Condensación Estática y Rigidez Lateral

7/25/2019 Condensacin Esttica y Rigidez Lateral

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CONDENSACION ESTTICA Y MATRIZ DERIGIDEZ LATERAL

ATERIA:

Anlisis estructura II

ROFESOR:

Ing. Jos Chuquillanqui

NTEGRANTES:

Conislla Limaylla JessLuna Lopez Marco

ECHA DE PRESENTACION:

2 ! "# ! $#

O:

2"$#

CONDENSACION EST TICA Y

MATRIZ DE RIGIDEZ

UNIVERSIDAD NACIONAL MAYOR DE

SAN MARCOS

Facultad de IngenieraGeolgica! Minera!

E&cuela Acad'(ico)ro*e&ional de Ingeniera


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a,la de contenidoCONDENSACIN ESTTICA DE LA MATRIZ DE RIGIDZ.....................2

1.1 DEFINICIN...............................................................................2

1.2 MTODO GENERAL DE LA CONDENSACIN DE LA MATRIZ DE RIGIDEZ 2

1.3 MATRIZ DE RIGIDEZ PARA ANLISIS LINEAL..............................3

1.3.1 Anlisis sin nu! "#$i!.......................................................3

1.3.2 Anlisis %!n nu! "#$i!......................................................&

1.3.3 Ens'()l'*+ + l' (',"i- + "i$i+-.....................................

1./ CONDENSACIN DE LA MATRIZ DE RIGIDEZ.............................

1./.1 C!n+ns'%in ' l's %!!"+n''s '................................10

1./.2 C!n+ns'%in ' l's %!!"+n''s )................................11

1. CONDENSACIN MEDIANTE SOL4CIN DE EC4ACIONES........12

1..1 C's! +n 5u+ 6)7 0...........................................................12

1..2 C's! +n 5u+ 6'7 0...........................................................13

1.& CONDENSACIN MEDIANTE ELIMINACIN DE GA4SS.............13

1.8 9ARRA ARTIC4LADA:R;GIDA....................................................1

RIGIDEZ LATERAL...........................................................................1

2.1 INTROD4CCION.........................................................................1

2.2 RIGIDEZ LATERAL DE 4N PRTICO SIMPLE................................1

2.3 MATRIZ DE RIGIDEZ LATERAL DE 4N PRTICO DE .........................28

3./ ESTR4CT4RACION CON M4ROS DE CORTANTE.........................2

3. COMPORTAMIENTO DE LOS M4ROS DE CORTANTE....................2

3.& COMPORTAMIENTO DE LOS M4ROS DE CORTANTE EN


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-. CONDENSACI/N ESTTICA DE LA MATRIZ DE RIGID0Z

- DEFINICI/N

%!n+ns'%in +s,,i%' + l' (',"i- + "i$i+-@ +s l' )'s+ un'(+n,'l B'"' +l 'nlisis s#s(i%! +

"u%,u"'s.B"+s+n,'n ,"+s !"('s + +n%!n,"'" l' (',"i- + "i$i+- %!n+ns''@ ' s')+"> l' B"i(+"' in!lu%"' l'+"sin + un' (',"i-@ l' s+$un' i(Bli%' l' s!lu%in + un %!n*un,! + +%u'%i!n+s lin+'l+s l' ,+"%+"'i'n,+ l' +li(in'%in + G'uss.

" !,"' B'",+@ s+ B"+s+n,' l' (',"i- + "i$i+- + l!s +l+(+n,!s B'"' +l 'nlisis s#s(i%! + B",i%!s Bl'n!s@!s ('n+"'s@ l' B"i(+"' sin %!nsi+"'" nu!s "#$i!s l' s+$un' %!nsi+"'n! nu!s "#$i!s.

1 M0TODO GENERAL DE LA CONDENSACI/N DE LA MATRIZ DE RIGIDEZ

' un sis,+(' + N $"'!s + li)+",' %'"'%,+"i-'! (+i'n,+ su (',"i- + "i$i+- + ,'('! N H N

+ "+l'%i!n' lin+'l(+n,+ +sBl'-'(i+n,!s 4 %!n s!li%i,'%i!n+s F ,'l 5u+>

(J,!! + %!n+ns'%in + $"'!s + li)+",' B+"si$u+ !),+n+"@ B'",i+n! + +s,+ (!+l!@ l' (',"i- ++- %!""+sB!ni+n,+ 'l %'s! + K')+" +li(in'! un! ! (s $"'!s + li)+",'. Eli(in'" n $"'!s +

+",' i(Bli%' 5u+ %i+",!s +sBl'-'(i+n,!s ! $i"!s +*'"n + 'B'"+%+" +n +l +%,!" 4@ sus""+sB!ni+n,+s s!li%i,'%i!n+s +s'B'"+%+"n +l +%,!" F@ +*'n! B!" l! ,'n,! nu+s,"! (!+l! %!(!>

n+ +i+n,+(+n,+ l' (',"i- +)+ s+" !),+ni' + !"(' 5u+ +s,+ (!+l! %!n+ns'! s+ %!(B!",+'%,'(+n,+ i$u'l 5u+ +l !"i$in'l. Es i(B!",'n,+ n! B+"+" + is,' 5u+ +l sis,+(' %!n+ns'! si$u+""+sB!ni+n! 'l (is(! sis,+(' #si%!@ !n+ s!l! K+(!s i(Bu+s,! 'l$un' %!ni%in +,"'. En l!s %'s!s+ '(!s ' +s,ui'"@ i(B!n"+(!s 5u+ l's u+"-'s l!s (!(+n,!s +n l!s $l %!n+ns'!s l!s 5u+s'B'"+%+n s+'n nul!s. D+ +s,' !"('@ si +n!,'(!s l!s su)%!n*un,!s + F 4 5u+ 'n ' s+"n+ns'!s %!(! F% 4%@ "+sB+%,i'(+n,+@ B!+(!s "++s%"i)i" l' E%. 2..1 %!(!>

n+ l!s suB+"#ni%+s % n us'!s B'"' l's su)(',"i%+s ini%'n l!s ,J"(in!s "+l',i!s ' l!s $l 5u+ K'n! 5u+ n! K'n si! %!n+ns'!s@ "+sB+%,i'(+n,+. D+sB+*'n! 'K!"' l' +%u'%in %!""+sB!ni+n,+ ' F%

!" si(Bl+ +s'""!ll! +l B"!u%,! (',"i%i'l@ ,+n+(!s>

2


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!(! i(B!n+(!s l' %!ni%in + u+"-'s (!(+n,!s %!n+ns'!s F% nul!s l!s +sBl'-'(i+n,!s""+sB!ni+n,+s ' i%K!s $l s!n li)"+s@ B!+(!s +sB+*'" +l 'l!" + l!s +sBl'-'(i+n,!s $i"!s +nK!s $l>

+ "+sul,'! s+ Bu++ 'K!"' sus,i,ui" +n l' +%u'%in %!""+sB!ni+n,+ ' F s+$n l' E%. 2..3@ ll+$'n! 'lul,'!>

(! s+ +@ l' (',"i- + "i$i+- %!""+sB!ni+n,+ 'l sis,+(' %!n+ns'! n! s+ !),i+n+ si(Bl+(+n,+,"'+n! l' B'",+ 5u+ n!s in,+"+s' + l' (',"i- !"i$in'l @ sin! 5u+ K' 5u+ "+s,'"l+ un ,J"(in! 'i%i!n'l+ (!+l' +l ++%,! 5u+ ,i+n+n l!s $l %!n+ns'!s +n l!s +(s.

2 MATRIZ DE RIGIDEZ )ARA ANLISIS LINEAL

'nlisis lin+'l s+ %!nsi+"' 5u+ l' "i$i+- ' Q+in (EI)o@ +s %!ns,'n,+ l! B"!Bi! su%++ %!n l' "i$i+- 'l",+ (GA)o. En %!ns+%u+n%i'@ l' (',"i- + "i$i+- + un +l+(+n,! +s %!ns,'n,+ l! (is(! su%++ %!n l',"i- + "i$i+- + l' +s,"u%,u"'@ %!(! s+ K' is,! +n l!s %'B#,ul!s 'n,+"i!"+s.

2.- An#li&i& &in nudo rgido

l' $u"' 1&.1@ s+ ini%' +l sis,+(' + %!!"+n''s l!%'l+s + un +l+(+n,! K!"i-!n,'l + un B",i%!n!@ +n +l 5u+ n! s+ %!nsi+"' l' +!"('%in 'i'l@ KiB,+sis + %l%ul! 5u+ s+ Bu++ u,ili-'" +n +llisis s#s(i%! + +s,"u%,u"'s B'"' l!s +l+(+n,!s K!"i-!n,'l+s.

Figura -3.- C!!"+n''s l!%'l+s B'"' un +l+(+n,! 'i'l(+n,+ "#$i!."' +l +l+(+n,! K!"i-!n,'l ini%'! +n l' $u"' 1&.1@ s+ ,i+n+ 5u+ +l sis,+(' + %!!"+n''s l!%'l+s +s'l 'l sis,+(' + %!!"+n''s $l!)'l+s. P!" !,"' B'",+@ s+ "+%u+"' 5u+ l's +s,"u%,u"'s s+ "+su+l+n +n

!"+n''s $l!)'l+s.

(',"i- + "i$i+- +l +l+(+n,!@ +s si(J,"i%' %!n "+sB+%,! ' l' i'$!n'l B"in%iB'l@ "'-n B!" l' %u'l s!l!B"+s+n,' l' (',"i- ,"i'n$ul'" suB+"i!". C!n "+l'%in 'l sis,+(' + %!!"+n''s l!%'l+s + l' $u"' 1&.1@

3


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(',"i- + "i$i+- +s l' si$ui+n,+.

!"(' + l' (',"i- + "i$i+-@ ini%'' 1&.1 +s li' B'"' +l+(+n,!s + s+%%in %!ns,'n,+ ! +"i')l+. P'"' +l+(+n,!s + s+%%in %!ns,'n,+@ s+ ,i+n+>

n+ E +s +l (ul! + +l's,i%i' +l (',+"i'l@ I +s l' in+"%i' ' Q+in + l' s+%%in ,"'ns+"s'l@ +s +l,!" + !"(' B!" %!",+ + l' s+%%in@A +s +l "+' + l' s+%%in ,"'ns+"s'l@ G +s +l (ul! + %!",+ Ll' l!n$i,u +l +l+(+n,!.

EM)LO N.5 -

%!n,"'" l' (',"i- + "i$i+-@ sin %!nsi+"'" nu!s "#$i!s@ B'"' un' i$' + s+%%in %!ns,'n,+ + 30 %(.)'s+ B!" 30 %(. + 'l,u"' ,i+n+ un' l!n$i,u + 3.8 (. P!" !,"' B'",+@ %!nsi+"'" E72100000 T/m2/0000 T/m2.

SOL6CI/N

/


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"' un +l+(+n,! +",i%'l@ +n l' $u"' 1&.2@ s+ ini%' +l sis,+(' + %!!"+n''s $l!)'l+s B'"' +l %'s! ++ +l +l+(+n,! s+' ,!,'l(+n,+ Q+i)l+.


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(',"i- + "i$i+- +l +l+(+n,! +",i%'l@ +n %!!"+n''s $l!)'l+s +s l' si$ui+n,+.

s "+s,'n,+s ,J"(in!s + l' (',"i- + "i$i+-@ u+"!n ini%'!s +n l's +%u'%i!n+s 'n,+"i!"+s.

2.1 An#li&i& con nudo rgido

+l 'nlisis +s,"u%,u"'l s+ Bu++ %!nsi+"'" 5u+ l!s nu!s s!n %!(Bl+,'(+n,+ "#$i!s. En %!ns+%u+n%i'@!n$i,u + l!s +l+(+n,!s 5u+ in$"+s' 'l nu!@ ,i+n+n "i$i+- 'i'l inni,' "i$i+- ' Q+in inni,'.

'n c1 c2l's l!n$i,u+s + "i$i+- inni,' + un +l+(+n,!@ %!(! +l ini%'! +n l' $u"' 1&.3.

gura -3.2 C!!"+n''s l!%'l+s B'"' un +l+(+n,! 'i'l(+n,+ "#$i! %!n !s s+%,!"+s + "i$i+-ni,'

!"'@ l' (',"i- + "i$i+- +l +l+(+n,!@ +s l' si$ui+n,+>

n+@ l!s ,J"(in!s + "i$i+- k, a, k', b, b', t@ s!n l!s ini%'!s +n l's +%u'%i!n+s 1&.2.1 ' 1&.2.8.

&


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E4EM)LO N.5 1+s+' +n%!n,"'" l' (',"i- + "i$i+-@ B'"' l' i$' + s+%%in %!ns,'n,+ +l +*+(Bl! 'n,+"i!"@

nsi+"'n! nu!s "#$i!s@ B'"' +l %'s! + l' $u"' 1&./.

SOL6CI/N

"++(Bl'-'" c17 c27 0.1 l!s "+s,'n,+s ',!s ini%'!s +n +l +*+(Bl! 'n,+"i!"@ +n 1&.@ s+ !),i+n+>

l' $u"' 1&.@ s+ ini%' +l sis,+(' + %!!"+n''s $l!)'l+s@ + un +l+(+n,! +",i%'l@ +n +l %u'l s+

nsi+"'n !s s+%,!"+s + "i$i+- inni,' + l!n$i,u+s c1@ B'"' +l nu! ini%i'l c2@ B'"' +l nu! n'l. L',"i- + "i$i+- +l +l+(+n,!@ +n +s,+ %'s! +s l' ini%'' +n 1&.&.

8


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2.2 En&a(,la7e de la (atri8 de rigide8

(',"i- + "i$i+- + un' +s,"u%,u"' K@ s+ !),i+n+ B!" +ns'()l'*+ i"+%,! + l's (',"i%+s + "i$i+- +n!"+n''s $l!)'l+s + %'' un! + l!s +l+(+n,!s + l' (is('. P'"' +ll! +s n+%+s'"i! +ni" +l +%,!" +!%'%in 9C@ +l (is(! 5u+ +s, %!n!"('! B!" l!s $"'!s + li)+",' +l nu! ini%i'l n'l + un(+n,!.

E4EM)LO N.5 2

%!n,"'" l' (',"i- + "i$i+- + l' +s,"u%,u"' ini%'' +n l' $u"' 1&.&@ %!nsi+"'n! nu!s "#$i!s.!s l!s +l+(+n,!s s!n + 30U30. S+ %!nsi+"'n l!s (is(!s 'l!"+s + E G@ + l!s +*+(Bl!s 'n,+"i!"+s.

Figura -3.3 G+!(+,"#' $"'!s + li)+",' + B",i%! Bl'n!@ u,ili-'! +n +*+(Bl!3.

SOL6CI/N

' l' %!lu(n' i-5ui+"'@ +l +l+(+n,! n(+"! 1@ l' i$' +l 2 l' %!lu(n' +"+%K' +l 3. L!s +%,!"+s +!%'%in + +s,!s +l+(+n,!s@ s!n>

(',"i- + "i$i+- +l +l+(+n,! !s@ s+ ini% +n +l +*+(Bl! 'n,+"i!" l' + l!s +l+(+n,!s un! ,"+s@ 'li%'" 1&.&@ s+ !),i+n+>


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: CONDENSACI/N DE LA MATRIZ DE RIGIDEZ

l' $u"' 1&.8@ s+ B"+s+n,' nu+'(+n,+ l' +s,"u%,u"' 5u+ s+ K' +ni! 'n'li-'n! %u!s $"'!s ++",' s+ ini%'"!n +n l' $u"' 1&.&. A l' i-5ui+"' s+ ini%'n ,!!s l!s $"'!s + li)+",' ' l' +"+%K'ini%' ni%'(+n,+ l' %!!"+n'' ' l' %u'l s+ ' ' %!n+ns'" l' (',"i- + "i$i+-.

+l sis,+(' + %!!"+n''s + un' +s,"u%,u"'@ s+ Bu++ i+"+n%i'" un $"uB! + %!!"+n''s ' l's 5u+

+n!(in' VV%!!"+n''s 'WW@ 5u+ +n +l +*+(Bl! + l' $u"' 1&.8 +s l' un! l's "+s,'n,+s@ ' l's 5u+ s+n!(in' %!!"+n''s )WW. Al K'%+" +s,!@ ,'n,! +l +%,!" + %'"$'s $+n+"'li-''s Q@ %!(! +l +%,!" +!"+n''s $+n+"'li-''s q@ +s,n B'",i%i!n'!s + l' si$ui+n,+ !"('>

" !,"' B'",+@ l' +%u'%in )si%' + 'nlisis +s,,i%!@ 5u+ "+l'%i!n' +l +%,!" + %'"$'s $+n+"'li-''s Q@n +l +%,!" + %!!"+n''s $+n+"'li-''s q@ B!" (+i! + l' (',"i- + "i$i+- + l' +s,"u%,u"' K@ +s>


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Q 7K q 1&.

"++(Bl'-'" 1&.8.1 1&.8.2 +n 1&. 'l ,"')'*'" %!n su)(',"i%+s@ l' (',"i- + "i$i+- + l'"u%,u"'@ ,'()iJn +s,'" B'",i%i!n''@ + l' si$ui+n,+ !"('>

%!n+ns'%in +s,,i%' + l' (',"i- + "i$i+- s+ ' %u'n! 6' ! 6) s!n %+"!s@ l!s !s %'s!s s+s'""!ll'n ' %!n,inu'%in>

:.- Conden&acin a la& coordenada& ;a;

+ %'s! s+ B"+s+n,' %u'n! +l +%,!" .

!n+>

+$!>

' KX l' (',"i- + "i$i+- %!n+ns'' ' l's %!!"+n''s '.

:.1 Conden&acin a la& coordenada& ;,;

B"+s+n,' %u'n! +l +%,!" + %'"$'s Qa 7 >. P"!%+i+n! +n !"(' si(il'" s+ !),i+n+>

' KX l' (',"i- + "i$i+- %!n+ns'' ' l's %!!"+n''s ).

10


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E4EM)LO N.5 :

%!n,"'" l' (',"i- + "i$i+- %!n+ns'' ' l' %!!"+n'' l',+"'l 1@ ini%'' +n l' $u"' 1&.8@ 5u+"+sB!n+ ' l' +s,"u%,u"' + l' $u"' 1&.&@ 5u+ s+ K' +ni! 'n'li-'n!.

SOL6CI/N

+s,+ %'s!@ l' B'",i%in + l' (',"i- + "i$i+- + l' +s,"u%,u"' Ks+ l' "+'li-' +n l' B"i(+"' l' B"i(+"'u(n'@ ,!' +- 5u+ +is,+ un' s!l' %!!"+n'' '. P!" l! ,'n,! l's su)(',"i%+s@ s!n>

su)(',"i- )' +s l' ,"'nsBu+s,' + l' su)(',"i- '). P'"' 'Bli%'" l' +%u'%in 1&.10.3 +s n+%+s'"i!%ul'" l' in+"s' + )'.

? CONDENSACI/N MEDIANTE SOL6CI/N DE EC6ACIONES

,"')'*'" %!n l' +%u'%in 1&.10.3 ! %!n l' +%u'%in 1&.11.3 i(Bli%' %'l%ul'" un' (',"i- in+"s'@ l! %u'l('n' )'s,'n,+ ,i+(B! + %l%ul!@ "'-n B!" l' %u'l l! 5u+ s+ K'%+ +n l' B"%,i%'@ +s ,"'ns!"('" +l%ul! + l' (',"i- in+"s' B!" un sis,+(' + +%u'%i!n+s lin+'l+s@ %!(! s+ + ' %!n,inu'%in.

?.- Ca&o en @ue

l' +%u'%in 1&.10.3 s+ "+'li-'@ s+ +n+ l' (',"i- T+ l' si$ui+n,+ ('n+"'>

11


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(ul,iBli%'" '()!s l'!s + l' +%u'%in 1&.12.1 B!" K bb s+ !),i+n+>

ra encontrar la matriz T , se debe resolver n con!nto de ecaciones lineales, c"a matriz de

e#cientes es la sbmatriz ,," los t$rminos inde%endientes son las di&erentes colmnas de la sbmatriz

n +l %'()i! + '"i')l+ "+'li-'!@ l' +%u'%in 1&.10.3 s+ ,"'ns!"(' +n>

E4EM)LO N.5 ?

%!n,"'" l' (',"i- + "i$i+- %!n+ns'' +l +*+"%i%i! 'n,+"i!"@ B!" in,+"(+i! + l' (',"i- T.

SOL6CI/N

sus,i,ui" l's su)(',"i%+s@ +l +*+(Bl! 'n,+"i!" +n 1&.12.2@ s+ !),i+n+>

s!lu%in +l sis,+(' + +%u'%i!n+s lin+'l+s@ "+B!",'

?.1 Ca&o en @ue

B"!%++ +n !"(' si(il'" 'l ini%'! +n +l 'B'",'! 1&.3.1@ %!n l! 5u+ s+ !),i+n+>

12


14/37

!"'@ l' (',"i- Ts+ !),i+n+ "+s!li+n! un %!n*un,! + +%u'%i!n+s lin+'l+s 5u+ ,i+n+n un' s!l' (',"i- ++%i+n,+s 5u+ +s aaB+"! i+"+n,+s ,J"(in!s in+B+ni+n,+s 5u+ s!n l's i+"+n,+s %!lu(n's + l',"i- Ba,.

3 CONDENSACI/N MEDIANTE ELIMINACI/N DE GA6SS

)i+n +s %i+",!@ (+i'n,+ l' s!lu%in + un %!n*un,! + +%u'%i!n+s lin+'l+s@ s+ !B,i(i-' l' !),+n%in +(',"i- + "i$i+- %!n+ns''. N! +s (+n!s %i+",!@ 5u+ ,!'#' s+ Bu++ !B,i(i-'" +l B"!%+s! + %l%ul!%'(+n,+ ,"i'n$ul'"i-'n! l' (',"i- + "i$i+-@ ,+(' 5u+ s+ ,"',' ' %!n,inu'%in +s li! ni%'(+n,+"' +l %'s! + 5u+ .

!n+>

' l' +%u'%in 1&.1/.1 (ul,iBli%'(!s B!" '':1@ +n Js,' s+ "++(Bl'-' l' +%u'%in1&.13.1@ s+ !),i+n+>

!"'@ si ' l' +%u'%in 1&.1/.3 (ul,iBli%'(!s B!" :)' su('(!s ' l' +%u'%in 1&.1/.2@ s+ +n%u+n,"'>

'%u+"! ' 1&.13.2@ l' +%u'%in +n,"+ B'"Jn,+sis +s l' (',"i- + "i$i+- %!n+ns'' .

"+s%"i)i" +n !"(' (',"i%i'l l's +%u'%i!n+s 1&.1/.3 1&.1/. s+ K'll'.

" %!nsi$ui+n,+@ '' l' (',"i- + "i$i+- ,!,'l@ s+ 'Bli%' l' +li(in'%in + G'uss =!"'n K's,' +li(in'" l!s(+n,!s %!""+sB!ni+n,+s ' l's %!!"+n''s ' l! 5u+ s+ !),i+n+n s!n l's (',"i%+s T .

E4EM)LO N.5 3

%!n,"'" l' (',"i- + "i$i+- %!n+ns''@ + l' +s,"u%,u"' 5u+ s+ K' +ni! 'n'li-'n!@ B+"! 'Bli%'n! l'(in'%in + G'uss =!"'n.

SOL6CI/N

13


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(+"! s+ +)+ +n%!n,"'" l' (',"i- + "i$i+- + l' +s,"u%,u"'@ B'"' l' nu+' nu(+"'%in + l!s $"'!s ++",'@ 5u+ s+ ini%'n +n l' $u"' 1&.. N,+s+ 5u+ l' %!!"+n'' l',+"'l@ s+ K' nu(+"'! 'l l,i(!.

gura -3. Nu(+"'%in + l!s $"'!s + li)+",' B'"' B!+" 'Bli%'" l' +li(in'%in + G'uss

,"i'n$ul'"i-'" l' (',"i- + "i$i+-@ s+ !),i+n+>

'l(+n,+@ 'l ll+'" ' l' !"(' + l' +%u'%in 1&.1/.@ s+ +n%u+n,"'>

s 'l!"+s + l's %u',"! B"i(+"'s l's + l' 5uin,' %!lu(n'@ %!""+sB!n+n ' l' (',"i- 5 T@ l' i+"+n%i'

+ +is,+ +s +)i! 'l "+!n+!. El l,i(! 'l!" +s l' (',"i- + "i$i+- %!n+ns'' ' l' %!!"+n''+"'l@ + l' +s,"u%,u"' 'n'li-''.

1/


16/37

ARRA ARTIC6LADA5RGIDA

' )'""' +n l' 5u+ un! + sus +,"+(!s i s+' '",i%ul'! s+ %'"'%,+"i-' B!" ,+n+" $l. A i+"+n%i' +ls! + )'""' '",i%ul'' +n '()!s +,"+(!s@ '5u# s# B!"#'n 'B'"+%+" u+"-'s %!",'n,+s.

",i+n! + l' +%u'%in %!""+sB!ni+n,+ ' un' )'""' )i+(B!,"''@ l' E%. 2.3.@ B!+(!s !),+n+" l',"i- B'"' un' )'""' '",i%ul'':"#$i' (+i'n,+ %!n+ns'%in +l ,+"%+" $"'! + li)+",' Y'i@ +l $i"! +l

! i@ !"-'n! 5u+ +l (!(+n,! (i s+' nul!>

ui+n! +l "+sul,'! !),+ni! +n l' E%. 2..&@ B!+(!s !),+n+" l' %!""+sB!ni+n,+ (',"i- + "i$i+- + >

1


17/37

$u'l 5u+ +n 'n,+"i!"+s (!+l!s + )'""'s@ l!s +%,!"+s + s!li%i,'%i!n+s +sBl'-'(i+n,!s l' (',"i-"i$i+- +n %!!"+n''s $l!)'l+s s+ "+l'%i!n'n %!n l's + %!!"+n''s l!%'l+s (+i'n,+ un' (',"i- +'%in T' B"!Bi' + %'' )'""' (+i'n,+>

1&


18/37

2. RIGIDEZ LATERAL

1INTROD6CCION.

"'n,+ +l (!i(i+n,! + un' +i%'%in B!" l' '%%in s#s(i%'@ l'si%i,'%i!n+s s!)"+ '5u+ll' s!n "+'l(+n,+ + i"+%%in+"s'. S+ K' ll+$'! ' %!nsi+"'" 5u+ +l (!i(i+n,! +l su+l!n+ s+is %!(B!n+n,+s + (!i(i+n,! in+B+ni+n,+s@ ,"+ssl'%i!n'l+s ,"+s "!,'%i!n'l+s. D+n,"! + +s,'s %!(B!n+n,+s@ ,"'sl'%i!n'l+s +n l's i"+%%i!n+s K!"i-!n,'l+s su+l+n s+"

(''s +n %u+n,'@ +n !"(' in+B+ni+n,+@ B'"' n+s ,+n+" %!ni%i!n+s + %'"$' +n l!s 'nlisis@ '! 5u+ B!" l!

n+"'l s!n l!s (s i(B!",'n,+s.

+l %'s! + un B",i%! Bl'n!@ l' s!l' %!nsi+"'%in + uni(i+n,! ,"'sl'%i!n'l + l' )'s+ i(Bli%'"#' l' 'B'"i%in +

%i!n+s + in+"%i' ,"'sl'%i!n'l+s "!,'%i!n'l+s. Sin +()'"$!@ l!s $i"!s !%'si!n'!s s!n "+l',i'(+n,+5u+!s@ B!" l! 5u+ l's '%%i!n+s "!,'%i!n'l+s ,'()iJn l! s!n B"%,i%'(+n,+ n! inQu+n +n l!s ++%,!s'l+s s!)"+ l' +s,"u%,u"'@ ,'n,! ' ni+l + +sBl'-'(i+n,!s %!(! + u+"-'s in,+"n's. P!" +s,' "'-n@ s+nsi+"' un' '%%in + in+"%i' ,"'sl'%i!n'l@ B!" l! 5u+ l' u+"-' s#s(i%' ,i+n+@ B'"' n+s + 'nlisis@ unn,i! K!"i-!n,'l.

1 RIGIDEZ LATERAL DE 6N )/RTICO SIM)LE' +l B",i%! Bl'n! si(Bl+@ + un' %"u*#'@ (!s,"'! +n l' Fi$u"' 1@ s!(+,i! ' l' '%%in + un' u+"-'"i-!n,'l F@ 5u+ "+B"+s+n,' l' '%%in s#s(i%'. L' +!"('%in 'i'l + l!s +l+(+n,!s n! s+ %!nsi+"'"+%i')l+@ + (!! 5u+ l!s ,"+s $"'!s + li)+",' +l sis,+(' %!nsis,+n +n un +sBl'-'(i+n,! l',+"'l s $i"!s +n l!s nu!s suB+"i!"+s.

+%u'%in )si%' +l 'nlisis (',"i%i'l + +s,"u%,u"'s +s l' si$ui+n,+>

18


19/37

F

0

0 }=

[24EIc

h3

6EIc

h2

6EIc

h2

6EIch

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4EIch +

4EIvL

2EIvL

6EIc

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notandoH

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F

0 }=[K11 K12K21 K22]{U}

ndeH

11 > su)(',"i- %!n ,"'sl'%i!n+s !"i$in''s B!" l!s $"'!s + li)+",' + ,"'sl'%in. 22 > su)(',"i- %!n "!,'%i!n+s !"i$in''s B!" l!s $"'!s + li)+",' + "!,'%in. 12 > su)(',"i- %!n ,"'sl'%i!n+s !"i$in''s B!" l!s $"'!s + li)+",' + "!,'%in. 21 > su)(',"i- %!n "!,'%i!n+s !"i$in''s B!" l!s $"'!s + li)+",' + ,"'sl'%in.

&arrollando la& ecuacione&H

=K11U+K12

1


20/37

=K21U+K22

la &egunda ecuacin! &e de&e7aH

=K221K21U

+(Bl'-'n! l' +B"+sin + +n l' B"i(+"' +%u'%in@ s+ ,i+n+>

=K11UK12K221

K21U

=(K11K12K221K21)U

!n+>

KL=(K11K12K221K21)

nal(ente! &e o,tiene ara el rticoH

KL=12EIc

h3 ( 1+62+3)

!B+"'%in "+'li-'' s+ +n!(in' C!n+ns'%in Es,,i%'. Ti+n+ B!" !)*+,! "+u%i" l' (',"i- + "i$i+-n l!s ,J"(in!s 's!%i'!s +%lusi'(+n,+ ' l's u+"-'s '%,u'n,+s s!)"+ l' +s,"u%,u"'. D+ +s,' ('n+"' s+n%+n,"' l' l')!" +n l' !),+n%in + %i+",!s +sBl'-'(i+n,!s B'"'@ ' B'",i" + Js,!s@ %'l%ul'" l!ssBl'-'(i+n,!s "+s,'n,+s@ sin l!s %u'l+s n! B!"#'n %'l%ul'"s+ l's u+"-'s in,+"n's %!(Bl+,'(+n,+. En+ %'s!@ (+i'n,+ un' %!n+ns'%in +s,,i%' l' (',"i- + "i$i+- !"i$in'l u+ "+u%i' ' un' (',"i- ++- l',+"'l + un ,J"(in! B'"' !),+n+" +l +sBl'-'(i+n,! l',+"'l + Bis! %'us'! B!" un' u+"-'

"i-!n,'l.

2 MATRIZ DE RIGIDEZ LATERAL DE 6N )/RTICO DE 9ARIOS )ISOS.

un B",i%! + '"i!s Bis!s@ l' (',"i- + "i$i+- ,!,'l +s un' !B+"'%in "+B+,i,i' + +ns'()l+s +,"i%+s + l!s +l+(+n,!s@ s+'n +s,'s i$'s@ %!lu(n's@ (u"!s ! '""i!s,"+s@ %!(! s+ (u+s,"' +n l' Fi$u"'P'"' !),+n+" l' (',"i- + "i$i+- l',+"'l s+ K'"n l's (is('s suB!si%i!n+s 5u+ +n l' si,u'%in 'n,+"i!"@" +*+(Bl!@ l!s +sBl'-'(i+n,!s l',+"'l+s s!n i$u'l+s ' ni+l + %'' Bis! +!"('%i!n+s 'i'l+s n!nsi+"''s l's '%%i!n+s + in+"%i' "!,'%i!n'l+s n! s!n ,!(''s +n %u+n,'@ s!l'(+n,+ l's '%%i!n+s"i-!n,'l+s. A+(s@ +l (!+l! s+"#' (s 'B"!Bi'! B'"' +i%i!s + )'*' ' (+i'n' 'l,u"'@ +n l!s %u'l+s++%,!s + l's +!"('%i!n+s 'i'l+s s!n B!%! %!nsi+"')l+s.

1


21/37

(',"i- + "i$i+- ,!,'l +sB"+s+n,'' B!" un' s+"i+ +)(',"i%+s@ 5u+ ,i+n+n +l (is(!ni%'! 5u+ +n +l '%Bi,+ 'n,+"i!".

U1

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K21 K22]

+$! s+ ll+$' ' l' (is(' +B"+sin.

F

0 }=[K11 K12K21 K22]{U}s'""!ll'n! (',"i%i'l(+n,+ l's B'",i%i!n+s ++%,u'n! l' %!n+ns'%in +s,,i%'>

=(K11K12K221K21)U

=(KL)U

+$!@ l' "i$i+- l',+"'l +s, '' B!" l' +B"+sin (',"i%i'l>

KL=(K11K12K221K21)

: E4EM)LO DE A)LICACI/N

+s'""!ll'" +l 'nlisis + un' +s,"u%,u"' 'B!",i%'' + %!n%"+,! '"('! E7210& ,U( + un Bis!@nsi+"'n! 5u+ +l i'"'$(' + Bis! +s su%i+n,+(+n,+ "#$i! B'"' "+'li-'" +l 'nlisis %!n un (!+l!

+u!:,"ii(+nsi!n'l. L' u+"-' 'Bli%'' F +s + 10 ,!n+l''s@ %!(! s+ ini%' +n l' $u"'.

20


22/37

O),+n%in + l' Ri$i+%+s L',+"'l+s + %'' B",i%!

P",i%! TiB! I

I 7 ./ 10:3 (/I% 7 &.8 10:/ (/L 7 &.0 (K 7 3. (E 7 E% 7 2 10:& ,U(

= 5.4 /66.75/3.5

=0.4667

,"i- + Ri$i+->

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21


23/37

KL=(K11K12K221K21)

+"'n!>

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6.848 T/cm

P",i%! TiB! II

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+"'n!>

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2

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22


24/37

23


25/37

2. M6ROS DE CORTE

- DEFINICION.

n +n ++%,! vias en voladizo vertical de ran %eralte e %ro%orcional estabilidad lateral a lastrctras al resistir las &erzas cortantes " momentos *e+ionantes en ss %lanos casados %or las

erzas laterales@ l' "+sis,+n%i' + l!s (u"!s %!",'n,+s +s %'si si+(B"+ %!n,"!l'' B!" sus "+sis,+n%i's 'in@ sin +()'"$! +n 'l$un's !%'si!n+s Bu++n "+5u+"i" 'l$n "+u+"-! %!",'n,+ B'"' B"++ni" l's 'll's" ,+nsin i'$!n'l.

s (u"!s +)+n s+" su%i+n,+(+n,+ u+",+s B'"' li(i,'" l's +Q+i!n+s ' 'l!"+s "'-!n')l+s is+'"s+('n+"' 5u+ l!s +su+"-!s + ,+nsin %'us''s B!" l's u+"-'s l',+"'l+s n! +%+'n l!s +su+"-!s +

(B"+sin %'us'!s B!" +l B+s! +l +i%i!.

s (u"!s %!",'n,+s son sados comnmente en edi#cios con losas de %iso de %laca %lana +s,'()in'%in + l!s's (u"!s +s +l ,iB! (s %!(n + %!ns,"u%%in us'! +n +i%i!s 'l,!s +'",'(+n,!s !,"!s ,iB!s + +i%i!s "+si+n%i'l+s.

s (u"!s + %!",'n,+ s'l'n l's is,'n%i's +",i%'l+s +n,"+!s@ si s!n %ui'!s'(+n,+ si(J,"i%'(+n,+ %!l!%'!s Bl'n,'@ "+sis,i"n +%i+n,+(+n,+ si in,+"+"i"

nsi+"')l+(+n,+ %!n l!s "+5uisi,!s '"5ui,+%,ni%!s. En l'+%%in K!"i-!n,'l Bu++n us'"s+ (u"!s + %!",+ 5u+ %!""'n s!)"+' l' l!n$i,u + l!s B'n+l+s ! %"u*#'s %u'n! l's u+"-'s s!n

n!"+s +ll!s ,i+n+n 5u+ %!""+" s!)"+ l!n$i,u+s B'"%i'l+s + l!sn+l+s.

l' $u"' / (u+s,"' un (u"! + %!",'n,+ s!(+,i! ' un'+"-' l',+"'l 9u. El (u"! +s +n "+'li' un' i$' +n !l'i-! +%K! J B"!uni' ,!,'l LK.+n l' B'",+ ' + l' $u"' +l (u"!' Q+i!n'! B!" 9u + i-5ui+"' ' +"+%K'@ B!" l! 5u+ s+ui+"+n )'""'s + ,+nsin ' l' i-5ui+"' ! l'! + ,+nsin. Si 9u s+i%' +s+ l' +"+%K' %!(! +n l' B'",+ ) s+ "+5u+"i"n )'""'s +sin +n +l +,"+(! +"+%K!. P!" +n+ un (u"! + %!",'n,+

%+si,' "+u+"-! + ,+nsin +n '()!s l'!s ' 5u+ 9u Bu++

,u'" +s+ %u'l5ui+" i"+%%in. P'"' %l%ul!s B!" Q+in@ l' B"!uni' + l' i$' +l +,"+(! +(B"+sin +l (u"! 'l %+n,"! + $"'+' + l's )'""'s + ,+nsin s+ +s,i(' 5u+ +s 'B"!i(''(+n,+0@ +%+s l' l!n$i,u LK +l (u"! + '%u"! 'l ACI. El (u"! + %!",'n,+ '%,' %!(! un' i$' +",i%'l +n'i-! 'l B"!B!"%i!n'" s!B!",+ l',+"'l 5u+' s!(+,i! ' Q+in u+"-'s %!",'n,+s. P'"' ,'l (u"!@ l'

+"-' %!",'n,+ (i(' 9u +l (!(+n,! Q+i!n'n,+ (i(! Mu s+ %'l%ul' +n l''s+. Si s+ %'l%ul'nu+"-!s + Q+in@ l' ('$ni,u + +s,!s s+" '+%,'' B!" l' %'"$' 'i'l + is+! N 4 %u! ++%,! +)+lui"s+ +n +l 'nlisis.

2/


26/37

Figura :.5Mu"! + %!",'n,+

1 RE> )ARA M6ROS DE CORTANTE

-. L' u+"-' %!",'n,+ '%,!"i-'' + l' i$' +)+ s+" i$u'l ! (+n!" 5u+ l' "+sis,+n%i' + is+! B!"%!",'n,+ +l (u"!.

9u Vn

1. L' "+sis,+n%i' + is+! B!" %!",'n,+ +l (u"! +s i$u'l ' l' "+sis,+n%i' + is+! B!" %!",'n,+ +l%!n%"+,! (s l' +l "+u+"-! B!" %!",'n,+.

Vu Vc Vs

2. L' "+sis,+n%i' n!(in'l B!" %!",'n,+ -n+n %u'l5ui+" s+%%in K!"i-!n,'l +n +l Bl'n! +l (u"! n! +)+

,!('"s+ ('!" 5u+ 10 f c .d11..3.

:. Al is+'" l's u+"-'s %!",'n,+s K!"i-!n,'l+s +n +l Bl'n! + un (u"!@ d+)+ ,!('"s+ i$u'l ' 0. l[@!n+ l[ +s l' l!n$i,u K!"i-!n,'l +l (u"! +n,"+ l's %'"'s + l!s 'B!!s@ ' (+n!s 5u+ Bu+'

*us,i%'"s+ un 'l!" ('!" B!" (+i! + un 'nlisis + %!(B',i)ili' + +!"('%i!n+s 11../.?. L' s+%%in 11... +l ACI +s,')l+%+ 5u+ ' (+n!s 5u+ s+ K'$' un %l%ul! (s +,'ll'! %!(! s+

+s%"i)+ +n +l si$ui+n,+ B""'!@ +l 'l!" us'! + l' "+sis,+n%i' n!(in'l B!" %!",'n,+ -c n! +)+

s+" ('!" 5u+ 2 f c .d +n l!s (u"!s s!(+,i!s ' un' %'"$' + %!(B"+sin 'i'l '%,!"i-''.

Si un (u"! +s, s!(+,i! ' un' %'"$' + ,+nsin@ +l 'l!" + -c n! +)+ s+" ('!" 5u+ +l 'l!"!),+ni! %!n l' si$ui+n,+ +%u'%in>

Vc=2(1+ Nu500Ag )%f'cbwd 0 Ecuacion--5! ACI

3. 4s'n! un' 'nlisis (s +,'ll'!@ +l 'l!" + -cs+ ,!('"' %!(! +l (+n!" 'l!" 5u+ s+ !),+n$' 'l

sus,i,ui" +n l' !s +%u'%i!n+s 5u+ si$u+n@ +n !n+ +s l' %'"$' 'i'l '%,!"i-'' n!"('l ' l'

2


27/37

s+%%in ,"'ns+"s'l 5u+ s+ B"+s+n,+ si(ul,n+'(+n,+ %!n -u. S+ ,i+n+ 5u+ s+ %!nsi+"'"'B!si,i' B'"' %!(B"+sin n+$',i' B'"' ,+nsin.

Vc=3.3%f'c hd+Nud

4 l w Ecuacion--51! ACI

)i+n

Vc=[0.6 %f' c+ lw (1.25 %f'c+

0.2Nu

lwh)Mu

Vu

lw

2]hd Ecuacion--51! ACI

B"i(+"' + +s,'s +%u'%i!n+s s+ !"(ul B'"' B"++%i" l' "+sis,+n%i' 'l '$"i+,'(i+n,! in%lin'! +n

'l5ui+" s+%%in + un (u"! + %!",'n,+@ 5u+ %!""+sB!n+" ' un +su+"-! B"in%iB'l + ,+nsin +"!i(''(+n,+ /

f' c +n +l %+n,"!i+ + l' s+%%in ,"'ns+"s'l +l (u"!. L' s+$un' +%u'%in u+

(ul'' B'"' %!""+sB!n+" ' l' B"+s+n%i' + un +su+"-! + ,+nsin B!" Q+in + &

f'c +n un'

%%in l[U2 '""i)' + l' s+%%in in+s,i$''. Si 0U- : l[U2 "+sul,' n+$',i!@ l' s+$un' +%u'%in n!n" s+n,i! n! s+" u,ili-''.

En uni'+s + SI@ +s,'s ,"+s l,i('s +%u'%i!n+s ,!('n l' !"('>

Vc=(1+0.3Nu

Ag )%f'c

6 bwd 0 Ecuacin --5P! ACI

Vc=1

4% f' c hd+

Nud

4 lw

Vc=

[

1

2%f' c+

lw (%f'c+2 Nulwh )Mu

Vu

lw

2

]

hd

10

. L!s 'l!"+s + -%'l%ul'!s %!n l's !s +%u'%i!n+s 'n,+"i!"+s ' un' is,'n%i' + l' )'s+ i$u'l ' l[U2! )i+n K[U2 l' 5u+ s+' (+n!"@ s!n 'Bli%')l+s ' ,!'s l's s+%%i!n+s +n,"+ +s,'s s+%%i!n+s l' + l')'s+ +l (u"! 11..8.

. Si l' u+"-' %!",'n,+ '%,!"i-'' - +s (+n!" 5u+ \


28/37

s +s l' s+B'"'%in +l "+u+"-! B!" ,!"sin ! B!" %!",'n,+@ +n un' i"+%%in B+"B+ni%ul'" 'l "+u+"-!K!"i-!n,'l 11...1.

Vs=Avfyd

s Ecuacin --51P! ACI

--. L' %'n,i' + "+u+"-! B!" %!",'n,+ K!"i-!n,'l B,%!(! B!"%+n,'*+ +l "+' ,!,'l +",i%'l +%!n%"+,! n! +)+" s+" (+n!" 5u+ 0.002 11...2.

-1. L' s+B'"'%in (i(' +l "+u+"-! s2B!" %!",'n,+ K!"i-!n,'l n! +)+" s+" ('!" K[U@ 3.!1Bl$ 11...3.

-2. L' %'n,i' + "+u+"-! B!" %!",'n,+ +",i%'l Bn%!(! B!"%+n,'*+ +l "+' ,!,'l K!"i-!n,'l +%!n%"+,! n! +)+" s+" (+n!" 5u+ +l 'l!" '! B!" l' si$ui+n,+ +%u'%in@ +n l' 5u+ K [ +s l' 'l,u"',!,'l +l (u"! 11.../.

!=0.0025+0.5

(2.5

hw

lw

)(h0.0025 ) Ecuacin --52>! ACI

+)+ s+" (+n!" 5u+ 0.0002 B+"! ,'(B!%! ('!" 5u+ +l "+u+"-! B!" %!",'n,+ K!"i-!n,'l "+5u+"i!%t.

l!s (u"!s 'l,!s@ +l "+u+"-! +",i%'l +s (u%K! (+n!s +%'- 5u+ +n l!s (u"!s )'*!s. Es,+ K+%K! s+Q+*' +n l' +%u'%in 'n,+"i!"@ !n+ B'"' (u"!s %!n un' "+l'%in + 'l,u"' ' l!n$i,u (+n!" 5u+ 0.@ l'n,i' + "+u+"-! +",i%'l n+%+s'"i' +s i$u'l 'l "+u+"-! K!"i-!n,'l "+5u+"i!. Si l' "+l'%in +s ('!"+ 2.@ s!l! s+ "+5ui+"+ un' %'n,i' (#ni(' + "+u+"-! +",i%'l +s +%i"@ 0.002s..

-:. L' s+B'"'%in (i(' +l "+u+"-! B!" %!",'n,+ +",i%'l. N! +)+" s+" ('!" 5u+ K[U3@ 3K !1 Bl$. 11....

-?. El %!",'n,+ +n +l Bl'n! +l (u"! +s i(B!",'n,+ B"in%iB'l(+n,+ B'"' (u"!s + %!",'n,+ %!nun' B+5u+' "+l'%in 'l,u"' ' l!n$i,u. D+ ('n+"' 'l,+"n',i'@ s+ B+"(i,+ is+'" (u"!s %!n un''l,u"' (i(' 5u+ n! +%+' 'l !)l+ +l l'"$! +l (u"! B'"' l's u+"-'s %!",'n,+s K!"i-!n,'l+s.

-3. S+$n +l NEC 2011

2 CLASIFICACION DE LOS M6ROS DE CORTANTEH

1.: Muro& ancJo&H%u' 'l,u"' n! +%+' l' ,+"%+"' B'",+ + l' l!n$i,u su )'s+ s+ +n%u+n,"++(B!,"''. A5u# +l ++%,! + l' u+"-' %!",'n,+ s+ %!nsi+"' B"i('"i!@ l!s ++%,!s + Q+in Bu++n's%+n+" + 10 ' 1] B'"' +l %l%ul! + l's +!"('%i!n+s.

2.: Muro& e&,elto&H P"+s+n,'n +!"('%i!n+s i(B!",'n,+s B!" +l +su+"-! %!",'n,+ n!"('l$+n+"'! B!" Q+in@ +s,! +s 5u+@ l!s +l+(+n,!s (+%ni%!s (!(+n,!s Q+i!n'n,+s u+"-'s%!",'n,+s %!n,"i)u+n ' l' +!"('%in +l sis,+('@ L' in,+"'%%in %!n l!s (u"!s + l' +s,"u%,u"''l,+"' l' "i$i+- B"in%iB'l(+n,+ %!n l!s (u"!s suB+"i!"+s.

sis,+(' +s,"u%,u"'l ,!,'l + un +i%i! + ii+ )si%'(+n,+ +n 2 $"uB!s>

28


29/37

Su)sis,+(' +",i%'l> S!n +s)+l,!s +n un' ! '()'s i"+%%i!n+s s+%%i!n'l+s B!" si (is(!s n!Bu++n s+" (u +s,')l+s@ +is,i+n! 3 ,iB!s + su)sis,+('s +",i%'l+s +s+n%i'l+s> Su)sis,+('s +Mu"!s@ C'!n+s


30/37

is,+n%i' K!"i-!n,'l@ ' l' "!,'%in B+"(i,+ 5u+ l's +s5uin's +l +i%i! s+ (u+'n B!" ++%,! +(B+"',u"'@ %!""i(i+n,! %!n,"'%%in. L' $u"' $ es n caso raro en e los mros %er%endiclares dan#ciente resistencia a las &erzas cortantes, %ero no a la torsin @ si(il'" ' l' $u"' ) +n 5u+ +l sis,+('su %!n*un,! B"!B!"%i!n' +s%'s' "+sis,+n%i' ' l' ,!"sin "+sB+%,! ' un' u+"-' K!"i-!n,'l 'si(J,"i%'

)"+ +l +i%i!. L!s (u"!s %u"!s + l' $u"' K Bu++n "+sis,i" l',+"'l(+n,+ +n i",u + su '%%in +

n%K'@ +sB+%i'l(+n,+ si l!s Bis!s si"+n %!(! i'"'$('s 5u+ "i$ii-'n i%K' %!n%K'.

? COM)ORTAMIENTO DE LOS M6ROS DE CORTANTE

)i! ' l's n+%+si'+s '%,u'l+s "+5ui+"+n 5u+ l!s +i%i!s 'l%'n%+n 'l,u"'s $"'n+s %!(B'"''s %!n +l+' 5u+ !%uB'n +)i! ' +s,!@ l's %'"'%,+"#s,i%'s + +s,+ ,iB! + !)"'s B"+s+n,'n 2 B"!)l+('s'(+n,'l+s>

1. R+5ui+"+n + un' "i$i+- +l+''@ B!" l! 5u+ s+ +(Bl+'n (u"!s + %!",'n,+ B'"' B"!B!"%i!n'" l''+%u'' "i$i+-@ %!l!%n!l!s +s,"',J$i%'(+n,+ +n %u)!s + +l+'!"+s ! +s%'l+"'s s+$n +lis+! !"i+n,'%in + l' +s,"u%,u"'.

2. El %!(B!",'(i+n,! + l!s +(s +l+(+n,!s 5u+ !"('n l' +s,"u%,u"' s+ + '+%,'! B!" l' "i$i+-5u+ 'n l!s (u"!s + %!",'n,+.

" +ll! +l 'nlisis is+! +s,"u%,u"'l +)+" +s,'" !"i+n,'! ' %!n*u$'" +l %!(B!",'(i+n,! +n,"+ '()!s(+n,!s +s,"u%,u"'l+s@ s!)"+ ,!! )'*! +l ++%,! B"!!%'! B!" l's u+"-'s K!"i-!n,'l+s +)i's ' l!si(i+n,!s s#s(i%!s + i+n,!.

+(Bl+! + (u"!s +n +i%i!s +)+ $'"'n,i-'" l' "+sis,+n%i' B'"' ')s!")+" l!s ++%,!s + l's u+"-'s@n%iB'l(+n,+ +)+" ,+n+"s+ +n %u+n,' l!s ++%,!s + "!,'%in.

s (u"!s + %!",'n,+ +n !"(' 'isl'' ,i+n+n !s (!!s + +!"('%in@ 5u+ +B+n+n +l (+%'nis(!!"('n,+ Q+in %!",'n,+@ si+n! +l B"in%iB'l +l Q+i!n'n,+@ +s +%i" un !l'i-! +",i%'l. T'()iJnlu+ l' u+"-' %!",'n,+ +l lu$'" !n+ s+ 'Bli%'.

2


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Figura .5P"in%iB'l+s +!"('%i!n+s +n (u"!s + %!",'n,+.

'l(+n,+ l!s +sBl'-'(i+n,!s +)i!s ' l' +!"('%in + l' %i(+n,'%in +B+n+n + l's"'%,+"#s,i%'s (+%ni%'s +l su+l!@ +ni!s B!" !s %!(B!n+n,+s>

4n! + ,"'nsl'%in _s B"!B!"%i!n'l 'l %!",'n,+ +n l' )'s+ +l +i%i! 4n! + "!,'%in B"!B!"%i!n'l 'l (!(+n,! Q+i!n'n,+ )'s+ +l +i%i!.

3 COM)ORTAMIENTO DE LOS M6ROS DE CORTANTE EN 9OLADIZO

(uro cortante en +oladi8o &i(le s+ %!(B!",' + l' (is(' ('n+"' 5u+ un' i$' + %!n%"+,!!"-'! l' s+%%in ,"'ns+"s'l +s,"+%K' ini%' 5u+ Bu++ Bl'n,+'"s+ +l B"!)l+(' + in+s,')ili' +l"+ + %!(B"+sin %!nsi+"'n! l' l!n$i,u %"i,i%' %!n "+sB+%,! 'l B'n+! i$u'l ' l' 'l,u"' +l Bis!s.

(uro crtate! &i act"a co(o un +oladi8o grande +s,'" su*+,! ' (!(+n,!s Q+i!n'n,+s u+"-'s

",'n,+s B"in%iB'l(+n,+ ' l's %'"$'s l',+"'l+s ' %!(B"+sin 'i'l B"!!%'' B!" l' $"'+'. Elu+"-! +",i%'l ! + Q+in +n l' B!"%in +l 'l(' +n un (u"! + %!",'n,+ s+ +)+ ,!('" +n %u+n,' 'l'lu'" l' %'B'%i' ' Q+in.

s "+5uisi,!s +s+n%i'l+s s!n %i(i+n,!s '+%u'!s 5u+ 'n *'%in ,!,'l ' l' )'s+ su%i+n,+ %!n+in +(u"!s %!",'n,+s B'"' ,"'ns(i,i" l' %'"$' K!"i-!n,'l

M6ROS DE CORTANTE EN 9OLADIZO CON )ATINES

K' "'-n B'"' +sB+"'" 5u+ l!s (u"!s 'l,!s + %!",'n,+ %!n B',#n s+ %!(B!",+n + ('n+"' is,in,' 5u+5u+ ,i+n+n s+%%i!n+s ,"'ns+"s'l+s "+%,'n$ul'"+s. Cu'n! l' u+"-' 'i'l +s B+5u+'@ B!+(!s

30


32/37

,i%iB'" 5u+ +s,!s (u"!s u,ili%+n +l B'" + '%+"! in,+"n! +n +l in,+"'l! in+ls,i%!@ B!" ,'n,! s+ B!"B!n+" + su%i+n,+ u%,ili'@ u"'n,+ un sis(!@ %!n ,'l 5u+ l' "+s,"i%%in %!n,"' +l B'n+! +l '%+"!' '+%u''.

'n l' %!(B"+sin 'i'l +s si$ni%',i'@ ,!! un B',#n B'",+ +l 'l(' Bu++n +s,'" +n %!(B"+sin. En

+s %'s!s s+ Bu++ %!nsi+"'" 5u+ l!s B',in+s s!n %!lu(n's %!n +s,"i)!s %'"$''s 'i'l(+n,+@ %!n un,!" (s )'*! + "+u%%in + %'B'%i' 70@8. P'"' l's B!si%i!n+s in,+"(+i's +l +*+ n+u,"!@ s+++ in,+"B!l'" +n,"+ 0@8 0@.

s B',in+s in%"+(+n,'n %!nsi+"')l+(+n,+ +l (!(+n,! + "+sis,+n%i' + (u"!s 'l,!s +n !l'i-! +",'n,+. P!" ,'n,! l' "+sis,+n%i' + l' u+"-' %!",'n,+ +n +l 'l(' Bu++ s+" (s %"#,i%' 5u+ +n l!s (u"!s+ ,+n$'n s+%%i!n+s ,"'ns+"s'l+s "+%,'n$ul'"+s. En un (u"! + %!",'n,+ )i+n is+'!@ n! s+ +sB+"'+ +l "+u+"-! ' %!",'n,+ %+' +n nin$un' +,'B' + l' %'"$'.

s (u"!s + %!",'n,+ %!",!s %!n B',in+s@ in%lus! %!n un' B+5u+' %'n,i' + "+u+"-! +",i%'l Bu++n(inis,"'" un' %'B'%i' ' Q+in 's!%i'' %!n un' %'"$' %!",'n,+ +%+si' +n +l 'l('@ %!n un' "+l'%in

K[Ul[+ 0@. T'()iJn un +(B'""ill'! + "+u+"-! +s ++%,i! B'"' is,"i)ui" uni!"(+(+n,+ l's $"i+,'s$!n'l+s B'"' %!n,"!l'" su 'n%K!. En l!s (u"!s + %!",'n,+ )'*!s %!n B',in+s@ l's *un,'s +

ns,"u%%in Bu++n %!ns,i,ui"s+ +n +l Bl'n! + 'll' %"#,i%'.

B',in' ' %!(B"+sin n! +s ++%,i! +n (u"! + %!",'n,+ )'*!s@ ' 5u+ n!s + Bu++ +s'""!ll'" +l )"'-!B'l'n%' $"'n+ i+'l@ n+%+s'"i! B'"' +l %!(B!",'(i+n,! + i$'.

5INTERACCION DE M6ROS DE CORTANTE Y MARCOS CON 46NTAS RIGIDAS

l' $u"' li,+"'l a s+ !)s+"' un (u"! +n !l'i-! + %!",'n,+ un ('"%!@ 5u+ ,"'ns(i,+n l' (is('"$' ' +,+"(in'' 'l,u"'@ l! 5u+ K'%+ 5u+ +l (u"! + %!",'n,+ su"' is,!"si!n+s Q+i!n'n,+s 5u+(+ un' B+ni+n,+ %!ns,'n,+ B!" '""i)' +l ni+l + %'"$'. El ('"%! +B+"i(+n,'l + +sBl'-'(i+n,! +sl'%in ,i+n+ ' K'%+"s+ +",i%'l B!" '""i)' +l ni+l + l' %'"$'.

gura5In,+"'%%in + (u"!s + %!",'n,+ ('"%!s "#$i!s.

'n! s+ +sB"+%i'n l!s '%!",'(i+n,!s + %!lu(n's@ l!s Bis!s B+"('n+%+n K!"i-!n,'l+s. D+)i! ' l'!(B',i)ili' + l's +!"('%i!n+s@ un (u"! + %!",'n,+s Bu++ !B!n+"s+ ' un ('"%! "#$i! +n l!s!s suB+"i!"+s l!s Bis!s in+"i!"+s@ s+ 'sis,+n (u,u'(+n,+ B'"' ,"'ns(i,i" l' %'"$' +,+"n'.

31


33/37

P M6ROS CORTANTE CON AERT6RAS

s +n,'n's Bu+",'s u%,!s + s+"i%i! "+5ui+"+n 5u+ l!s (u"!s in,+"i!"+s ! +,+"i!"+s + %!",'n,+n$'n ')+",u"'s. un' +s,"u%,u"' "'%i!n'l + (u"!s + %!",'n,+ +s ,'l 5u+ su %!(B!",'(i+n,! s+ Bu+''lu'" B!" si(Bl+ insB+%%in.

" l! $+n+"'l@ l's +s,"u%,u"'s i""'%i!n'l+s + (u"!s + %!",'n,+ +s'#'n l' s!lu%in (+i'n,+ 'nlisis"u%,u"'l+s n!"('l+s@ '5u# l's in+s,i$'%i!n+s + (!+l!s +s,ui!s + (!+l!s ni,!s Bu++n 'u'" '

'lu'" l's u+"-'s in,+"n's.

i(B+"',i! 5u+ l's ')+",u"'s in,+"+"'n l! (+n!s B!si)l+ %!n l' %'B'%i' + (!(+n,!s ,"'ns(isin%!",'n,+ + l' +s,"u%,u"'@ +n l' $u"' li,+"'l ' s+ (u+s,"' un )u+n +*+(Bl! + (u"! i""'%i!n'l +

",'n,+. El "+B+n,in! %'()i! + un' s+%%in + (u"! ' %!lu(n's "+u%+ "s,i%'(+n,+ l' "+sis,+n%i' 'in + l' +s,"u%,u"' + !l'i-!. En l' %"#,i%' s+%%in + l' )'s+. L' is,"i)u%in +s%'l!n'' + l's+",u"'s "+u%+ s++"'(+n,+ +l "+' + %!n,'%,! +n,"+ l!s 2 (u"!s@ +n +l li,+"'l ) s+ B!"#' ll+$'" ' uns! in+s+')l+ +n 5u+ +l (+%'nis(! + ,"'nsl'%in l',+"'l + l's %!lu(n's "!+' +l (u"! +n un'+%%in !Bu+s,' ' l' + l' ,"'sl'%in l',+"'l.

Figura.5Mu"!s i""'%i!n'l+s + %!",'n,+' In,+"+"+n%i' %!n l' "+sis,+n%i' ' Q+in %!",'n,+

) D+!"('%i!n+s in+s+')l+s +)i's ' l's %!lu(n's in%lin''s

-> M6ROS ACO)LADOS DE CORTANTE

%K!s (u"!s + %!",'n,+ %!n,i+n+n un' ! (s Kil+"'s +",i%'l+s + ')+",u"'s. 4n +*+(Bl! +s +l n%l+!%!",'n,+ + un +i%i! 'l,!@ 5u+ '%!(!' l!s %u)!s + +l+'!"+s@ B!-!s + +s%'l+"'s u%,!s +

"i%i!. L's Bu+",'s + '%%+s! + ,!!s +s,!s ',"'i+s'n l!s (u"!s. P!" ,'n,!@ l!s (u"!s s+ Bu++n+"%!n+%,'" +n,"+ i$'s %!",'s@ (u B+"'l,''s@ ll'('!s '%!Bl'!s B!" l' i$'. En l' $u"' li,+"'l ' s+s,"' un' +s,"u%,u"' ,#Bi%'. An,+s 5u+ s+ Bu+' +'lu'" +l %!(B!",'(i+n,! + un (u"! '%!Bl'!@ s+ +)+'(in'" 2 ,iB!s + +!"('%in. El sis,+(' + '%!Bl'(i+n,!@ %!nsis,+ +n un %!n*un,! + i$'s %!",'s +

!Bl'(i+n,!@ ,"'ns(i,+ u+"-'s %!",'n,+s + un (u"! ' !,"!@ l! 5u+ su*+,' ' l's i$'s + '%!Bl'(i+n,! '

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in ' %!",'n,+ +)i! ' l' B+5u+' "+l'%in +l %l'"!UB+"'l,+ + +s,'s i$'s@ l's +!"('%i!n+s '",'n,+ Bu++n !l+"s+ (u si$ni%',i's.

)i! ' 5u+ sus "i$i+%+s s!n $"'n+s@ l's i$'s + '%!Bl'(i+n,! s!n s+nsi)l+s ' l!s (!i(i+n,!s',i!s + sus 'B!!s +(B!,"'!s@ B!" +s,+ (!,i! l's +!"('%i!n+s 'i'l+s + l!s (u"!s '%!Bl'!s@

++n ,+n+" un ++%,! %!nsi+"')l+ +n +l %!(B!",'(i+n,! $l!)'l. L' "+l'%in %l'"!UB+ni+n,+ + l!s+()"!s l' %!nsi+"'%in + l's +!"('%i!n+s 'i'l+s@ si$ni%' 5u+ n! s+ Bu++ 'Bli%'" l's ,J%ni%'snu'l+s + 'nlisis + ('"%!s@ Bui+n! +n,!n%+s us'" +l +n!5u+ +l 'nlisis l'(in'" ! %!n,inu!.

Figura 5Mu"!s '%!Bl'!s + %!",'n,+ su (!+l! (',+(,i%!' Es,"u%,u"' B"!,!,iB!

--. MODELO DE CORTANTE )ARA EDIFICIOS

(!+l! + %!",'n,+ s+ +n+ %!(! na estrctra en la cal las rotaciones de na seccin .orizontal,nivel de cada %iso, no e+isten.C!n +s,' suB!si%in@ l' +s,"u%,u"' ,+n" (u%K's + l's %'"'%,+"#s,i%'sun' i$' +n !l'i-! +!"('' ni%'(+n,+ B!" '%%in + u+"-'s %!",'n,+s. A+(s s+ suB!n+ 5u+ l'ss's + l' +s,"u%,u"' +s,n %!n%+n,"''s +n l!s ni+l+s + Bis!@ l's i$'s + ,+%K! s!n inni,'(+n,+'s %!(B'"''s %!n l's %!lu(n's@ l' +!"('%in + l' +s,"u%,u"' +s in+B+ni+n,+ + l's u+"-'s'l+s +n l's %!lu(n's. D+ +s,' ('n+"' un +i%i! + ,"+s Bis!s@ B!" +*+(Bl!@ ,+n" ,"+s $"'!s +

+",'@ B'"' un' '%%in s#s(i%' +n un' i"+%%in K!"i-!n,'l +,+"(in''. N! !)s,'n,+@ +n l' li,+"',u"')"+ +l ,+(' s+ %u+n,' %!n (J,!!s B'"' +'lu'" l's "i$i+%+s + +n,"+Bis! ,!('n! +n %u+n,' l'i)ili' + l's i$'s l's B"!Bu+s,'s B!" `il)u" 9i$$s EE44 Mu,! ='Bn s!n +*+(Bl!s + +ll!.

l' Fi$u"' 3.' s+ B"+s+n,' un +s5u+(' "+B"+s+n,',i! + un (!+l! + un' +s,"u%,u"' + ,"+s Bis!s. S+++ ,"','" +l (!+l! %!(! un' %!lu(n' si(Bl+@ %!n ('s's %!n%+n,"''s 'l ni+l + %'' Bis!@,+ni+n! 5u+ l's ('s's %!n%+n,"''s '(i,+n s!l'(+n,+ ,"'sl'%i!n+s K!"i-!n,'l+s. L' "i$i+- + un,"+Bis!@ +n,"+ !s ('s's %!ns+%u,i's@ "+B"+s+n,' l' u+"-' %!",'n,+ "+5u+"i' B'"' B"!u%i" unsBl'-'(i+n,! uni,'"i! "+l',i! +n,"+ !s Bis!s ''%+n,+s.

l' Fi$u"' 3.) s+ (u+s,"'n l!s i'$"'('s + %u+"B! li)"+ %!n l!s 5u+ s+ !),i+n+n l's +%u'%i!n+s +

i(i+n,! B'"' +s,+ (!+l!.

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i%'n! +l B"in%iBi! + DWAl+()+",@ s+ !),i+n+n l's +%u'%i!n+s + (!i(i+n,!>

M1 a1 b 1 b 2 41 : 2 42 7 F1, M2 a2 : 2 41 b 2 b 3 42 : 3 43 7 F2,

M3 a3 : 3 42 b 3 43 7 F3,

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M a b 4 7 F ,

s (',"i%+s + ('s's + "i$i+- s!n@ "+sB+%,i'(+n,+>

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K1+K2 K2 0K2 K2+K3 K3

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ILIOGRAFIA

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Bs>UU'%'+(i%.uB"(.+u]2Fl'%%+i]2Fin+.BKB]2FRIDNAIC]2F'",i%l+]2Fi+[]2F30&]2F328
http://ingmec.ual.es/~jlblanco/papers/blanco2012calculo_matricial_estructuras.pdfhttp://www.espe.edu.ec/portal/files/libros/analisis/capi16p.pdfhttps://prezi.com/on_viiuwvaqx/condensacion-de-la-matriz-de-rigidez/http://civilgeeks.com/2014/09/02/manual-de-calculo-matricial-de-estructuras/http://ingmec.ual.es/~jlblanco/papers/blanco2012calculo_matricial_estructuras.pdfhttp://www.espe.edu.ec/portal/files/libros/analisis/capi16p.pdfhttps://prezi.com/on_viiuwvaqx/condensacion-de-la-matriz-de-rigidez/http://civilgeeks.com/2014/09/02/manual-de-calculo-matricial-de-estructuras/

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