Viga 40 Lorenzo

(1)

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

hh:=:=2.2.22

Ancho ala inferior

bb_b_b

:=

8080cc

Ancho ala superior

bb_tt

:=

1.21.2

Espesor ala inferior

tt_b_b

:=

1919cc

Espesor ala superior

tt_tt

:=

1111cc

Espesor ala inclinada inferior

t´t´_b_b

:=

1515cc

Espesor ala inclinada

t´t´_tt

:=

1010cc

Espesor del alma

bb_w_w

:=

2020cc

Peso especifico del H°

γ

cc 200200 kgf kgf m m!!

:=

"ongitud de calculo de la viga

LL:=:=!9.0!9.0⋅⋅

Altura de la losa

hh_f_f

:=

0.0.22

⋅⋅

#umeor de vigas

NN_vigas_vigas

:=

22

$eparacion entre vigas

SS:=:=2.2.55⋅⋅

%raccion de carga

(2)

Area

A_cp b_b

⋅

t_b bb

−

bw 2

⋅

t´b

+

b_t

⋅

t_t bt

−

bw 2

⋅

t´t

+

(

h t

−

_t

−

t_b

)

⋅

b_w

:=

A_cp

=

&590 cm

⋅

2 A_cp

=

11&'.52 in

⋅

2

Peso propio

W_o

:=

A_cp

⋅

γ

c W_o 1821.' kgf m

⋅

=

Localizacion del eje neutro de la seccion

Area

A1

:=

b_b

⋅

t_b A1

=

0.152m2

(ra)o

y1 h tb 2

−

:=

y1

=

2.105m

*omento estatico

A1 y1

⋅

=

0.!2 m! A1 y1

⋅

2

=

'&!51580 cm

⋅



+nercia

I1 b_b tb ! 12

⋅

:=

I1

=

5&2'.''& cm

⋅



Area

A2 bb

−

bw 2

⋅

t´b

:=

A2= 0.05m2

(ra)o

y2 h t

−

_b t´b !

−

:=

y2 =1.9' m

*omento estatico

A2 y2⋅ =0.088 m! A2 y2⋅ 2 =1&28&200 cm⋅ 

+nercia

I2

(

b_b

−

b_w

)

t´b ! !'

⋅

:=

I2

=

5'25 cm

⋅



Area

A3

:=

b_t

⋅

t A3= 0.1!2m2

(ra)o

y3 tt 2

:=

y3

=

0.055m

*omento estatico

A3 y3⋅ =&2'0 cm⋅ ! A3 y3⋅ 2 =!99!0 cm⋅ 

(3)

+nercia

I3 b_t tt ! 12

⋅

:=

I3=1!!10 cm⋅ 

Area

A4 bt

−

bw 2

⋅

t´

:=

A4

=

0.05 m2

(ra)o

y4 t_t t´t !

+

:=

y4=0.1!m

*omento estatico

A4 y4⋅ =&1''.''& cm⋅ ! A4 y4⋅ 2 =102&22.222 cm⋅ 

+nercia

I4

(

b_t

−

b_w

)

t´t ! !'

⋅

:=

I4

=

2&&&.&&8 cm

⋅



Area

A5

:=

(

h t

−

_t

−

t_b

)

⋅

b_w A5= 0.!8 m2

(ra)o

y5 t_t h t

−

t

−

tb 2

+

:=

y5

=

1.0' m

*omento estatico

A5 y5

⋅

=

0.0! m! A5 y5

⋅

2

=

2'9'800 cm

⋅



+nercia

I5

(

h t

−

t

−

tb

)

! 12

⋅

bw

:=

I5=11!1'''.''& cm⋅ 

,oordenadas del centro

de gravedad

c1p A1 y1

⋅

+

A2 y2

⋅

+

A3 y3

⋅

+

A4 y4

⋅

+

A5 y5

⋅

A1 A2

+

A3

+

A4

+

A5

:=

c_1p

=

108.&& c

⋅

c_2p

:=

h c

−

_1p c_2p

=

111.25! c

⋅

Momento de inercia

I_cp

:=

I1 I2

+

I3

+

I4

+

I5

+

A1 y1

⋅

2

+

A2 y2

⋅

2

+

A3 y3

⋅

2

+

A4 y4

⋅

2

+

A5 y5

⋅

2

−

c_1p2

⋅

A_cp I_cp

=

9219!&.'2 cm

⋅



(4)

Modulo de la seccion

S_1p Icp c_1p

:=

S1p 52'05.8'8 cm !

⋅

=

S_2p Icp c_2p

:=

S_2p

=

20&.5'& cm

⋅

!

Resumen de las propiedades

h

=

2.2 m b_b

=

0.8m W_o 1821.' kgf m

⋅

=

c_2p

=

111.25! c

⋅

γ

c 200 kgf m!

⋅

=

b_t

=

1.2m t_b

=

0.19 m A_cp

=

0.&59 m2 S_1p

=

0.5! m! I_cp

=

0.92 m t_t

=

0.11 m t´_b

=

0.15 m c_1p

=

108.&& c

⋅

S_2p

=

0.2 m! t´_t

=

0.1 m b_w

=

0.2m

Propiedades de los Materiales

Resistencia a la rotura de la losa:

f´_closa 210 kgf cm2

⋅

:=

Resistencia a la rotura de la viga:

f´_cviga !50 kgf cm2

⋅

:=

Factor de Corrección de resistencia:

η

c f´_cviga f´_closa

:=

η

c

=

1.291 Cables de preesfuerzo f _p 18&29 kgf cm2

⋅

:=

-2&0 /

f _pi

:=

0.&0 f

⋅

_p f _pi 1!110.! kgf cm2

⋅

=

f _pe

:=

0.82 f

⋅

_p f _pe 10&50.' kgf cm2

⋅

=

(5)

Propiedades de la sección compuesta

L la longitud de la viga, h 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: L=!9. m h_f

=

0.2m S =2.5m b_e L 

:=

b_e

=

9.85 m NN L  12 h

⋅

_f

+

b_t S









:=

NN 9.85 !.' 2.5









m

=

b_e

:=

12 h

⋅

_f

+

b_t b_e

=

!.'m b_e

:=

S b_e

=

2.5m

entonces

b_e

:=

min NN- / b_e

=

2.5 m Area "fectiva de la losa:

b_e

=

2.5m h_f

=

0.2m

A_L b_e hf

ηc

⋅

:=

A_L

=

0.!8&m2 Para la seccion compuesta

c_2c A_cp

⋅

c_2p A_L h hf 2

+









⋅

+

A_cp

+

A_L

:=

c_2c

=

1.51 m

,oordenadas del centro

de gravedad de la seccion

compuesta

c_1c

:=

h c

−

_2c c_1c

=

0.'8' m c_3c

:=

c_1c

+

h c_3c

=

0.88' m

+nercia de la

seccion

compuesta

I_cc I_cp

+

A_cp

⋅

(

c_2c

−

c_2p

)

2 be

ηc

h_f! 12

⋅

+

A_L c_1c hf 2

+









2

⋅

+

:=

I_cc

=

0.855 m

*odulo de la seccion de

la seccion compuesta

S_1c Icc c_1c

:=

S_1c

=

1.2' m! S_2c Icc c_2c

:=

S_2c

=

0.5'5 m! S_3c Icc c_3c

:=

S_3c

=

0.9'5 m! f _e

=

1.92

(6)

Cargas

rae prefaricada

Wo 1821.' kgf m

⋅

=

"osa

W_!p 200 kgf m!

⋅

h_f

⋅

S

:=

W_!p 1200 kgf m ⋅ =

Carga viva AA()*+ Factor de impacto 15.2 m⋅ L+ !8 m⋅ =0.19& L=!9. m "_inpacto 1 15.2 m⋅ L+ !8 m⋅ + := "_inpacto

=

1.19& Momentos Ma,imos

Por peso propio

#_o W_o L

2

8

⋅

:=

#_o

=

!5!&2.!&2 k gf

⋅

Por losa #umeda

#_!p W_!p L 2 8

⋅

:=

#_!p

=

2!285 kgf

⋅

Por diafragma h´

:=

(

h t

−

_b

)

h=2.2m

Segn las lineas !e inflencia tenemos%

#_& h´ 0.20

⋅

m S 1

⋅

-5m

+

10m

+

5m/

⋅

200kgf m!

:=

#_&

=

820 m kgf

⋅

Por capa de rodadura

#_'o! S 5

⋅

kgf m2

⋅

L 2 8

⋅

:=

#_'o!

=

218!0.0'! k gf

⋅

Por bordillo aceras y pasamanos

(_ace'a 180 kgf m ⋅ :=

acera

(_bo'!illo 25!.8 kgf m

⋅

:=

ordillo

!00 kgf m

⋅

arandado

(7)

#_sp (_ace'a

+

(_bo'!illo !00 kgf m

⋅

+









L2 8

⋅

2 N_vigas

⋅

:=

#_sp

=

12!90.221 k gf

⋅

Por carga viva

#

:=

1.25 288800

⋅

kgf

⋅

)S * 25

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

#_imp # 1

2

⋅

"_inpacto

⋅

f _e

:=

#_imp

=

!22!!2.1! m kgf

⋅

#_imp

=

!22!!2.1! k gf

⋅

"_inpacto

=

1.19& -umero re.uerido de torones

e_c

:=

c_2p

−

0.1

⋅

e_c

=

0.89! m f ₂

−

(e A_cp (_e e

⋅

_c S_2p

−

#o S_2p

+

#!p S_2p

+

#imp S_2c

+

#& S_2c

+

#'o! S_2c

+

#sp S_2c

+

:=

(e

#_sp

+

#_'o!

+

#_&

+

(

#_!p

)

+

( )

#_o

=

&98&8'.'55 k gf

⋅

f´_cviga !50 kgf

cm2

⋅

=

+,A-  #

⋅

(

sp

+

#'o!

+

#&

+

#!p

+

#o

+

#imp

)

L

:=

f _ts 1.' f´_cviga kgf cm2 ⋅ ⋅ := f _ts 29.9!! kgf cm2

⋅

=

+,A-=125.' tonf ⋅ (_e 1 1 A_cp e_c S_2p

+

#_o S_2p #_!p S_2p

+

#imp S_2c

+

#& S_2c

+

#'o! S_2c

+

#sp S_2c

+

−

f _ts





⋅

:=

(_e

=

591521.!!8 k gf

⋅

. ! +,A-

⋅

+

0tonf 8.m 1

⋅

kgf cm2 .9& m

=

:=

(_i (e 0.82

:=

(_i

=

&21!'&.85 kgf

⋅

A_p (i f _pi

:=

A_p

=

55.02! cm

⋅

2 Nt Ap 0.98cm2

:=

Nt=5'.1' Nt:='0 /sa' A_p

:=

Nt 0.98

⋅

cm2 A_p

=

58.8 cm

⋅

2

(8)

"sfuerzo de Fle,ion

(_i

:=

f _pi A

⋅

_p (_i

=

&&0885.' k gf

⋅

(_e

:=

f _pe A

⋅

_p (_e

=

'!212'.225 k gf

⋅

"sfuerzos del Concreto en la transferencia t/0

"n el e,tremo e_e

:=

20 c

⋅

f ₁

−

(i A_cp (i ee S_1p

⋅

+

:=

f ₁

−

'&.502 kgf cm2

⋅

=

f ₂

−

(i A_cp (i e_e S_2p

⋅

−

:=

f ₂

−

1!'.1' kgf cm2

⋅

=

"n el centro del claro

f ₁

−

(i A_cp (i e_c S_1p

⋅

+

#o S_1p

−

:=

f ₁

−

2&.'5 kgf cm2

⋅

=

f ₂

−

(i A_cp (i ec S_2p

⋅

−

#o S_2p

+

:=

f ₂

−

1&&.191 kgf cm2

⋅

=

Contra los siguientes esfuerzos admisibles

f _ci

:=

−

0.8

⋅

0.'

⋅

f´_cvig f _ci

−

1'8 kgf cm2

⋅

=

-omp'esion f _ti 1.1 kgf cm2

⋅

:=

0ension

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 A_cp (e e_c 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 cm2

⋅

=

(9)

Parte inferior de la seccion prefabricada f ₂

−

(e A_cp (e e_c S_2p

⋅

−

#o S_2p

+

#!p S_2p

+

#imp S_2c

+

#& S_2c

+

#'o! S_2c

+

#sp S_2c

+

:=

f ₂ 1'.!92 kgf cm2

⋅

=

f ₃

−

#imp S3c η

⋅

_c

:=

f ₃

−

25.8&8 kgf cm2

⋅

=

f ₃

Parte superior de la losa

f ₄

−

#imp S1c η

⋅

_c

:=

f ₄

−

20.0!8 kgf

cm2

⋅

=

f

Parte inferior de la losa

Contra los siguientes esfuerzos admisibles

f _cs

de la viga -comp./

−

0.0

⋅

f´_cviga

−

10 kgf cm2

⋅

=

f _cs

de la losa -comp./

−

0.0

⋅

f´_closa

−

8 kgf cm2

⋅

=

f _ts

de la viga -tens./

1.' f´_cviga kgf cm2

⋅

29.9!! kgf cm2

⋅

=

Los esfuerzos calculados despues de las perdidas son satisfactorios

Momento de agrietamiento f _' 1.989 f´_cviga kgf cm2

⋅

:=

ecuacion 2

f _' !&.211 kgf cm2

⋅

=

#_c' (_e S2c A_cp

⋅

(_e

⋅

e_c S2c S_2p

⋅

+

f _' S

⋅

_2c

:=

#_c'

=

10100.859 k gf

⋅

Calculo del factor de seguridad contra el agrietamiento:

"_c' #c'

−

#o

−

#!p

−

#&

−

#'o!

−

#sp #_imp

:=

(10)

Resistencia a fle,ion 4 Momento ultimo f _pe

f _p

=

0.5&

>

0.50

Por lo tanto usar ecuacion 2' para

f _ps !

:=

c_1p

+

h_f

+

e_c b_e

=

2.5m !

=

2.18 m

ρ

ρ A_p b_e

⋅

!

:=

f _ps f _p 1

ρ

_ρ f p 2 f´

⋅

_cviga

⋅

−





⋅

:=

f _ps 18188.!55 kgf cm2

⋅

=

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 cm2

⋅

=

b_e

=

2.5 m A_p f ps b_e

η

c !

⋅





f´_closa

⋅

=

0.121

ecuacion !'

0.0&5

<

0.!0

usar ecuacion para vigas surefor)adas

#_n A_p

⋅

f _ps ! a 2

−





⋅

:=

#_n

=

2255'!.5!' k gf

⋅

φ _:=1 #_ 1.!0

φ

#o

+

#!p

+

#&

+

#'o!

+

#sp 5 !

⋅

#imp

+





⋅

:=

_{ecuacion 18}

#_

=

1&!'808.9'1 k gf

⋅

#_n

>

# ok

!

ecuacion 1'

(11)

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

φ

#n #_c'

⋅

>

1.2 #_n

=

2255'!.5!' k gf

⋅

φ

#n #_c'

(12)

Cortante en el alma

Cortante de los cuartos del claro Por peso propio

_o W_o 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´ 0.2

⋅

m S 1

⋅

200 kgf m!

⋅





⋅

-1

+

0.&5

+

0.5

+

0.25/ 1 L

⋅

:=

W_& 15!.0' kgf m

⋅

=

_& W_& L  ⋅ := _&

=

150&.5 k gf

⋅

Por capa de rodadura

W_'o! 125 kgf m

⋅

:=

_'o! W_'o! L 

⋅

:=

_'o!

=

12!1.25 k gf

⋅

Por acera bordillo y pasamanos

W_sp (_ace'a + (_bo'!illo !00 kg f m ⋅ + := W_sp &!!.8 kgf m ⋅ = _sp W_sp L  ⋅ := _sp

=

&22&.9! k gf

⋅

Por carga viva

5_:=1.25 !0850_⋅ _⋅_{kg f}

)S *25

_AAS)06

_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

⋅

_c

:=

0.0' f´

⋅

_cviga

⋅

b_w

⋅ ⋅

77 _c 0.0' f´⋅ _cviga⋅b_w ! a 2 −





⋅ := _c

=

8850.!05 k gf

⋅

A_v

:=

2 0.&9

⋅

cm2 φ18 f _y 2800 kgf cm2

⋅

:=

(13)

s_'e 2 f

⋅

_y

⋅

A_v ! a 2

−

_

−

_c

⋅

:=

s_'e

=

9.'2 c

⋅

s_ma9 A_v f y &.0! kgf cm2

⋅

b_w

⋅

:=

s_ma9

=

!1.'5 c

⋅

e10 c 420

Cortante #orizontal h_f

=

0.2m b_t

=

1.2m : be

η

c h_f

⋅

c_3c hf 2

−









⋅

:=

:= 0.!05 m! v_ _ : I_cc

⋅

b_t

⋅

:=

v_ !.&& kgf cm2

⋅

=

!.8'8<21.1 co''ecto! Armadura de piel h=2.2m 100 As b_w

⋅

-2 !

⋅

−

h/

⋅

≥

0.05 As 0.05 b⋅ w 2 !⋅ −h 100 ⋅ := A_s

=

2.1' cm

⋅

2

φ

18

c 420

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

+

#_!p

+

#_&

+

#_'o!

+

#_sp

=

&98&8'.'55 k gf

⋅

L:=000 c⋅ δ_a!m L 800 :=

δa!m

=

5 c

⋅

f´_c !50 kgf cm2

⋅

:=

I_cp

=

9219!&.'2 cm

⋅

 γ :=2. ,_c γ 1.5⋅200 f´_c kgf cm2 ⋅ ⋅ :=

(14)

,_c 2921'.155 kgf cm2

⋅

=

5ebido a las cargas muertas: _! 8 L2 #_o

+

#_!p

+

#_&

+

#_'o!

+

#_sp

(

)

⋅

:=

_! !9.9!9 kgf cm

⋅

=

_δ

! 5 !8

⋅

! L ,_c I

⋅

_cp

⋅

:=

δ

!

=

9.259 c

⋅

eido a la carga viva

#=!'1000 kgf ⋅ ⋅

_L 8 # 2 L2

:=

_L 9.025 kgf cm

⋅

=

δ

L 5 !8

⋅

L L ,_c I

⋅

_cp

⋅

:=

δL

=

2.092 c

⋅

5ebido al preesfuerzo  (_e e

⋅

_c 8 L2

⋅

:=

 2820.9&1 kgf m

⋅

=

δ

pi 5 

⋅

L !8 1 ,_c I

⋅

_cp

⋅

:=

δ

pi

=

'.5!9 c

⋅

Flec#a final

δ

fi

:=

δ

!

+

δ

L

−

δ

p

δ

fi

=

.811 c

⋅

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

:=

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

=

'0 ;1 y cm c

:=

+

_2p ;1

=

1&1.25! c

⋅

;2

:=

y cm

−

!0cm

+

c_2p ;2 = 11.25! c⋅ ;3

:=

y cm

−

'0cm

+

c_2p ;3 = 111.25! c⋅ ;4

:=

y cm

−

90cm

+

c_2p ;4

=

81.25! c

⋅

;5

:=

y cm

−

120cm

+

c_2p ;5 = 51.25! c⋅ c_2p

=

1.11! m

(15)

*rayectoria de los cables

La ecuación general es

; 2 L2 ;_a

−

2 ;

⋅

_b

+

;_c

(

)

⋅

<2 1 L

⋅

(

−

!

⋅

;a

+

 ;

⋅

b

−

;c

)

⋅

<

+

;_a

6A+#A 1

;_a

:=

;1 ;_b

:=

0.09 ;_c

:=

;1

A - 0  0.0 /

;_a

=

1&1.25! c

⋅

;_c

=

1&1.25! c

⋅

( - 1!.5  0.09 /

L:=0

, - 2&  0.0 /

A 2 L2 ;_a

−

2 ;

⋅

_b

+

;_c

(

)

⋅

:=

. 1 L⋅

(

−!⋅;a+  ;⋅ b −;c

)

:= -

:=

;_a A=0.0005 m . =−0.1'181!m -=1.&125! m ; A <

:=

⋅

<2

+

. <

⋅

+

- ; 0.0005<2

−

0.1'181! <

⋅

+

1.&125!

6A+#A 2

;_a

:=

;2 ;_b

:=

0.1'& ;_c

:=

;2

A - 0  0.&0 /

( - 1!.5  0.1'& /

L=0 ;2 =1.1! m

, -2&  0.&0 /

A 2 L2 ;_a

−

2 ;

⋅

_b

+

;_c

(

)

⋅

:=

. 1 L

⋅

(

−

!

⋅

;a

+

 ;

⋅

b

−

;c

)

:=

-

:=

; A

=

0.00!1128 m .

=

−

0.1251!m -

=

1.125! m ; 0.00!1128 <⋅ 2− 0.1251! <⋅ + 1.125!

6A+#A !

;_a

:=

;3 ;_b

:=

0.1'& ;_c

:=

;3

A - 0  100 /

( - 1!5  0.1'& /

L

=

0 ;3

=

1.11! m

, -2&  100 /

(16)

A 2 L2 ;_a

−

2 ;

⋅

_b

+

;_c

(

)

⋅

:=

. 1 L

⋅

(

−

!

⋅

;a

+

 ;

⋅

b

−

;c

)

:=

-

:=

; A

=

0.002!'28 m .

=

−

0.0951!m -

=

1.1125! m ; 0.002!'28 <⋅ 2− 0.0951! <⋅ + 1.1125!

6A+#A 

;_a

:=

;4 ;_b

:=

0.1'& ;_c

:=

;4

A - 0  100 /

( - 1!5  0.1'& /

L=0 ;4 =0.81! m

, -2&  100 /

A 2 L2 ;_a

−

2 ;

⋅

_b

+

;_c

(

)

⋅

:=

. 1 L

⋅

(

−

!

⋅

;a

+

 ;

⋅

b

−

;c

)

:=

-

:=

; A

=

0.001'128 m .

=

−

0.0'51!m -

=

0.8125! m ; 0.001'128 <⋅ 2 −0.0'51! <⋅ + 0.8125!

6A+#A 5

;_a

:=

;5 ;_b

:=

0.1'& ;_c

:=

;5

A - 0  100 /

( - 1!5  0.1'& /

L=0 ;5 =0.51!m

, -2&  100 /

A 2 L2 ;_a

−

2 ;

⋅

_b

+

;_c

(

)

⋅

:=

. 1 L⋅

(

−!⋅;a +  ;⋅ b−;c

)

:= -

:=

;_a A=0.0008'28 m . =−0.0!51!m -=0.5125! m ; 0.0008'28 <

⋅

2

−

0.0!51! <

⋅

+

0.5125!

(17)

! continuación se presenta las coordenas correspondientes a las vainas, "ue estan redondeadas a los tres decimales.

PR+6R"(78A +R5"-A5A +R5"-A5A +R5"-A5A +R5"-A5A +R5"-A5A CA5A 10 cm 8A7-A 2 8A7-A 9 8A7-A  8A7-A 3 8A7-A 1

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$'.## #.*'# #.*% #.*)* #.**) #.$() $'.)# #.*$% #.*+* #.*& #.*$+ #.$&) $).## #.$) #.*') #.**+ #.*#( #.$(# $).)# #.$%+ #.*&# #.*$) #.*## #.$(' $+.## #.$) #.*$% #.*#) #.$& #.$($* $+.)# #.$'' #.*#+ #.$+ #.$(% #.$%(# $%.## #.$&$ #.$) #.$( #.$(* #.$%)* $%.)# #.$*# #.$(% #.$(* #.$%% #.$%*( $(.## #.$$# #.$(# #.$%% #.$%' #.$%#(

$(.)# #.$#& #.$%' #.$%& #.$%$ #.$+&

$.## #.#( #.$%$ #.$%# #.$+ #.$+(&

$.)# #.#) #.$+( #.$+( #.$+( #.$+%+

*#.## #.#' #.$+% #.$+% #.$+% #.$+%'