CompuertasCompuertas

F R=∫AdF=∫A

PdA=∫Aρ gydA

P=ρ gh

P prom=ρ gh2

F R=P prom A=ρghA

2

Como la presión solo varía en y

F R=∫0

h

ρ gy a dy

A=ah

F R=ρgah2

2

F R=aρ g∫0

h

y dy

F R=∫AdF=∫A

PdA=∫Aρ gydA

F R=∫0

3

ρ gy a dy

F R=aρ g∫0

3

y dy

F R=1.5m 1000kg

m3 9.81m

s2 3m 2

2 =66.22 kN

F R=aρ g [ y2

2 ]0

3

Fuerza sobre áreas planas: Fuerza sobre áreas planas: superficies inclinadassuperficies inclinadas

αO

dF h

Superficie libre

y

hsen =α

αysenh =

∫=AdAgysenF αρ

y

F R=∫AdF=∫

APdA=∫

Aρ gy sen α dA

FR=aρg∫0

1.5

1 ysen α dy

F R=ρ gysen α∫AdA

F R=aρg [ yy2

2sen α ]

0

1 . 5

F R=3m 1000kg

m3 9.81m

s2 [1.5m1.5m 2

2sen 30 ]=60.7 kN

a=ancho de la compurta=3m

F R=∫AdF=∫A

PdA=∫Aρ gydA

m5.1=y

1.3m)30cos(5.1 ==x

F R=1000kg

m3 9.81m

s2 [1m1.5m

2sen 30 ]3m 1.5m =60.7 kN

F R=1000kg

m3 9.81m

s2 [1m0.75m

2 ]3m 1.5m=60.7 kN

h=1.5msen 30=0.75

30.35kN)7.60)(60cos( ==xF

52.56kN)7.60)(60sin( ==yF

wvvy gVgVwFF ρρ +=+=

52.6kN14.34kNkN25.38 =+=yFF R=60.7 kN

F x=30.35 kN

F y=52.56 kN

F x=F H= 1000kg

m3 9.81m

s2 [1m0.75m

2 ]3m 0.75m =30.35 kN

Fv= 1000kg

m3 9.81m

s2 1.3m1m 3m =38 .26 kN

w= 1000kg

m3 9 .81m

s2 [ 1.3m 0.75m

2 ]3m =14 .34 kN

F R=∫AdF=∫A

PdA=∫Aρ gydA

F x=∫Aρ gydA y

F y=∫Aρ gydA x

F x=∫0

1

ρ gy a dy

F x=aρ g∫0

1

y dy

F x=aρ g [ y2

2 ]0

1

F x=0.6m 1000kg

m3 9.81m

s2 [ 12

2 ]=2943 NF x=aρg∫0

1

1− y dy

F x=aρg [ y−y2

2 ]0

1

F x=0.6m 1000kg

m3 9.81m

s2 [1−12

2 ]=2943 N

F y=aρg∫0

1

1− y dx

Para la dirección y

F y=aρg∫0

1

1−x2 dx

F y=aρg [ x− x3

3 ]0

1

F y=0 .6m 1000kg

m3 9 .81m

s2 [1−13

3 ]=3924 N

F y=aρg∫0

1

y dy

F y=aρg [ 2y3 /2

3 ]0

1

F y=0 .6 1000kg

m3 9.81m

s2 [ 213/2

3 ]=3924 N

VgwFy ρ==

F y=w= 1000kg

m3 9. 81m

s2 21m 1m

3 0 .6m =3924N

V=A a

A=2h x

3

F y=w=ρ g2h x

3a

F R=F x2F y

2

F R=294323924 2

=4905N

o1.53N 2943

N 3924 tan 1 ==−θ

Centroides y Momentos de Inercia de Figuras comunes

Por estática

ydFFycp =

Por estática

ydFFycp =

Pero

∴=PdAdF

PdAyFyAcp ∫=

αρgysenP =

∫=Acp dAgsenyFy αρ2

∫=Acp dAygsenFy 2αρ

Pero

∫=Ax dAyI 2 Segundo momento de área ó

Momento de inercia de área

I x= I y2 A

ycp

F=ρ gsen α I y2 A

ycp ρg y sen αA =ρ gsen α I y2 A

yCP= yI

y A

Por estática

∫=Acp xPdAFx

∫=Acp dAgsenxyFx αρ

∫=Axy xydAI

AyxII xyxy +=

Producto de Inercia de área

xcp ρg y sen αA =ρ gsen α I xyx y A

xcp= xI xy

y A

∑ =0M

F1 .5−∫AyPdA=0

F1 .5=∫0

1 . 5

yρ g y a dy

F1 .5=a ρg [ y 3

3 ]0

1 . 5

F1 .5=a ρg∫0

1 .5

y2 dy

F1 .5=1.5ρg [ y3

3 ]0

1 . 5

F=1000kg

m3 9.81m

s2 [ 1.53

3 ]=11.03 kN

∑ =0M

F1 .5−∫AyPdA=0

F1 .5=∫0

1 . 5

yρ g 3 .6−1.5 y a dy

F1 .5=a ρg [2.1y 2

2

y 3

3 ]0

1 . 5

F1 .5=a ρg∫0

1 . 5

2.1y y 2 dy

F1 .5=1.5ρg [2.1y2

2

y3

3 ]0

1 . 5

F=1000kg

m3 9.81m

s2 [2.11 .52

2

1.53

3 ]=34 .21 kN

b = 2 m

a = 2.5 m

water8 m

4 m

FR

abA π=

F R=ρ gysen α∫AdA

F R=ρg 8 ysen α πb∫0

5 dy2

F R=ρgπb [4yy 2

4sen α ]

0

5

F R=1000kg

m3 9.81m

s2 π 2 [4 5 52

4sen 53.1 ]=1540. 82kN

F R=9800 N

m3 8 m2m π 2 .5 m 2 m =1540 .82kN

Ancho =1m2m

ghAFF Hx ρ==

4m

1m

2m

gVghAwFF vy ρρ +=+=

arAaV 2

41 π==

109.3kN30.8kNkN5.78 =+=yF

kN9.146109.398.1 22 =+=rF

F x=F H= 1000kg

m3 9.81m

s2 4112=98 .1kN

Fv= 1000kg

m3 9.81m

s2 4 12=78 .5 kN

w=1000kg

m3 9 .81m

s2 1422 1=30.8 kN

98.1kN=xF

109.3kN=yF kN9.146=rF

AyI

yycp +=

Ay

Ixx xy

cp +=

o4898.1kN

kN3.109 tan 1 ==−θ

cpy

cpx

m067.5)2)(1(4

22

12)2(1

422

3

=

+

+

+=cpy

1.067m4m-m067.5 ==cpy

42

+=Hy

12

3bHI =

Por estática

xwxFFx vvycp +=

y

vvcp F

xwxFx

+=

m957.03.109

)849.0(8.30)1(5.78 =+=cpx

m849.0)2(3

4

3

4 ===ππ

rx

1m2

2

2=== B

xv

o480.957m

m067.1 tan 1 ==−θ

98.1kN=xF

109.3kN=yF kN9.146=rF

m957.0=cpx

m067.1=cpy

Ancho =1m

2m

1m

2m

19.62kN)2)(1)(1(sm

81.9mkg

1000 23 =

== Hx FF

30.8kN)1()2(41

sm

81.9mkg

1000 223 =

== πwFy

kN51.3630.819.62 22 =+=rF

19.62kN=xF

30.8kN=yF

kN51.36=rF

o5.5719.62kN

kN8.30 tan 1 ==−θ

cpy

cpx

m333.1)2)(1(

22

12)2(1

22

3

=

+

=cpy

m849.08.30

)849.0(8.30 ==cpx

o5.570.849m

1.333m tan 1 ==−θ

2m 4m

1m

98.1kN)2)(1)(14(sm

81.9mkg

1000 23 =+

== Hx FF

gVghAwFF vy ρρ +=+=

78.5kN)2)(1)(4(sm

81.9mkg

1000 23 =

=vF

19.62kN)1(2

)2)(2(sm

81.9mkg

1000 23 =

=w

aBH

AaV2

==

98.1kN19.62kNkN5.78 =+=yF

kN7.13898.198.1 22 =+=rF

Ancho =1m

98.1kN=xF

98.1kN=yF

kN7.138=rF

AyI

yycp +=

Ay

Ixx xy

cp +=

o4598.1kN

kN1.98 tan 1 ==−θ

cpy

cpx

m067.5)2)(1(4

22

12)2(1

422

3

=

+

+

+=cpy

1.067m4m-m067.5 ==cpy

42

+=Hy

12

3bHI =

Por estática

xwxFFx vvycp +=

y

vvcp F

xwxFx

+=

m067.11.98

)33.1(6.19)1(5.78 =+=cpx

m33.13

)2(2

3

2 === Bx

1m2

2

2=== B

xv

AgysenF αρ=

Ancho=3 m

m5.1=y

1.3m)30cos(5.1 ==x

F R=1000kg

m3 9.81m

s2 [1m1.5m

2sen 30 ]3m 1.5m =60.7 kN

F R=1000kg

m3 9.81m

s2 [1m0.75m

2 ]3m 1.5m=60.7 kN

h=1.5msen 30=0.75

m4.1)75.0)(3)(

2

75.01(

12)75.0(3

)275.0

1(

3

=+

++=CPy

Por estática

xwxFFx vvycp +=

y

vvcp F

xwxFx

+=

m7098.06.52

)866.0(34.14)65.0(26.38 =+=cpx

m866.03

)3.1(2

3

2 === Bx

0.65m2

3.1

2=== B

xv