Viga 40 Lorenzo

DISEÑO DE LA VIGA POSTENSADA "PROYECTO PUENTE VEHICULAR Y ACCESOS SAN LORENZO " L=40 mts. (lu ! #t$%&' HS)* AASHTO  Propiedades:de la viga

Altura de la viga

2.2 h := 2.2

Ancho ala superior 

bt

Espesor ala superior 

tt := 11c 11c

Espesor ala inclinada inferior  t´b := 15c 15c

Espesor ala inclinada

10c t´t := 10c

Espesor del alma

:= 1.2

Peso especifico del H°

Ancho ala inferior 

bb

Espesor ala inferior 

19c tb := 19c

 kgf 

γ c := 200

!

m

"ongitud de calculo de la viga

L := !9.0 ⋅

Altura de la losa

0.2 ⋅ hf  := 0.2

 #umeor de vigas

Nvigas := 2

$eparacion entre vigas

2.5 ⋅ S := 2.5

%raccion de carga

f e := 1.92

bw

80c := 80c

20c := 20c

Area Acp := bb ⋅ tb

+

bb

− bw 2

⋅ t´b + bt ⋅ tt +

bt

− bw

⋅ t´t + ( h − tt − tb) ⋅ bw

2

Acp

= &590 cm ⋅ 2

Acp

= 11&'.52 in ⋅ 2

Peso propio Wo := Acp ⋅ γ c

= 1821.' ⋅

Wo

kgf  m

Localizacion del eje neutro de la seccion Area (ra)o

A1 := bb ⋅ tb

A1

tb

y1 := h

−

A1 ⋅y1

= 0.!2 m!

A1 ⋅y1

2

I1 := bb ⋅

A2 :=

= 2.105m

= '&!51580 cm ⋅ 

+nercia

Area

y1

2

*omento estatico

= 0.152m2

bb

tb

!

I1

12

− bw 2

= 5&2'.''& cm ⋅ 

2

⋅ t´b

A2 = 0.05m

(ra)o y2 := h

− tb −

t´b

y2 = 1.9' m

!

*omento estatico !

A2 ⋅y2 = 0.088 m 2



A2 ⋅y2 = 1&28&200 ⋅ cm

+nercia I2 := ( bb

− bw ) ⋅

t´b

! 

⋅ I2 = 5'25 cm

!'

Area 2

A3 := bt ⋅ t

A3 = 0.1!2m

(ra)o y3 :=

*omento estatico

tt

y3

2 !

A3 ⋅y3 = &2'0 ⋅ cm 2



A3 ⋅y3 = !99!0 ⋅ cm

= 0.055m

+nercia

!

tt

I3 := bt ⋅

12

Area

− bw

bt

A4 :=



I3 = 1!!10 ⋅ cm

2

(ra)o y4 := tt

+

⋅ t´

A4

t´t

= 0.05 m2

y4 = 0.1!m

!

*omento estatico

!

A4 ⋅y4 = &1''.''& ⋅ cm 2



A4 ⋅y4 = 102&22.222 ⋅ cm

+nercia

!

t´t



I4 := ( bt

− bw ) ⋅

A5 := ( h

− tt − tb) ⋅ bw

⋅ I4 = 2&&&.&&8 cm

!'

Area 2

A5 = 0.!8 m

(ra)o h − tt

+

A5 ⋅y5

= 0.0! m!

A5 ⋅y5

+nercia I5 :=

2

12

A1 ⋅y1

c2p := h

= 1.0' m

= 2'9'800 cm ⋅ 

( h − tt − tb )

c1p :=

y5

2

*omento estatico

,oordenadas del centro de gravedad

− tb

y5 := tt

− c1p

+

! 

⋅ bw

I5 = 11!1'''.''& ⋅ cm

A2 ⋅ y2

+ A3 ⋅ y3 + A4 ⋅ y4 + A5 ⋅ y5 A1 + A2 + A3 + A4 + A5 c2p

c1p

= 108.&& c ⋅

= 111.25! c ⋅

Momento de inercia Icp :=   I1

+

I2

+ I3 + I4 + I5 + A1 ⋅ y12 + A2 ⋅ y22 + A3 ⋅ y32 + A4 ⋅ y42 + A5 ⋅ y52 − c1p2 ⋅ Acp Icp

= 9219!&.'2 cm ⋅ 

Modulo de la seccion S 1p :=

S 2p :=

Icp c1p

Icp c2p

S 1p

= 52'05.8'8 cm ⋅ !

S 2p

= 20&.5'& cm ⋅ !

Resumen de las propiedades h

= 2.2 m

bb

= 0.8m

bt

= 1.2m

tb

tt

= 0.11 m

t´b

= 0.15 m

= 0.1 m

bw

= 0.2m

t´t

= 0.19 m

Wo

= 1821.' ⋅

kgf  m

c2p

γ c = 200⋅

= 111.25! c ⋅

kgf  !

m

Acp

= 0.&59 m2

S 1p

= 0.5! m!

c1p

= 108.&& c ⋅

S 2p

= 0.2 m!

Icp

= 0.92 m

 Propiedades de los Materiales Resistencia a la rotura de la losa: kgf 

f´closa := 210 ⋅

2

cm

Resistencia a la rotura de la viga: kgf 

f´cviga := !50⋅

2

cm

Factor de Corrección de resistencia:

ηc :=

f´cviga

ηc = 1.291

f´closa

Cables de preesfuerzo f p

:= 18&29⋅

kgf 

-2&0 /

2

cm

⋅p f pi := 0.&0 f  f pi

= 1!110.! ⋅

kgf  2

cm

⋅p f pe := 0.82 f  f pe

= 10&50.' ⋅

kgf  2

cm

 Propiedades de la sección compuesta h

 L la longitud de la viga,  el espesor de la losa y s la separación entre vigas todo en metros! "l anc#o efectivo del pat$n %be& ser' el menor de: hf  = 0.2m

L = !9. m

be :=

L

  L       NN := ⋅ f  + bt    12 h

be

= 9.85 m

⋅ f  + bt be := 12 h

be

= !.'m

be := S

be

= 2.5m

entonces

be := min - NN/



S = 2.5m

 9.85   NN = !.' m    2.5

S be

= 2.5 m

Area "fectiva de la losa: be

= 2.5m

AL := be ⋅

hf  = 0.2m

hf 

AL

ηc

= 0.!8&m2

Para la seccion compuesta Acp ⋅ c2p c2c :=

,oordenadas del centro de gravedad de la seccion compuesta

+nercia de la seccion compuesta

Icc := Icp

+

Acp ⋅ ( c2c − c2p )

− c2c

c1c

= 0.'8' m

c3c := c1c + h

c3c

= 0.88' m

be hf 

ηc

  2  

hf 

= 1.51 m

c1c := h

+

 h +  

c2c

!

2

+ AL ⋅

⋅

12

Acp

  + AL ⋅ c1c +  

*odulo de la seccion de la seccion compuesta

+ AL

2

  2  

hf 

S 1c :=

S 2c :=

S 3c :=



Icc = 0.855 m

Icc c1c Icc c2c Icc c3c

f e = 1.92

!

S 1c = 1.2' m

!

S 2c = 0.5'5 m

!

S 3c = 0.9'5 m

Cargas

= 1821.' ⋅

kgf 

:= 200⋅

kgf 

rae prefaricada

Wo

"osa

W!p

m

!

⋅ hf ⋅ S

W!p = 1200 ⋅

m

Carga viva AA()*+ Factor de impacto 15.2 ⋅ m

= 0.19&

L + !8⋅ m

"inpacto := 1 +

L = !9. m

15.2 ⋅ m

"inpacto

L + !8⋅ m

= 1.19&

Momentos Ma,imos Por peso propio 2

#o := Wo ⋅

L

#o

8

= !5!&2.!&2 k ⋅ gf ⋅

Por losa #umeda 2

#!p

:= W!p ⋅

L

8

#!p

= 2!285 kgf  ⋅ ⋅

Por diafragma h´ := ( h − tb)

h = 2.2m

Segn las lineas !e inflencia tenemos% #& := h´⋅ 0.20m⋅

S 1

⋅ - 5m + 10m + 5m/ ⋅ 200

kgf 

#&

!

= 820 m⋅ kgf 

m

Por capa de rodadura 2

#'o! := S ⋅ 5⋅

kgf  L 2

m

⋅

8

#'o!

= 218!0.0'! k ⋅ gf ⋅

Por bordillo aceras y pasamanos (ace'a := 180 ⋅

kgf 

(bo'!illo := 25!.8 ⋅ !00⋅

kgf  m

acera

m

kgf  m

 ordillo  arandado

kgf  m

2

2 kgf    L   ⋅ ⋅ #sp := (ace'a + (bo'!illo + !00 ⋅ m   8 Nvigas  

#sp

= 12!90.221 k ⋅ gf ⋅

Por carga viva

# := 1.25 ⋅ 288800kgf ⋅

,arga estandar para camiones de la AA$H3

)S * 25 #imp := #⋅ #imp

1 2

⋅ "inpacto ⋅ f e

#imp

= !22!!2.1! k ⋅ gf ⋅

= !22!!2.1! m⋅ kgf 

"inpacto

= 1.19&

-umero re.uerido de torones ec := c2p

f 2 :=

−(e Acp

f´cviga

− 0.1 ⋅

−

ec = 0.89! m

(e ⋅ e c S 2p

#o

+

+

#!p

+

#imp

2

+,A- :=

cm

f ts := 1.'⋅ f´cviga ⋅

kgf 

f ts

2

 ⋅ ( #sp

1 1 Acp

+

ec

⋅

  #o +  S 2p

#!p S 2p

+

+

#'o!

+

#sp

#imp S 2c

+

+ #'o! + #& + #!p + #o +

= 29.9!! ⋅

kgf  2

cm

#& S 2c

+

#'o! S 2c

+

#sp S 2c

+,A- = 125.' ⋅ tonf 

⋅591521.!!8 k + 0tonf  ⋅ gf  ( =! +,A. :=e kgf  8.m⋅ 1

− f ts

S 2p

2

cm

(i :=

Ap :=

Nt :=

(e

(i

0.82

(i

Ap

f pi

Ap

= &21!'&.85 kgf  ⋅

= 55.02! cm ⋅ 2

Nt = 5'.1'

2

0.98cm

Nt := '0 2

Ap := Nt⋅ 0.98cm

Ap

#imp )

L

cm

(e :=

#&

S 2p S 2p S 2c S 2c S 2c S 2c ⋅ gf ⋅ #sp + #'o! + #& + ( #!p ) + ( #o) = &98&8'.'55 k

kgf 

= !50 ⋅

+

 

= 58.8 cm ⋅ 2

/sa'

= .9& m

"sfuerzo de Fle,ion

⋅ p (i := f pi  A

(i

(e := f pe ⋅ Ap

= &&0885.' k ⋅ gf 

⋅ gf  (e = '!212'.225 k

 "sfuerzos del Concreto en la transferencia t/0 "n el e,tremo

⋅ ee := 20 c f 1 :=

f 2 :=

−(i Acp

−(i Acp

+

(i ⋅

− (i ⋅

ee

f 1

S 1p

= −'&.502 ⋅

kgf  2

cm

ee

f 2

S 2p

= −1!'.1' ⋅

kgf  2

cm

"n el centro del claro f 1 :=

f 2 :=

−(i

+ (i ⋅

Acp

−(i

− (i ⋅

Acp

ec S 1p

ec S 2p

−

+

#o

f 1

S 1p

= −2&.'5 ⋅

kgf  2

cm

#o

f 2

S 2p

= −1&&.191 ⋅

kgf  2

cm

Contra los siguientes es fuerzos admisibles f ci := −0.8⋅ 0.'⋅ f´cvig

f ci

kgf 

= −1'8   ⋅ -omp'esion 2

cm f ti := 1.1 ⋅

kgf 

 

2

0ension

cm

 Los esfuerzos calculados en la transferencia son satisfactorios "sfuerzos del Concreto despues de las perdidas con carga viva en el centro del claro */infinito "cuaciones 12 y 13 Parte superior de la seccion prefabricada 

f 1 :=

−(e Acp

+ (e ⋅

ec S 1p

−

#o S 1p

−

#!p S 1p

−

#imp S 1c

−

#& S 1c

−

#'o! S 1c

−

#sp S 1c

f 1

= −1!1.09 ⋅

kgf  2

cm

Parte inferior de la seccion prefabricada

f 2 :=

−(e Acp

− (e ⋅

f 3 :=

f 4 :=

ec S 2p

+

#o S 2p

+

#!p S 2p

+

−#imp S 3c ⋅ η c

f 3

−#imp S 1c ⋅ η c

f 4

#imp S 2c

+

#& S 2c

= −25.8&8 ⋅

#'o!

+

S 2c

kgf  2

+

#sp

f 2

S 2c

= 1'.!92 ⋅

kgf  2

cm

f 3

Parte superior de la losa

f 

Parte inferior de la losa

cm

= −20.0!8 ⋅

kgf  2

cm

Contra los siguientes esfuerzos admisibles

de la viga -comp./

f cs

−0.0 ⋅ f´cviga = −10⋅

kgf  2

cm

de la losa -comp./

f cs

−0.0 ⋅ f´closa = −8⋅

kgf  2

cm

de la viga -tens./

f ts

1.'⋅ f´cviga ⋅

kgf  2

= 29.9!! ⋅

cm

kgf  2

cm

 Los esfuerzos calculados despues de las perdidas son satisfactorios Momento de agrietamiento

f ' := 1.989 ⋅ f´cviga ⋅

kgf 

ecuacion 2

2

cm f ' = !&.211 ⋅

#c' := (e ⋅

kgf  2

cm S 2c Acp

+ (e ⋅ e c⋅

S 2c S 2p

+ f '⋅ S 2c

⋅ gf ⋅ #c' = 10100.859 k Calculo del factor de seguridad contra el agrietamiento:

"c' :=

#c' − #o

"c' = 1.8'8

− #!p − #& − #'o! − #sp #imp

Resistencia a fle,ion 4 Momento ultimo f pe f p

= 0.5&

! := c1p be

+

> 0.50

Por lo tanto usar ecuacion 2' para f ps

hf  + ec

= 2.5m

! = 2.18 m

ρ ρ :=

Ap be ⋅ !

1 − ρ ⋅ ρ

f ps

:= f p ⋅

f ps

= 18188.!55 ⋅

f p

 

⋅ cviga 2 f´

kgf  2

cm a := ρ ρ ⋅ f ps ⋅

! 0.85 f´ ⋅ cviga

ecuacion 25 o !5

a = 1.!&9 ⋅ c a < 20

Por lo tanto usar ecuacion para vigas rectangulares f´closa

= 210⋅

kgf 

be

2

= 2.5 m

cm

Ap ⋅

f ps

 ⋅ ! ⋅ f´ closa be

= 0.121

ecuacion !'

ηc

0.0&5 < 0.!0

usar ecuacion para vigas surefor)adas



#n := Ap ⋅ f ps ⋅ ! − #n

a

 

2

= 2255'!.5!' k ⋅ gf ⋅

φ := 1 # :=

1.!0

φ

⋅  #o +

#!p

#

= 1&!'808.9'1 k ⋅ gf ⋅

#n

>#

+

#&

+ #'o! +

#sp

ok !

  + ⋅ #ecuacion imp 18 ! 5

ecuacion 1'

Comparacion del re.uisito de la AA()*+

φ⋅

#n

#n

φ⋅

>

#c'

1.2

= 2255'!.5!' k ⋅ gf ⋅ #n

#c'

= 1.'09

mayo' e 12 ok

 Cortante en el alma Cortante de los cuartos del claro Por peso propio o := Wo ⋅

L

o



= 1&92.&' k ⋅ gf 

Por losa #umeda !p := W!p ⋅

L

!p



= 11820 k ⋅ gf 

Por diafragmas

Segn lineas !e inflencia W& := h´⋅ 0.2m⋅

& := W& ⋅

S 1

⋅  200 ⋅

kgf  !

 ⋅ -1 + 0.&5 + 0.5 + 0.25/ ⋅ 1

W&

L

m

L

&



= 15!.0' ⋅

= 150&.5 k ⋅ gf 

Por capa de rodadura W'o! := 125 ⋅

kgf 

'o! := W'o! ⋅

m L

'o!



= 12!1.25 k ⋅ gf 

Por acera bordillo y pasamanos Wsp := (ace'a + (bo'!illo + !00 ⋅

kg f 

Wsp = &!!.8 ⋅

m

sp := Wsp ⋅

L

sp



kgf  m

= &22&.9! k ⋅ gf 

Por carga viva

)S *25 AAS)06

5 := 1.25⋅ !0850 ⋅ kg f 

imp :=  ⋅ 0.5⋅ "inpacto ⋅ f 

imp

= !!1.98 k ⋅ gf 

Cortante ultimo φ := 1  :=

1.!0

φ

⋅  o + !p + & + 'o! + sp +

5 !

⋅ imp   

= 12'250.82' k ⋅ gf 

⋅ cviga ⋅ bw ⋅ 7 ⋅ c := 0.0' f´



c := 0.0'⋅ f´cviga⋅ bw ⋅ ! −

a

 

⋅ gf  c = 8850.!05 k

2 2

Av := 2⋅ 0.&9 ⋅ cm f y := 2800 ⋅

kgf  2

cm

φ18

kgf  m

a

!−

⋅ y ⋅ Av ⋅ s 'e := 2 f  s ma9



2

s 'e

− c

f y

:= Av ⋅ &.0! ⋅

kgf  2

= 9.'2 c ⋅

s ma9

= !1.'5 c ⋅ e10 c 420

⋅ bw

cm

Cortante #orizontal

hf  = 0.2m

: :=

bt be

ηc

⋅ hf ⋅

= 1.2m

 c − hf    3c 2     !

: = 0.!05 m

:

v

:=  ⋅

v

= !.&&⋅

Icc⋅ bt kgf 

!.8'8 < 21.1   co''ecto!

2

cm

Armadura de piel h = 2.2m

100 ⋅

As

As := 0.05 ⋅ bw ⋅

As

≥ 0.05

⋅ − h/ bw ⋅ - 2 !

⋅ −h 2 ! 100

= 2.1' cm ⋅ 2 c 420

φ18

 Por cara

 5eterminación de Flec#as El cálculo de las deflexiones debidas a las cargas externas es similar al de las vigas no preesforzadas. Mientras el concreto no se agriete, la viga puede tratarse como un cuerpo homogeneo y se le aplica la teoría elástica usual para el cálculo de deflexiones. 5efle,ión Admisible #o L := 000 ⋅ c

f´c := !50 ⋅

+ #!p + #& + #'o! + #sp = &98&8'.'55 k ⋅ gf ⋅ δ a!m :=

L 800

kgf  2

cm

γ  := 2. 1.5

,c := γ 

δ a!m = 5 c ⋅ Icp

⋅ 200 ⋅ f´c ⋅

= 9219!&.'2 cm ⋅ 

kgf  2

cm

kgf 

,c = 2921'.155 ⋅

2

cm

5ebido a las cargas muertas: 8

! :=

2

⋅ ( #o + #!p + #& + #'o! +

#sp )

L !

kgf 

= !9.9!9 ⋅

cm

δ ! :=



5 !8

⋅ ! ⋅

L

δ ! = 9.259 c ⋅

,c ⋅ Icp

eido a la carga viva

# = !'1000 ⋅ kgf ⋅

#

L := 8

2

L

2

= 9.025⋅

L

δ L :=

kgf  cm



5 !8

⋅ L ⋅

L

,c ⋅ Icp

δ L = 2.092 c ⋅

5ebido al preesfuerzo

 := (e ⋅ e c ⋅

8

 = 2820.9&1 ⋅

2

L



δ pi := Flec#a final

δ fi := δ! + δ L − δ p

5  ⋅ L !8

⋅

kgf  m

1 ,c ⋅ Icp

δ pi = '.5!9 c ⋅

δ fi = .811 c ⋅

- − '0/ + 11.&'- y − 90/ + 11.&'- y − 120/ ⋅ + 11.&'- y − !0/ + 11.&' y f - y/ := 11.&' y

y := ! y := 'oot - f - y/ , y/

y = '0 ;1 := y cm + c2p

;1

= 1&1.25! c ⋅

;2 := y cm − !0cm + c2p

;2 = 11.25! ⋅ c

;3 := y cm − '0cm + c2p

;3 = 111.25! ⋅ c

;4 := y cm − 90cm + c2p

;4

;5 := y cm − 120cm + c2p

;5 = 51.25! ⋅ c

= 81.25! c ⋅ c2p

= 1.11! m

*rayectoria de los cables La ecuación general es 2

;

2

⋅ ( ;a − 2 ; ⋅ b + ;c) ⋅