esfuerzo deformación según el criterio de Mander

CONCRETO CONCRETO CONFINADO POSGRADO UNI-FIC - 15/05/2015   MTorres 1. Obteng !s gr"#$%s &e es#'er(o &e#or)%$*n &e#or)%$*n &e ! s e%%$*n &e % o!')n re%tng'!r re%tng'!r +'e se )'estr en ! #$g'r. #$g'r. E),!ee !os %r$ter$os &e ent  Pr )o&$#$%&o Pr Pr et ! 1324  e! &e Mn&er et ! 1334 h := 400

Per!te &e ! se%%$on en ))

 b := 400

400 

 An%o &e ! se% %$*n en ))

d b := 20

D$")etro &e ! brr en ))

d bt := 9.2

D$")etro &e! estr$bo en ))

s := 80

       0        4

Es,%$)$ento &e estr$bos en ))

f yt := 440

ρ :=  3.14 ⋅

       0        0        4

Es#'er(o &e #!'en%$ &e estr$bo en MP 1

6 &e! re#'er(o !ong$t'&$n!

100

f´c := 50

Res$sten%$  %o),res$*n &e! %on%reto en MP

Ec := 32570

Mo&'!o e!"st$%o &e! %on%reto en MP

Cc := 20.8

Re%'br$)$ento Re%'br$)$ento  %r e7t er$or &e! &e! ess tr$bo en ))

1.1 C"!%'!o &e ! % 'r8 es#'er(o &e#or)%$*n &e#or)%$*n seg9n ent  Pr )o&$#$%&o 1.1.1C"!%'!o &e ! % 'nt: 8o!');tr$% &e! re#'er(o re#'er(o 1.1.1. C"!%'!o &e! 8o!')en &e re#'er(o re#'er(o &e % on#$n)$ento π 2 Ast := Ast Ast = 66.4 66.476 76 ⋅ d bt 4

(

)

Lhb := 2 ⋅ h + b − 2 2 ⋅ Cc + d bt

(

)

Lhd := 4 ⋅ 0. 0.5⋅ h − 2 ⋅ Cc + d bt ⋅ 2

(

)

Vst := Ast⋅ Lhb + Lhd

.25⋅ Lhb  = 349.2

Lhb =  1.3 1.397 × 10

.25⋅ Lhd  = 246.922

Lhd =  987.687

Vst Vst = 1585 158511 11.4 .4

1.1.1.b C"!%'!o &e! 8 o!9)en &e! %on%reto %on#$n&o hcc := h − ( 2 ⋅ Cc)

hcc hcc = 358. 358.4 4

 bcc := b − ( 2 ⋅ Cc)

bcc bcc = 358. 358.4 4

Vcc := hcc⋅ bcc⋅ s

< %'nt: 8o!');tr$% ser" Vst ρs := Vcc

ρs =  0.015

Vcc Vcc = 1027 102760 6044 44.8 .8

3

Lhb + Lhd   = 2384.5

1.2 C"!%'!o &e !os #%tores ,r ! %'r8 &e %on%reto %on#$n&o 1.2.1 C"!%'!o &e! #%tor = f yt

κ := 1 + ρs⋅ f´c

f yt = 440

κ =  1.136

f´c = 50

1.2.2 C"!%'!o &e !s &e#or)%$ones ε o := if  f´c < 20   , 0.002 ,

 

 

300000

⋅ f´c +

3 + 0.29⋅ f´c

ε 50u :=

3 4

⋅ ρs⋅

0.005  3

 

−3

=  2.5 × 10

ε 50u =  0.0028

145⋅f´c  − 1000

ε 50h :=

Z :=

5

hcc

ε 50h =  0.024

s

0.5

ε 50u + ε 50h − κ ⋅ ε o



( )

fcc ε c :=

κ ⋅ f´c⋅ 2 ⋅



εc κ⋅ ε o

Z =  20.452

  ε c    2

−

κ⋅ ε o

 

  

(

if  0 < ε c ≤ ε o ⋅ κ

)

!" κ⋅ f´c ⋅ 1 − Z⋅ ε c − κ⋅ ε o , 0.2⋅ κ ⋅ f´c

ε cu := κ ⋅ ε o +

0.8 Z

ε cu =  0.042

κ ⋅ ε o  = 0.0028

(

)

fcc κ ⋅ ε o =  56.787

ε c := 0 , 0.000001 .. 1.2ε cu

0.2⋅ κ ⋅ f´c  = 11.357

60

40

( )

fcc εc

20

0

0

0.02

0.04

εc

if  ε c ≥ ε o ⋅ κ

1.2 C"!%'!o &e ! %'r8 es#'er(o &e#or)%$*n seg9n e! %r$ter$o &e Mn&er  f´co := 50

Res$sten%$  %o),res$*n &e! %on%reto en MP4

ε co := ε o =  0.0025 De#or)%$*n  res$sten%$ )"7$) &e! %on%reto no %on#$n&o f yh := 440

Res$sten%$ &e #!'en%$ &e! %ero &e estr$bos MP4

s = 80

Es,%$)$ento  e>e &e estr$bos en e! sent$&o !ong$t'&$n! &e! e!e)ento))4

d bt = 9.2

D$")etro &e ! 8r$!! &e estr$bos ))4

s´ := s − d bt  = 70.8 d b = 20

Es,%$)$ento !$bre entre estr$bos ))4

D$")etro &e 8r$!! &e re#'er(o ,r$n%$,! ))4

Cc = 20.8

Re%'br$)$ento  %r &e estr$bos ))4

(

)

#´i := 0.5 h − 2 ⋅ Cc − 2 ⋅ d bt − d b − d b = 140

D$stn%$ !$bre entre brrs !ong$t'&$n!es %on restr$%%$*n ))4

 bc := b − 2 ⋅ Cc − 2 ⋅ d bt − d b + d bt + d b  = 349.2

 An%o &e n9%!eo %on#$n&o  %entro &e estr$bo ,er$#er$%o en ! &$re%%$*n 7 ))4

dc := h − 2⋅ Cc − 2 ⋅ d bt − d b + d bt + d b  = 349.2

 An%o &e n9%!eo %on#$n&o  %entro &e estr$bo ,er$#er$%o en ! &$re%%$*n  ))4

π 2 As b := ⋅ d b   = 314.159 4

 Are &e 'n 8r$!! &e re#'er(o !ong$t'&$n!

As$ := 16⋅ As b  = 5026.5

 Are tot! &e! re#'er(o !ong$t'&$n! ))2 4

As% := bc⋅ dc  = 121940.6

 Are tot! &e! n9%!eo %on#$n&o ))24

As$ = 0.041 ρcc := As%

Re!%$*n &e "re &e re#'er(o  ! &e! n9%!eo %on#$n&o

En este %so ,rt$%'!r ! se%%$*n t$ene estr$bos en &$gon  + %!%'!r e! "re #e%t$8 en %& &$re%%$*n dc θ" := !t!%       bc 

θ"⋅

180

π

A" := ( 2 + 2 cos( θ") ) ⋅ Ast = 227 Ay := ( 2 + 2 cos( θy) ) ⋅ Ast = 227

C"!%'!o &e ! %'nt$ ?s

= 45

 bc θy := !t!%     dc 

θy⋅

180

π

= 45

 Are &e re#'er(o &e %on#$n)$ento en ! &$re%%$*n 7  Are &e re#'er(o &e %on#$n)$ento en ! &$re%%$*n 

A" ρ" :=   = 0.008 s⋅ dc

Ay ρy :=   = 0.008 s⋅ bc

2

 #´i  3 8⋅ = 4.356 × 10   6  

ρs := ρ" + ρy  = 0.016

#´i = 140

C"!%'!o &e es#'er(os e#e%t$8os &e %on#$n)$ento % := 8

ΣAi := % ⋅

400 

n9)ero &e "res no %on#$n&s

 #´ 2  i

 

6

 

 

       0        4

= 26133        0        4        1        0        0        4

ρcc =  0.041

 1 − s´   ΣAi     s´     2 ⋅ dc  ⋅ 1− ⋅ &' :=  1 −    bc⋅ dc    2⋅ bc  (1 − ρcc)

       0        4        1

140

&' = 0.662

140  

1 f´$ := ⋅ &'⋅ ρs⋅ f yt  = 2.37 2

Res$sten%$ &e! %on%reto %on#$n&o f´$ f´$     f´cc := f´co⋅   −1.254 + 2.254⋅ 1 +  7.94⋅ − 2⋅ = 64.753 f´co f´co    

De#or)%$*n &e! %on%reto  )"7$) res$sten%$  %o),res$*n &e! %on%reto %on#$n&o   f´cc ε cc := ε co⋅ 1 + 5 ⋅       − 1 = 0.0062   f´co  

ε su := .12

De#or)%$*n 9!t$) &e! %ero &e %on#$n)$ento

ε cu :=  0.004 +

(1.4⋅ ρ s⋅ f yt⋅ εsu)   f´cc

= 0.023

ε c := 0 , 0.000001 .. ε cu 4

Ec := 5000⋅ f´co = 3.536 × 10   f´cc 4 Es'c := = 1.046 × 10 ε cc  :=

Ec

  = 1.42 Ec − Es'c

De#or)%$*n 9!t$) &e! %on%reto %on#$n&o

fcc( εc ) :=

εc f´cc⋅     ⋅   ε cc

   

  εc   −1+  εcc 

 

80

60

( )

fcc εc 40

20

0

0

0.01

0.02

εc 80

60

( ) 40 fcc( εc) fcc εc

20

0

0

0.01

0.02

εc

Re#eren%$s b$b!$$ogr"#$%s@ 1.- PrR Pr$st!e N.  G$!!  . 1324 BD'%t$!$t o# S+'re-Con#$ne& Con%rete Co!')nsB  o'rn! Str'%t'r! D$8$s $on ASCE Abr$! 132 8o! 103 ,,.2-50 2.- Mn&er  . Pr$est!e N.  Pr R. 1334 BTeoret$%! St ress Mo&e! #or Con#$ne& Con%reteB o'rn! Str'%t'r! D$8$s$on ASCE 1134,, 130 -132