρ
•
ρ < ρ cy &&.
f y
'
' s
c − d
•
f = 0,003 E s
•
%"+. As f y = 0,85 f β 1cb + A 0,003 E s
c '
' c
' s
c − d c
f y
4
•
As' 600.000 − As f y 600.000 As' d ' c + c = 0 #%, − ' ' β β f b f b 0 , 85 0 , 85 c 1 c 1
•
' M u ≤ φ M n = φ 0,85 f c β 1cb d −
2
β 1c ' ' ' + As f s (d − d ) 2
%&%$%#%%. b, d , h, As , As' , f c' , f y , M u As
•
ρ =
•
ρ ≤ ρ max #5&&!%#% As'
•
ρ > ρ max #5%+&!%# ρ =
bd
) ρ max = ρ 0,004
'
'
'
•
ρ cy = 0,85 β 1
•
ρ ≥ ρ cy f s' =
(
0,003
f y d 0,003− ∈ y
bd
'
) + ρ
f y ,
o
. ρ max = ρ 0, 004 + ρ '
o
ρ ≤ ρ max φ M n = φ As' f y (d − d ' ) + φ ( As − As' ) f y (d −
o
•
f c d '
As
2
)
ρ > ρ max $
ρ < ρ cy f s' < o
a
f y #.
As' 600.000 − As f y 600.000 As' d ' c + c =0 − ' ' β β f b f b 0 , 85 0 , 85 c 1 c 1 2
'
o
' s
f = 0,003 E s
c − d c
'
o
'
ρ max = ρ 0,004 + ρ
f s
f y
o
o
' ρ ≤ ρ max M u ≤ φ M n = φ 0,85 f c β 1cb d −
ρ > ρ max $
%&%%%#%%. b, d , h, f c' , f y , M u
β 1c ' ' ' + As f s (d − d ) 2
•
&6φ&-. ( As − As' ) = ρ maxbd
•
%6φ78#98-. ( As − As' ) = ρ 0,005bd
•
M n 2 = ( As − A ) f y ( d − ) a = 2
•
M n1 =
•
" f s' =
•
:+. As = ( As − As' ) + As'
•
. ρ =
•
ρ ≥ ρ cy f s' =
f y #:;!&+)%
•
ρ < ρ cy f s' <
f y #.
a
' s
o
o
M u
φ
a=
c=
( As − As' ) f y '
0,85 f c b
− M n 2 #/&%"" f y ,
. As' =
M n1 '
f y ( d − d )
'
As
) ρ cy = 0,85 β 1
bd
f c d '
(
0,003
f y d 0,003− ∈ y
'
) + ρ
( As − As' ) f y '
0,85 f c b
a
β 1 '
o
o
' s
f = 0,003 E s '
'
Asrev = As
f y '
f s
c − d c
#$*%
,&<$. • • • •
<!.B?)B> f c' = 21 /7'888; f y = 420 /78'888; &&%
•
=%. As 74>?@7'94&& 79#48 &
•
=%&. As' 78@7'88&& 78#88&
•
•
ρ =
As bd
ρ min =
=
19,35 x10
−4
= 0,0258
0,25 * 0,30
0,25 f c'
f y
=
0,25 21 420
= 0,0027 ≥
ρ max = ρ 0,004
•
ρ > ρ max #5%+&!% '
•
ρ =
As bd
=
f y
=
1,4 420
= 0,0033
f c' 0,003 21 0,003 * 78#8 = 0,85 β 1 = 0,85 * 0,85 * f y 0,003 + 0,004 420 0 , 003 + 0 , 004
•
'
1,4
10,20 x10
−4
0,25 * 0,30 '
f c d '
= 8#84A 0,003
'
0,003 ) + 0,0136 =8#84?? 420 420 0,25 0,003 − 200.000
ρ cy = 0,85 β 1
•
ρ < ρ cy f s' < f y #.
•
As' 600.000 − As f y 600.000 As' d ' c + c = 0 #7& − ' ' 0 , 85 β 0 , 85 β f b f b c 1 c 1
(
f y d 0,003− ∈ y
) + ρ = 0,85 * 0,85
21 0,5
•
2
'
•
(
' s
f = 0,003 E s
c − d
= 0,003 * 200 x10 6
c
0,12 − 0,05
'
'
f s
0,12 350
= 48'888;748/
= 8#8?
•
ρ max = ρ 0, 004 + ρ
•
β 1c ' ' ' ' cb d A f d d − + ( − ) ρ ≤ ρ max M u ≤ φ M n = φ 0,85 f c β 1 s s 2
f y
= 0,0155 + 0,0136
420
A •
φ = 0,9
•
0,85 * 0,12 −4 φ M n = 0,9 0,85 * 21.000 * 0,85 * 0,12 * 0,25 0,25 − + 10,20 *10 * 350.000(0,25 − 0,05) 2
•
φ M n = #?>;(&
•
ρ 0, 005
f c' 0,003 21 0,003 78#84 = 0,85 β 1 * = 0,85 * 0,85 * f y 0,003 + 0,005 420 0 , 003 + 0 , 005 '
'
f s
350
•
ρ 0, 005 = ρ 0, 005 + ρ
•
ρ ≤ ρ 0, 005 &+"φ = 0,9 6;-
f y
= 0,01355 + 0,0136
420
= 8#8
,&%. "%$&2&&0&%%/' • • •
/7488;(& f c' = 24,5 /7'88; f y = 420 /78'888;
•
M u = 300 ;(&
•
f c' M u f c' − ρ = − 1,18 f y φ f y2 0,59bd 2 1,18 f y
•
300,0 * 24.500 24.500 ρ = − = 0,0275 − 2 2 1,18 * 420.000 1,18 * 420.000 0,9 * 420.000 * 0,59 * 0,25 * 0,40
2
f c'
2
24.500
'
•
•
ρ min =
0,25 f c
f y
ρ max = ρ 0, 004
=
0,25 24,5 420
= 0,0029 ≥
1,4
f y
=
1,4 420
= 0,0033 #&8#8844
f c' 0,003 24,5 0,003 = 0,85 β 1 = 0,85 * 0,85 * * 78#8> f y 0,003 + 0,004 420 0 , 003 + 0 , 004
?
f c' 0,003 24,5 0,003 78#8> = 0,85 β 1 * = 0,85 * 0,85 * f y 0,003 + 0,005 420 0 , 003 + 0 , 005
•
ρ 0, 005
•
ρ > ρ max #"%&$%+&!%
•
. ( As − As' ) = ρ max bd = 0,0158 * 0,25 * 0,40 = #>88 &7'>8&& '
( As − As ) f y
−4
* 420.000
= 8#?&
•
a=
•
a 0,127 −4 ' M n 2 = ( As − As ) f y ( d − ) = 15,80 x10 * 420.000 * (0,40 − ) = 4#4;(& 2 2
•
φ = 0,9
•
M n1 =
•
" f s' =
•
As =
•
' ' As = ( As − As ) + As = 15,80 x10
' c
0,85 f b
M u
φ
•
ρ =
•
'
'
f y ( d − d )
As bd
=
'
ρ =
As bd
=
0,85 * 24.500 * 0,25
300
− M n 2 =
M n1
'
=
15,80 x10
=
0,9
f y ,
420.000 * (0,40 − 0,05)
−4
0,25 * 0,40 7,49 x10
−4
0,25 * 0,40 '
•
ρ cy = 0,85 β 1
•
ρ < ρ cy f s' < f y
. 110,03
23,29 x10
f c d '
− 223,3 = 8#84;(&
(
−4
= ?#98 &78#8?9&
+ 7,49 x10 −4 = 4#98 &
= 8#844 = 8#88?9 0,003
f y d 0,003− ∈ y
'
) + ρ = 0,85 * 0,85 *
24,5 0,05
0,003 ) + 0,00745 = 420 420 0,40 0,003 − 200.000 (
'
( As − As ) f y
•
a=
•
c=
•
' s
f = 0,003 E s
•
Asrev = As
'
0,85 f c b
a
β 1
12,70
=
0,85
78#?& = 8#9& '
'
'
f y ' s
f
c − d c
= 0,003 * 200 x106 *
= 7,49 x10
−4
399 420
0,149 − 0,05 0,149
= 499'888;7499/
= ?#8 &7?&&
8#88>
> • • •
'
'
Asrev = ( As − As ) + Asrev = 15,80 x10
−4
+ 7,11x10
−4
= #98 &7'9&&
5% φ = 0,9 ρ 0, 005
f c' 0,003 24,5 0,003 = 0,85 β 1 = 0,85 * 0,85 * * 78#8> f y 0,003 + 0,005 420 0 , 003 0 , 005 + '
'
f s
399
= 8#89
•
ρ 0, 005 = ρ 0, 005 + ρ
•
ρ = ρ 0,005 &+"φ = 0,9 6;-C"D%2%E
•
ρ =
•
'
ρ =
•
:%! As .4B>FB?68@4F4>?@7'48&& -B9FB>6A@F8@7'48 &&-
•
% % As . 4B> F B? 64#9?4@4F4#8@7>#884 ;(3&- B9 F B> 6#8A8@F4#9?4@7>#8AA;(3&-
•
:%! As' .B?64>?@7??&& -4BA6>@47>&& -
•
%% As' .B?64#8@7A#8>;(3&-4BA6#4@47A#?8;(3&-
Asrev bd
=
'
Asrev bd
=
f y
22,91 x10
= 0,0158 + 0,00749
−4
0,25 * 0,40 7,11x10
−4
0,25 * 0,40
420
= 8#89 = 8#88?
9 DISEÑO DE VIGAS T
8
6-
&6&G+- &!&!#)"%%$%' %&+H*%)"+,' &!&#%%+,#&%%"*,%& &%+,&'
I ) (< " ) & %+ &2 ) % % " +, 61$-$+,%,&.
4
- +J!3 +- 6++H-3J>1 - 6++H-3JK6+H- s
$L%".
$L%.
%'('%%%%&#)&' %%
a ≤ h f #). a =
As f y '
0,85 f c b
a ≤ h f -$1+'+%+$1$%' •
a=
As f y
≤ h f #%. a =
' c
0,85 f b
a > h f
•
ρ w > ρ min #
ρ w =
As bw d
% L&# '
•
Asf =
0,85 f c (b − bw ) h f
f y
ρ f y d ' c
0,85 f
ρ =
As bd
A
•
M 1 = Asf f y ( d −
h f 2
)
# ( As − Asf ) f y
•
a=
•
M 2 = ( As − Asf ) f y (d −
•
h f a φ M n = φ ( M 1 + M 2 ) = φ Asf f y (d − ) + ( As − Asf ) f y (d − ) 2 2
'
0,85 f c bw
a 2
)
!"#)&&#+% . c
0,003
•
&+%%.
•
"+. As f y = 0,85 f c' cβ 1bw + 0,85 f c' (b − bw )h f
•
Lo que es equivalente a: As f y = 0,85 f c cβ 1bw + Asf f y
d
=
0,003+ ∈ y
'
$%%. dbw f y '
f c
0,003
•
ρ wbal = ρ bal + ρ f , entonces:
•
ρ w max = ρ max + ρ f , esto 1"!%&"%%%#
bw d
y ρ f =
Asf
ρ w = 0,85 β 1
f y 0,003+ ∈ y
+ ρ f ,donde: ρ w =
As
•
bw d
&2&+%' •
%+&" ρ w ≤ ρ w max
•
El factor de φ se debe ajustar de la misma manera que para vigas rectangulares.
<"%(<. <!/2&.
?
+,$%!#&&%$
,&'<$. • • •
<!.B>FB? f c' = 21 /7'888; f y = 420 /78'888;
5%%&#$L%' •
beff ≥ 4bw ≥ b = @8#88#>8 beff = 0,80 m (Ok.)
> bw
•
= 0,10 m, en este caso no cumple se debería aumentar al ancho del ala. Vamos a calcular de 2 todos modos el momento resistente.
•
=%. As 768?F4>?-@7'?>>&& 7?#>>8 &
h f ≥
",%%% •
a=
As f y
=
' c
0,85 f b
17,88 x10
−4
* 420.000
0,85 * 21.000 * 0,80
= 0,053m > h f = 0,05m
,%+,%#1)"%&$L' •
As
ρ w =
bw d
17,88 x10
=
0,20 * 0,30 '
•
ρ min =
−4
0,25 f c
f y
=
= 8#89>
0,25 21 420
'
•
0,85 f c (b − bw ) h f
Asf =
f y
( As − Asf ) f y
1,4
f y
=
1,4 420
= 0,0033 #&8#8844
0,85 * 21.000 * (0,80 − 0,21)0,05 420.000
(17,88 x10
−4
= 12,54 x10 − 4 &
−
− 12,54 x10 4 ) * 420.000
= 0,063m
•
a=
•
h f a φ M n = φ ( M 1 + M 2 ) = φ Asf f y (d − ) + ( As − Asf ) f y (d − ) 2 2
•
φ = 0,9
•
0,05 0,063 −4 −4 φ M n = 0,9 12,54 x10 − 4 * 420.000(0,30 − ) + (17,88 x10 − 12,54 x10 ) * 420.000 * (0,30 − ) = 184,55 2 2
'
0,85 f c bw
=
=
= 0,0028 ≥
0,85 * 21.000 * 0,20
;(& Asf
12,54 x10
•
ρ f =
•
ρ w max = ρ max + ρ f
bw d
=
−4
0,20 * 0,30
f c' 0,003 21 0,003 = 0,85 β 1 * + 0,0209 78#84A + ρ f = 0,85 * 0,85 * f y 0,003 + 0,004 420 0 , 003 + 0 , 004
•
ρ w max
•
ρ min < ρ w < ρ w max #:M
•
5% φ = 0,9
•
ρ 0, 005
= 0,0209
f c' 0,003 24,5 0,003 78#8> = 0,85 β 1 = 0,85 * 0,85 * * f y 0,003 + 0,005 420 0 , 003 + 0 , 005
9 •
ρ 0,005 = ρ 0 , 005 + ρ f = 0,0158 + 0,0209 = 8#84A?
•
ρ w < ρ 0 , 005 &+"φ = 0,9 6;-
,&%.
• • • • •
f c' = 28 /7>'888; %>#8& f y = 420 /78'888; 5$.#8;(3& +%./2&0(<+
$0%. • . 8#8?@#@9#> • $. 8#48@8#4>@#@9#>3#88 • +%1%)/2&0(<. • +%%%/2&0(<. • /. • 5$. • <. F • 0&. #F#AN# 0&$ wu .#44@7>#A>;(3& *%$.
6;(3& - #A #4 8#>8 #A8 #49 #9 8#4 #44N?#
8
5%%&*#1$%&&$.
• • •
+J!37>#837#8:; 6++H-3J>1#.+7@>1F+H7#&#$%& 6++H-3JK6+H-#6#8#48-378#A&J6#88#48-378#>&#:;
&&$. •
",%%# a ≤ h f
•
%%%&&$.+ 7#&#%78#8&# M u = 129.06 ;(&
•
φ = 0,9 2
f c' M u f c' • ρ = − − 1,18 f φ f 2 0,59bd 2 1,18 f y y y f c'
2
129,06 * 28.000 28.000 • ρ = − = 8#88 − 2 2 1,18 * 420.000 1,18 * 420.000 0,9 * 420.000 * 0,59 *1,42 * 0,40 28.000
• As = ρ bd = 0,0015 * 1,42 * 0,40 = >#A8 &7>A&& •
a=
As f y ' c
0,85 f b
=
8,65 x10
ρ min =
0,25 f c
f y
* 420.000
0,85 * 28.000 *1,42 '
•
−4
=
0,25 28 420
78#88?&# a < h f $#$L
= 0,00315 ≥
1,4
f y
=
1,4 420
= 0,0033 #&8#8844
• As min = 0,0033dbeff = 8#8844@8#8@#7>#948
&7'>94 &&#
+
M u
#
" !
&2&' f c' 0,003 28 0,003 78#88A = 0,85 β 1 = 0,85 * 0,85 * * f y 0,003 + 0,004 420 0 , 003 + 0 , 004
•
ρ max
•
ρ < ρ max #:;
•
5% φ = 0,9 f c' 0,003 24,5 0,003 = 0,85 β 1 = 0,85 * 0,85 * * 78#8> f y 0,003 + 0,005 420 0 , 003 + 0 , 005
•
ρ 0, 005
•
ρ < ρ 0,005 &+"φ = 0,9 6;-
•
:%! As .4B96A@47'94&& -B>68@7'88&& -
•
%% As .4B96#8A8@47#>8;(3&-B>64#9?4@7#>9;(3&-
&&$. • •
%%%&&$.%78#8&#+H78#48# M u = 229.06 ;(& φ = 0,9 2
f c' M u f c' • ρ = − − 1,18 f φ f 2 0,59bd 2 1,18 f y y y f c'
2
229,06 * 28.000 28.000 • ρ = − = 8#8 − 2 2 1,18 * 420.000 1 , 18 * 420 . 000 0 , 9 * 420 . 000 * 0 , 59 * 0 , 30 * 0 , 40 28.000
• As min =
0,25 f c'
f y
d 2bw ≥
1,4
f y
d 2bw =
0,25 28 420
d 2bw ≥
1,4 420
d 2bw = 0,0033 * 0,40 * 2 * 0,30 = >#888&#
M u− f c' 0,003 28 0,003 78#88A = 0,85 β 1 = 0,85 * 0,85 * * f y 0,003 + 0,004 420 0 , 003 + 0 , 004
•
ρ max
•
ρ min < ρ < ρ max #:;
•
5% φ = 0,9 f c' 0,003 24,5 0,003 = 0,85 β 1 = 0,85 * 0,85 * * 78#8> f y 0,003 + 0,005 420 0 , 003 + 0 , 005
•
ρ 0, 005
•
ρ < ρ 0,005 &+"φ = 0,9 6;-
• As = ρ bw d = 0,0145 * 0,30 * 0,40 = ?#4?8 &7?4?&& •
:%! As .B>FB?68@F4>?@7'?9A&&-B9FB>6A@F87'>88 &&-
•
%% As .B>FB?64'9?4@F4'8@7'848;(3&-B9FB>6'8A8@F 4'9?47'894;(3&-
4 CURVA MOMENTO – CURVATURA – (M – ψ )
<&&%%$)$ %1'(!%&%#& *%%%%%&# %%& %*) %%+*%&&""% '
<!&) '$.$*%%%%%,%&&% '
ψ =
1
r
$/&$
% ψ =
∈1 c1
=
∈r c2
M cr =
f r I t c2
% ψ =
M cr M el
M cr =
f r I t c2
42&%%%%& ψ el =
∈el kd
f 1 =
'
M el =
f c 2
kjbd 2
##A)?&* ψ el =
∈1 c1
M inel = Cz = Tz
f c' 2
