1 Superficie-Curvas

8/19/2019 1 Superficie-Curvas

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MECANICA DE FLUIDOS I

INDICE

PRESENTACION..................................................................................

OBJETIVOS.........................................................................................

INTRODUCCIÓN................................................................. ................

FUERZAS DE PRESIÓN SOBRE SUPERFICIES CURVAS.......................................................................................... ................

DEFINICIONES.......................................................................................

o CENTRO DE PRESIONES.............................................................o EL CENTRO DE MASA.................................................................

o EL CENTRO DE GRAVEDAD.........................................................

o EL CENTROIDE...........................................................................

FUERZAS DE SUPERFICIES CURVAS:....................................................

DE FORMA ANALÍTICA................................................................

DE MANERA INTEGRAL.............................................................

- FUERZA HORIZONTAL ( F H ).................................................

- FUERZA VERTICAL ( F V )........................................ ..............EJERCICIOS........................................................................ ..............

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# $


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PRESENTACION

E" %# #&'#%!# *% !"*"!*+o *,* %&%#+ %# *+/# *0*+&!# 1o+%o 2!o &o" *% 3" * 1o*+ !*4#+ #'!#&'o+!#5*"'* %#*'+&'+# 6* %o &o"'!*"*".

E% 1+**"'* '+#,#0o &o51+*"* *% *#++o%%o * "# '*5á'!,##o *" %# *+/# 7!+o'á'! *" %# 1*+3&!* &+8# 9# 6**'* '*5# * 59 !51o+'#"'* 1#+# "*'+# &o51+*"!"; *" '#%5#"*+# 6* "o #9#+# # &o51+*"*+; &o5o * *% "&!o"#5!*"'o* %# 1+*# * ## (*"'+o * "*'+o 3o %#,o+#%); &o5o *6* 1**" o1o+'#+ '#"'# 1+*!" * ## o &#%6!*+ o'+o 2!o.

P#+# %# *%#,o+#&!" *% '+#,#0o 7*5o *ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# >


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OBJETIVOS

D#+ # &o"o&*+ %o &o"&*1'o 6* !51%! "*'+o '*5#.

V!#%!/#+ %# !'+!,&!" * *+/# o,+* "# 1*+3&!*&+8#

C#%&%#+ %# *+/# +*%'#"'* 'o'#% o,+* "# 1*+3&!*&+8#.

C#%&%#+ %# !+*&&!" *" 6* #&'?# %# *+/# +*%'#"'* 95o'+#+ %="*# * #&&!" *" " !#+#5# * %#1*+3&!*.

A"#%!/#+ 9 #+ o%&!" # %o 1+o,%*5# 1%#"'*#o;7#&!*"o o * %o &o"o&!5!*"'o #6!+!o *" *'#*


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INTRODUCCION

T*"*5o *% #+#o * !+!!+"o 7#&!# '** &o" *% 1+o1!'o *#51%!#+ "# 1*6*4# 1#+'*; 1*+o !"!3'!8#; * &o"o&!5!*"'o*"'+o * %# +#5# * INGENIERIA CIVIL; *'* '*5# * 59 !51o+'#"'*1#+# "*'+# &o51+*"!"; *" '#% 5#"*+# 6* "o #9#+# &o51+*"*+;&o5o * *% "&!o"#5!*"'o * %# 1+*# * ## (*"'+o * "*'+o3o %#,o+#%); &o5o * 6* 1**" o1o+'#+ '#"'# 1+*!" * ## o&#%6!*+ o'+o 2!o.

T*"!*"o &o5o ,#* "#5*"'#% %# '*o+=# =! 5#'*5á'!;

'#%* &o5o:

FUERZA HIDROSTÁTICA SOBRE SUPERFICIES CURVAS

H#,!*"o &o51+*"!o %# '*o+=# * F*+/# 7!+o'á'! o,+*1*+3&!* 1%#"# 5*+!#; !+*5o 6* %# !*+*"&!# ,á! *" *%&á%&%o * %# *+/# 6* #&'?# o,+* "# 1*+3&!* &+8# +*1*&'o* "# 1%#"# +#! *" *% 7*&7o * *+ 1*+1*"!&%#+ *" 'oo5o5*"'o # %# 1*+3&!*; *"'o"&* # !*+*"&!#% * *+/# '!*"*"# !+*&&!" !*+*"'*.

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"#


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FUERZAS DE PRESION SOBRESUPERFICIES CURVAS

1. DEFINICIONES

1.1 CENTRO DE PRESIONES

E% &*"'+o * 1+*!o"* * *% 1"'o 1o+ *% &#% * *0*+&*" %# %="*# *#&&!" * %# *+/# 6* *0*+&*" 1+*!" o,+* " &*+1o 5*+!o *"" %=6!o

1.2 EL CENTRO DE MASA

E *% 1"'o *" *% &#% * 1** &o"!*+#+ &o"&*"'+## 'o# %#5## * " o,0*'o o * " !'*5#.A" ! *% o,0*'o *'# *" +o'#&!"; *% &*"'+o * 5## * 5*8*&o5o ! *+# 1#+'=&%#. A%"# 8*&* *% &*"'+o * 5## **&+!,* &o5o ! *'8!*+# *" *% 1"'o * *6!%!,+!o * " o,0*'o%!o. Po+ *0*51%o; ! '* *6!%!,+# " 5*'+o o,+* *o; *%&*"'+o * 5## * %# 8#+!%%# * 5#*+# *'á %o%!/#o!+*&'#5*"'* o,+* *o 9 'o# %# 5## 1#+*&* *'#+&o"&*"'+## #7=.

1.3 EL CENTRO DE GRAVEDADE *% 1"'o *" *% 6* * *"&*"'+#" #1%!# %# *+/#+#8!'#'o+!# * " o,0*'o; o * *&!+ * *% 1"'o *" *% 6* #&'?#*% 1*o. S!*51+* 6* %# #&*%*+#&!" * %# +#8*# *#&o"'#"'*; *% &*"'+o * +#8*# * *"&*"'+# *" *% 5!5o1"'o 6* *% &*"'+o * 5##.

1.4 EL CENTROIDE

E% C*"'+o!* * " 1"'o 6* *3"* *% &*"'+o *o5'+!&o * "o,0*'o. S %o%!/#&!" 1** *'*+5!"#+* # 1#+'!+ * +5%#

*5*0#"'* # %# '!%!/## 1#+# *'*+5!"#+ *% &*"'+o *+#8*# o *% &*"'+o * 5## *% &*+1o. S* &o"!*+#" '+*o *1*&=3&o: 8o%5*"; á+*#; %="*#.



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2. FUERZAS DE SUPERFICIES CURVAS

2.1 DE FORMA ANALÍTICA:

L# R*%'#"'* 'o'#% * %# *+/# * 1+*!" 6* o,+#" o,+* "#1*+3&!* &+8#; *'á o+5## 1o+ %# 5# * %o *%*5*"'o!*+*"&!#%* * *+/# ( dF = pdA ) "o+5#%* # %# 1*+3&!*. L#5#"!' 9 1o!&!" * %# R*%'#"'* * *'# *+/#*%*5*"'#%*; "o 1** *'*+5!"#+* á&!%5*"'* 1o+ %o 5'oo#o 1#+# 1*+3&!* 1%#"#. S!" *5,#+o; * 1**"*'*+5!"#+ &o" #&!%!# %# &o51o"*"'* 7o+!/o"'#% 9 8*+'!% *%# R*%'#"'* 1#+# %*o &o5,!"#+%# 8*&'o+!#%5*"'*

Co"!+*"* %# *+/# 6* o,+#" o,+* *% 1+!5# * %=6!o!%'+#o *" %# 3.(A); %!5!'#o 1o+ %# 1*+3&!* %!,+* #-o; 1o+ %#1*+3&!* 8*+'!% 1%#"# o-,; 9 1o+ %# 1*+3&!* &+8# #-,.

E% 1*o * *'* 8o%5*" * "# *+/# 8*+'!% 7#&!# #,#0o; 9#&'#"o * *+*&7# # !/6!*+#; o,+* o-, *'á %# *+/#

7o+!/o"'#% vGh Ah F γ = ; *" o"* A v * *% á+*# * %#1*+3&!* 1%#"# 8*+'!% !5#!"#+!#; "o * &9o ,o+* * o,.E'# *+/# * 5#"'!*"*" *" *6!%!,+!o 1o+ *+/# !#%* 9o1*'# * +*#&&!" * %# 1*+3&!* &+8# #-,.



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S* *&* *" &o"*&*"&!#; 6* %# &o51o"*"'* 7o+!/o"'#% * %#R*%'#"'* 'o'#% * %# 1+*!o"* o,+* "# 1*+3&!* &+8# *!#%; 9 *'á #1%!# *" *% 5!5o 1"'o; 6* %# *+/# 6* #&'?#o,+* %# 1*+3&!* 1%#"# 8*+'!% o+5## #% 1+o9*&'#+ *"!+*&&!" 7o+!/o"'#% %# 1*+3&!* &+8#.

Po+ o'+# 1#+'*; %# &o51o"*"'* 8*+'!% * !&7# R*%'#"'* 'o'#%o,+* %# 1*+3&!* &+8# * !#% #% 1*o *% %=6!o 6* **"&*"'+# *"&!5# * '#; 9 *'á #1%!# *" *% &*"'+o * %#+#8*# *% 8o%5*" %=6!o. U" +#/o"#5!*"'o *5*0#"'**5o'+#+á 6* &#"o *% %=6!o * *"&*"'+# *,#0o * %#1*+3&!* &+8#; %# &o51o"*"'* 8*+'!% * !#% #% 1*o *%8o%5*" !5#!"#+!o *% %=6!o 6* * *"&o"'+#+=# *"&!5# * %#1*+3&!* 9 *'á #1%!# 7#&!# #++!,# 1##"o 1o+ &*"'+o *+#8*#.

Po+ *0*51%o %# &o51o"*"'* 8*+'!% * %# R*%'#"'* 'o'#% *

1+*!o"*; *0*+&!# o,+* %# &o51o"*"'* +#!#% o * #,#"!&o *%# 3. (B); * !#% #% 1*o *% 8o%5*" +*1+**"'#o 1o+ LNM 9#&'?# 7#&!# #++!,# 1##"o 1o+ G &o5o * !"!.



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2.1.1 DIAGRAMA DE CUERPO LIBRE DE UN VOLUMENDE FLUIDO

U"# 5#"*+# * 8!#%!/#+ *% !'*5# * *+/# 'o'#%!"8o%&+## * #!%#+ *% 8o%5*" * 2!o 6* *'á!+*&'#5*"'* #++!,# * %# 1*+3&!* * !"'*+



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D"*:

F 1 : E %# *+/# +*%'#"'* o,+* %# 1#+'* 8*+'!%

!/6!*+# 9 * #"#%!/# !#% 6* %# 1#+** 8*+'!%*5*!# 7#'# "# 1+o"!# 7.

F 2a : E %# *+/# +*%'#"'* o,+* %# 1#+* 8*+'!%*+*&7# 9 * #"#%!/# !#% 6* %# 1#+** 8*+'!%*5*!# 7#'# "# 1+o"!# 7.



10/24


E" *'* !'*5# F 1= F 2a 1o+ '#"'o "o 7#&*" "!"?" **&'o(* &o"'+# +*'#"). F 2b : E %# *+/# 6* #&'?# o,+* %# 1#+'* *+*&7#; *" *%

á+*# 1+o9*&'## 1o+ %# 1*+3&!* &+8# *" *% 1%#"o

8*+'!%.L# 5#"!' * F 2b * *"&*"'+# ,#0o *% 5!5o1+o&*!5!*"'o *#++o%%#o 1#+# 1*+3&!* 1%#"#.

hc : E %# 1+o"!# *% C*"'+o!* *% á+*#1+o9*&'##

AHORA:

1. FUERZA HORIZONTAL ( F H !"#$%%"&' )#"*)'+,-

E !#% # %# *+/# "o+5#% 6* *% 2!o *0*+&* *% 2!oo,+* %# 1+o9*&&!" 8*+'!% * %# 1*+3&!* 9 %="*# *#&&!" 1## 1o+ *% &*"'+o * 1+*!" * !&7# 1+o9*&&!"

F H =γ hc A

D"*:

hc ; * %# 1+o"!# *% C*"'+o!* *% á+*#

1+o9*&'##; 1#+# "*'+o #"á%!! *% á+*# 1+o9*&'## *" +*&'á"%o

hc=h+s /2

rea= A=sw

D"*:

s=¿ E %# #%'+# * %# 1+o9*&&!" * %# 1*+3&!* &+8#.w=¿ E %# 1+o"!# o á"%o * %# 1*+3&!*

1+o9*&'##E"'o"&*:

F h=γ A (h+s /2 )= F 2b

F h=γ sw (h+s /2 )

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# $K


11/24


U/"%,%"&' !$ -, 0$#*, )#"*)'+,-:

S*?" %# +*%#&!o"* 8!'#:

h p−hc= I c

H c A

P#+# " +*&'á"%o *% á+*# 1+o9*&'## *:

I c=b (h)3

12

E% á+*# 8!*"* ## 1o+:

h p−hc= I c

H c A

h p−hc= h

2

12hc

2. FUERZA VERTICAL ( F V ) !"#$%%"&' $#+"%,-

L# &o51o"*"'* 8*+'!% * %# *+/# * 1+*!" o,+*"# 1*+3&!* &+8# * !#% #% 1*o *% %=6!o 6* **"&*"'+# 8*+'!%5*"'* 1o+ *"&!5# * !&7#1*+3&!* 9 * *


12/24


F v=γ AC ; A&'?# *" %# %="*# *% &*"'+o!* *% 8o%5*"

3. FUERZA RESULTANTE ( F R)

F =√ F h2+ F v

2

L# *+/# +*%'#"'* #&'?# *" " á"%o ϕ *"+*%#&!" &o" %# 7o+!/o"'#% *" !+*&&!" '#% 6* %="*#* #&&!" 1## 1o+ *% &*"'+o * &+8#'+# *%#1*+3&!*.

ϕ=tan−1( F v

F h)

2.2 DE MANERA INTEGRAL:

S!!*"o &o" *% '*5# * 1*+3&!* &+8# 5*+!#; *"

#%"o o *% &á%&%o * %# *+/# 'o'#%* * 7#&* "1o&o 5á &o51%*0o; &#"o *'# 9# "o #&'?#" *"1*+3&!* &o" á+*# &o"o&!#; &o5o * *% o * "#1*+3&!* &!+&%#+

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# $>


13/24


14/24


F$#*, V$#+"%,- (F


F !=γC ∫( "− !)d

F !=γC ∫( "−√ )d

γhCd=¿ γw∫hd F !=∫¿

Pero:

p=

dA=wd ∫dF !=∫ pdA


15/24


EJERCICIOS

1 S* '!*"* " &!%!"+o * > 5 * !á5*'+o; *% &!%!"+o '!*"* " 1*o * >KK

9 '!*"* "# %o"!' * $.K 5. D*'*+5!"#+ %# +*#&&!o"* *" A 9 B*1+*&!#"o *% +o/#5!*"'o.

DATOS

>.K 5+ $.K 5% $.K 5

+ K.KKPRESIÓN (FUERZA) (REA)FUERZA (PRESIÓN) < (REA)

SOLUCIN$. H#&!*"o *% !#+#5# * &*+1o %!,+* * %# +á3:



16/24


>. A"#%!/#"o%#&o51o"*"'* '#"'o 7o+!/o"'#%* &o5o 8*+'!%*:

#) L# &o51o"*"'* 7o+!/o"'#%

F H Q 7 A F H (KK 5@) < ($.K 5) < (>.K 5 < $.K 5) F H >KK (7#&!# %# *+*&7#)

E"'o"&* %# +*#&&!" *" A *+!#RA 5 2466 78 (,%", -, "*9"$#!,

,) L# &o51o"*"'* 8*+'!%

(H#&!# #++!,#) F V KK 5@ < $. 5 < REA (DOB)

REA (DOCE)

(H#&!# #++!,#) F V KK 5@

< $. 5 < ( ($)>

) 5>

($.K < $.K) 5>(H#&!# #++!,#) F V >$>.

(H#&!# #,#0o) F V KK 5@ < $. 5 < REA (DEC)

(H#&!# #,#0o) F V KK 5@ < $. 5 < REA (DOCE)

(REA DOC)(H#&!# #,#0o) F V KK 5

@ < $. 5 < ($.K < $.K) 5> ( ($)>) 5>(H#&!# #,#0o) F V >.>

&) L# &o51o"*"'* 8*+'!% "*'#

F V N*'# (H#&!# #++!,#) F8 (H#&!# #,#0o) F8

F V N*'# >$>. >.> F V N*'# $. (7#&!# #++!,#)

S5#'o+!# * *+/# 8*+'!%*



17/24


F V 5 6>KK -$. - RB K RB 5 ;1


18/24


L= 1.2m b=2.4m

Pamb=0.96 105 Pa

Ɣ=880#$.f

m3

P|¿|1.5 105 Pa

No'#: L * *% #"&7o

SOLUCION

Co" *% 1+**"'* 1+o,%*5# !%'+#+*5o %o 6* * %# #%'+#*6!8#%*"'*. L# #%'+# *6!8#%*"'* * %# 5*!# * %# &o%5"# *%=6!o # #+*#+ 9 *'* # %# 8*/ # %# 5!5# 1+*!"

5#"o5'+! 6**% # #'+#1#o

!"!&!#%5*"'*.

H e%= pman

γ li%

H e%=

(1.5−0.96 ) 105 Pa

880#$f

m3

❑❑❑❑



19/24


H e%=

6.26 &

m2

&

m3

H e%=6.26m

D* %# 3+#:

H , H e% >.5 .>5 .5

U'!%!/#"o *% 8o%5*" !5#!"#+!o

F8 γ li% V ima$=γ li% (dH + ' d2

8 ) (

F8

m60 10−2¿¿¿2

' (¿¿ 8¿) 1.2m60 10−2m 8.66m+¿880#$f

m3 9.81 & 1#$f

¿

m

60 10−2

¿¿¿2

' (¿¿ 8¿) 1.2m60 10

−2m8.66m+¿

FV =880#$f

m3

9.81 &

1#$f ¿



20/24


FV =8632.8 &

m3 (5.196m2+0.14m2 ) 1.2m

FV =8632.8 & m

3 6.4032m3

FV =55.278# &

3 U" &!%!"+o * >. 5 * !á5*'+o &!*++# " #0*+o +*&'#"%#+ *" "*1!'o * K.5 * #"&7o WCo" 6* *+/# 6*# 1+*!o"#o *% &!%!"+o&o"'+# *% o"o *% *1!'o 1o+ %# #&&!" * %o >.5 * 1+o"!# *

##X γ =1000#$f

m3

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# >K


21/24


DATOS- F CA F )*- ∅ >. 5

- F γ fluido.∀

SOLUCION

- ECUACION GENERAL:

F &*+A F C"* ( F CA F )* )

- HALLANDO F C"* :

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# >$


22/24


F C"* γ a$ua .∀

F C"* $KKK ,$f

m3<

[(2.1 2.4 0.9−

1

2

' . (1.2)2 0.9

)] <

m

3

F C"* >KK.>$ Y

- H#%%#"o F CA 6* '#5,!" * !#% # F )* :

F CA γ a$ua .∀

F CA $KKK ,$f

m3 <

[2.1 0.162 0.9+0.9 112 ' (1.2)2−12 (0.6 1.038 0.9)]m3

F CA @.>$ Y

E"'o"&* o,'*"!# %# *+/# +*6*+!# *" #5,o 1"'o; 1+o&**5o# +**51%#/#+ *" %# *&#&!" *"*+#%:

F &*+A

F C"*

( F CA

F )*

)

F &*+A >KK.>$ Y > (@.>$) Y

F &*+A $.Y

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# >>


23/24


) C#%&%#+ %# F H ,F V ! F R 1o+ 5*'+o * #"&7o *,!o # %# #&&!" *%## o,+* *% 5+o * +*'*"&!" 1#+#,%!&o 6* * 5*'+# *" %#3+#. E% 8+'!&* * %# 1#+á,o%# * *"&*"'+# *" *% 1"'o # 9 *!5'+!&o +*1*&'o #% *0* 9.

SOLUCION:

,CALCULO DE FUERZA HORIZONTAL

F H = γ . h$ . A pro!ec-ada

F H = $KKK#$

m3 ×>5 × (> ×> ) m2

F H = KKK

F H =¿ 8 tn

/ CALCULA DE LA FUERZA VERTICAL

ESCUELA PROFESIONAL DE INGENIERIA CIVIL Pá!"# >@


24/24


F V = A ancho

F V =1000#$

m3 × m2 × $5

F V =5000#$

F V =5 -n

% CALCULA DE LA FUERZA RESULTANTE

F R = √ 82-n2+52 -n2

F R = .@ '"

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