REPÚBLICA BOLIVARIANA DE VENEZUELA MINISTERIO DEL PODER POPULAR PARA LA EDUCACIÓN UNIVERSITARIA INSTITUTO UNIVERSITARIO POLITÉCNICO “SANTIAGO MARIÑO” CATEDRA: RESISTENCIA DE MATERIALES II SECCION: “C”
BACHILLER: VILLARROEL GEYSON RAUL
C.I: 24.800.0!
BARCELONA, JULIO DEL 2016.
INTRODUCCIÓN
Es en la je!a "e l#s a$%&'e('#s ) "e l#s &n!en&e#s $%e la *ala+a viga '&ene %n l%!a (en'al. La &!a es %n ele-en'# %n"a-en'al en la (#ns'%((&/n, sea s'a "e la n"#le $%e %ea. Se el '&*#, (al&"a" ) &n "e la (#ns'%((&/n l# $%e "e'e-&na -e"&"as, -a'e&ales "e la &!a, ) s#+e '#"#, s% (a*a(&"a" "e s#s'ene ) (#n'ene *es#s ) 'ens&#nes. Una &!a es' *ensa"a *aa s#*#'a n# s/l# *es&/n ) *es#, s&n# 'a-+&n le3&/n ) 'ens&/n, se!4n (%l &nal&"a" *e"#-&ne se el (#n(e*'# "e &!a *aa &n!en&ea # a$%&'e('%a, $%e *e"#-&ne. En *&n(&*&#, es &-*#'an'e "e&n& $%e en la 'e#a "e &!as se (#n'e-*la a$%ell# $%e es "en#-&na"# 5es&s'en(&a "e l#s -a'e&ales. As, es *#s&+le (al(%la la es&s'en(&a "el -a'e&al (#n $%e es' 7e(7a la &!a, ) a"e-s anal&8a la 'ens&/n "e %na &!a, s%s "es*la8a-&en'#s ) el es%e8# $%e *%e"e s#*#'a. Las &!as (#n'&n%as s#n a$%ellas $%e '&enes -s "e "#s a*#)#s !eneal-en'e se %san (%an"# la l%8 es -%) !an"e. El -'#"# "e C#ss Es'e -'#"# "esa#lla"# *# Ha") C#ss en 192, *a'e "e %na es'%('%a &"eal (%)#s n#"#s es'n *ee('a-en'e !&"#s, l# $%e #+l&!a $%e *aa lle!a a la es'%('%a eal es ne(esa&# eal&8a "#s *as#s: 1. D&s'&+%& l#s -#-en'#s "e "ese$%&l&+&# $%e se *esen'an en (a"a n#"#. 2. Es'#s -#-en'#s "e "ese$%&l&+&# "&s'&+%&"#s ae('an el #'# e3'e-# "e la +aa. En el -'#"# "e "&s'&+%(&/n "e -#-en'#s, (a"a a'&(%la(&/n "e la es'%('%a a se anal&8a"a, es &ja"a a &n "e "esa#lla l#s ;#-en'#s en l#s E3'e-#s <&j#s. Des*%s (a"a a'&(%la(&/n &ja es se(%en(&al-en'e l&+ea"a ) el -#-en'# en el e3'e-# &j# =el (%al al -#-en'# "e se l&+ea"# n# es' en e$%&l&+&#> s#n "&s'&+%&"#s a -&e-+#s a")a(en'es 7as'a $%e el E$%&l&+&# es al(an8a"#. El -'#"# "e "&s'&+%(&/n "e -#-en'#s en '-&n#s -a'e-'&(#s *%e"e se "e-#s'a"# (#-# el *#(es# "e es#le %na se&e "e s&s'e-as "e e(%a(&#nes *# -e"&# "e &'ea(&/n
VIGAS CONT"NUAS Se "en#-&nan (#n'&n%as las &!as s#*#'a"as *# -s "e "#s a*#)#s $%e n# *#seen a'&(%la(&#nes &n'e-e"&as. Las &!as (#n'&n%as *#seen n(%l#s s%*ea+%n"an'es a l#s $%e (#es*#n"en &n(/!n&'as es''&(a-en'e &n"e'e-&na"as. Las &!as (#n'&n%as es%l'an -s e(#n/-&(as $%e %na se&e "e 'a-#s &n"e*en"&en'es *#$%e, en &!%al"a" "e l%(es ) (a!as, se en(%en'an s%je'as a -#-en'#s le('#es -en#es. ?a-+&n *esen'an -a)# &!&"e8 a la a((&/n "e (a!as "&n-&(as. @# el (#n'a&#, (#-# '#"as las &!as 7&*ees''&(as, s'as s#n sens&+les a la (e"en(&a "e l#s a*#)#s, $%e *%e"e al'ea "e #-a *el&!#sa las (#n"&(&#nes es''&(as. El es'%"&# "e %na &!a (#n'&n%a, ) "e (%al$%&e es'%('%a (#-*leja en !eneal, se a(&l&'a ) se *%e"e eal&8a (#n -'#"#s sen(&ll#s, ) a e(es "e #-a &n-e"&a'a, (%an"# l#s n#"#s, *%"&en"# #'a, n# s%en "es*la8a-&en'#s. S& se (#ns&"ea %na &!a "e n#"#s !&"#s ) s&n "es*la8a-&en'#, e3(l%&"as las "e#-a(&#nes els'&(as, es *#s&+le (al(%lala a*l&(an"# el ;'#"# "e C#ss.
ECUACIÓN DE LOS TRES MOMENTOS El &n!en&e# an(s Cla*e)#n en 1 en%n(&# *# *&-ea e8 la e(%a(&/n %n"a-en'al "e l#s 'es -#-en'#s. La e(%a(&/n "e l#s 'es -#-en'#s es a*l&(a+le a 'es *%n'#s (%al$%&ea "e %n &!a, s&e-*e $%e n# 7a)a "&s(#n'&n%&"a"es, 'ales (#-# a'&(%la(&#nes, en esa *a'e "e la es'%('%aF. En'#n(es, es'e -'#"# s&e *aa 7alla l#s -#-en'#s en l#s a*#)#s "e %na &!a 7&*ees''&(a, # en *%n'#s (aa('es'&(#s # n#'a+les "e la &!a. Al a*l&(a la e(%a(&/n %n"a-en'al "e l#s 'es -#-en'#s, a 'es *%n'#s "e a*#)# (#nse(%'&#s &, j, , l#s '-&n#s "el (#&-&en'# "el se!%n"# -&e-+# "e la e(%a(&/n sen n%l#s # &!%ales a -#&-&en'#s (#n#(&"#s "e l#s *%n'#s "e a*#)# #+'en&en"# "e es'a -anea %na e(%a(&/n $%e (#n'&ene, (#-# 4n&(as &n(/!n&'as, a l#s -#-en'#s en l#s a*#)#s. Es'# s&!n&&(a, $%e *#"e-#s es(&+& %na e(%a(&/n en #-a &n"e*en"&en'e, *aa 'es *%n'#s "e a*#)# (#nse(%'&#s en %na &!a (#n'&n%a. De es'a -anea, se lle!a a %n s&s'e-a (#-*a'&+le nF e(%a(&#nes &n"e*en"&en'es (#n nF &n(/!n&'as $%e s#n l#s -#&-&en'#s en l#s a*#)#s, l#s (%ales se 7allan es#l&en"# el s&s'e-a. C%an"# en %na es'%('%a (#n'&n%a, 'ene-#s %n a*#)# e3'e-# e-*#'a"#, la #-a "e salal# l# ee-#s en l#s eje(&(&#s "e a*l&(a(&/n.
C%an"# se 'a+ajan (#n &!as (#n -s "e %n 'a-#, las ea((&#nes n# *%e"en se (al(%la"as es''&(a-en'e. Una #-a "e es#lelas es a*l&(an"# el ?e#e-a "e l#s ?es ;#-en'#s, el (%al *%e"e se %'&l&8a"# 'a-+&n *aa es#le &!as "e %n s#l# 'a-#. Es'a e(%a(&/n *%e"e se e3*esa"a "e la s&!%&en'e -anea:
L#s '-&n#s:
,*%e"en #+'enese (&l-en'e "e la s&!%&en'e 'a+la, $%e a!%*a l#s 6 '&*#s "e (a!as +s&(#s.
Es'#s '&*#s +s&(#s "e (a!a *%e"en (#-+&nase *aa #+'ene '&*#s -s (#-*lej#s, s%-n"#se # es'n"#se.
S& se a a 'a+aja (#n -s "e "#s 'a-#s, "e+en es(&+&se %na e(%a(&/n "e ?es ;#-en'#s *# (a"a *a "e 'a-#s (#nse(%'&#s. @# eje-*l#:
?a-#s 1 2
?a-#s 2
?a-#s
En es'e (as# 'en"a-#s e(%a(&#nes (#n &n(/!n&'as =;1, ;2, ;, ; ) ;>. Geneal&8an"#, s&e-*e a-#s a 'ene "#s &n(/!n&'as -s $%e las e(%a(&#nes "e ?es ;#-en'#s $%e a-#s a (#ns'%&. @e# l#s -#-en'#s en l#s e3'e-#s *%e"en se 7alla"#s "e a(%e"# a l#s s&!%&en'es (&'e&#s: 1K S& 'ene-#s %n a*#)# s&-*le, el -#-en'# en "&(7# e3'e-# se &!%al a (e#. @aa el "&a!a-a "e a&+a, ;1 0 ) ; 0. 2K S& 'ene-#s %n e-*#'a-&en'#, se *%e"e (#ns'%& %na e(%a(&/n a"&(&#nal "e ?es ;#-en'#s, (ean"# %n 'a-# &'%al en el $%e '#"#s l#s al#es sean &!%ales a (e#. @aa el "&a!a-a "e a&+a, s& s%*#ne-#s $%e el a*#)# es %n a*#)# e-*#'a"#, *#"a-#s es(&+& la s&!%&en'e e(%a(&/n "e ?es ;#-en'#s, en "#n"e '#"#s l#s '-&n#s (#n s%+n"&(e (e# alen (e#:
O sea:
K S& 'ene-#s %n #la"&8#, el -#-en'# en 'al e3'e-# se!%& al&en"# (e#. A"e-s, el -#-en'# s&!%&en'e al "e "&(7# e3'e-# se &!%al a la s%-a "e l#s *#"%('#s "e las (a!as *# s% +a8# "e *alan(a a es'e 4l'&-# a*#)#.
A*l&(an"# el ?e#e-a "e l#s ?es ;#-en'#s es (&l #+'ene l#s -#-en'#s le('#es en (a"a a*#)#. Halla las ea((&#nes en (a"a a*#)# es &!%al-en'e sen(&ll#, %'&l&8an"# la s&!%&en'e /-%la, *aa (a"a 'a-#:
@#s'e&#-en'e, las ea((&#nes e$%&alen'es "e (a"a 'a-# se s%-an. @# eje-*l#:
MOMENTOS DE DISTRIBUCIÓN
M#$%&'#( )% %$*#'+,$-%&'# %& %'+%$#( /-#( ;#-en'#s "e e-*#'a-&en'# en e3'e-#s &j#s s#n l#s -#-en'#s *#"%(&"#s al e3'e-# "el -&e-+# *# (a!as e3'enas (%an"# las j%n'as es'n &jas.
R-1-)% , 3, 3%-5& La R&!&"e8 a la
"e %n -&e-+# es e*esen'a"a (#-# el *#"%('# "el ;/"%l# "e Elas'&(&"a" =E> ) el Se!%n"# -#-en'# "e ea, 'a-+&n (#n#(&"# (#-# ;#-en'# "e Ine(&a =I> "&&"&"# *# la l#n!&'%" =L> "el -&e-+#, $%e es ne(esa&a en el -'#"# "e "&s'&+%(&/n "e -#-en'#s, n# es el al# e3a('# *e# es la Ra8/n a&'-'&(a "e &!&"e8 "e le3&/n "e '#"#s l#s -&e-+#s.
,6'#+%( )% D-('+-76-5&
L#s a('#es "e "&s'&+%(&/n *%e"en se "e&n&"#s (#-# las *#*#(&#nes "e l#s -#-en'#s n# +alan(ea"#s llea"#s *# (a"a %n# "e l#s -&e-+#s.
,6'#+%( )% A6,++%# 9T+,&(*#+'%
L#s -#-en'#s n# +alan(ea"#s, s#n llea"#s s#+e el #'# e3'e-# "el -&e-+# (%an"# la j%n'a es l&+ea"a. La a8/n "e -#-en'# a(aea"# s#+e el #'# e3'e-#, al -#-en'# en el e3'e-# &j# "el e3'e-# &n&(&al es el a('# "e a(ae#.
E('+6'+,( )% M,+6#( Es'%('%as "e -a(#s (#n # s&n sidesway *%e"en se anal&8a"as %'&l&8an"# el -'#"# "e "&s'&+%(&/n "e -#-en'#s.
REGLA DE LOS SIGNOS Un -#-en'# a('%an"# en sen'&"# 7#a&# es (#ns&"ea"# *#s&'&#. Es'# "&&ee "e la (#nen(&/n "e s&!n#s %s%al en &n!en&ea, la (%al e-*lea %n s&s'e-a "e (##"ena"as (a'es&anas (#n el eje *#s&'&# P a la "ee(7a ) el eje *#s&'&# Y 7a(&a a&+a, es%l'an"# en -#-en'#s *#s&'&#s s#+e el eje Q s&en"# an'&7#a&#s. S#n *#s&'&as las %e8as $%e a('4an 7a(&a a&+a ) *#s&'&#s l#s -#-en'#s $%e a('4an en sen'&"# 7#a&#. L#s al#es "e ; s#n *#s&'&#s a la "ee(7a "el 'a-# ) ne!a'&#s a la &8$%&e"a "el 'a-#.
;
MÉTODO DE CROSS
El ;'#"# "e C#ss es el -s s&-*le ) *&"# *aa (al(%la &!as 7&*ees''&(as "e n#"#s !&"#s $%e n# se "es*la8an. El -'#"# es &'ea'&# *# a*#3&-a(&#nes s%(es&as, *e-&'e #+'ene %n +%en !a"# "e e3a('&'%" ) (#ne!e *&"a-en'e. El *#(es# se *%e"e eal&8a "e la s&!%&en'e -anea: (#n#(&"as las (a!as a('%an'es s#+e (a"a 'a-#, se (al(%lan *&-e# l#s -#-en'#s "e e-*#'a-&en'# *ee('# ; en s%s e3'e-#s, (#n el s&!n# se!4n la (#nen(&/n =*#s&'&# en sen'&"# 7#a&#>. @#s'e&#-en'e se (al(%la el -#-en'# n# e$%&l&+a"# ; en (a"a n#"#, se (a-+&a "e s&!n# ) se e*a'e en *a'es *#*#(&#nales a las &!&"e(es "e (a"a 'a-# (#n(%en'e al n#"#. Se #+'&enen as *aa (a"a e3'e-# #*%es'# "e l#s n#"#s l#s -#-en'#s "e e*a'&(&/n, $%e a s% e8 *##(an -#-en'#s "e 'ans-&s&/n en l#s e3'e-#s #*%es'#s a ell#s, el -#-en'# "e 'ans-&s&/n es la -&'a" "el -#-en'# "e e*a'&(&/n.
C#n l#s -#-en'#s "e 'ans-&s&/n se (al(%lan "e n%e# l#s -#-en'#s n# e$%&l&+a"#s en (a"a n#"#, #+'en&en"# n%e#s -#-en'#s "e e*a'&(&/n, se (#n'&n4a el *#(es# an'e&# 7as'a $%e el -#-en'# n# e$%&l&+a"# en (a"a n#"# '&en"a a (e#. <&nal&8a"# el *#(es# &'ea'&#, *aa #+'ene el -#-en'# "e&n&'&# en (a"a a*#)# se s%-an l#s -#-en'#s "e e-*#'a-&en'#, l#s "e e*a'&(&/n ) l#s "e 'ans-&s&/n (#es*#n"&en'es a (a"a la"# "el a*#)#.
E
La &!a es''&(a-en'e &n"e'e-&na"a -#s'a"a en la &!%a sea anal&8a"a.
;&e-+#s AB, BC, CD '&enen la -&s-a l#n!&'%"
Las &!&"e(es a
.
Ca!as (#n(en'a"as "e -a!n&'%"
a('4an a %na "&s'an(&a
"es"e el s#*#'e A.
Ca!a %n&#-e "e &n'ens&"a"
;&e-+# CD es' (a!a"# a la -&'a" "e s% (la# (#n %na (a!a (#n(en'a"a "e -a!n&'%"
a('4a en BC.
.
En l#s s&!%&en'es (l(%l#s, l#s -#-en'#s an'&7#a&#s s#n *#s&'&#s.
M#$%&'#( %& E'+%$#( -#(
,6'#+%( )% D-('+-76-5&
L#s a('#es "e "&s'&+%(&/n "e las j%n'as A ) D s#n D AB D DC 1.
,6'#+%( )% A6,++%# 9T+,&(*#+'%
L#s a('#es "e a(ae# s#n , e3(e*'# *aa el a('# "e a(ae# "es"e D =s#*#'e &j#> a C el (%al es (e#.
D-('+-76-5& )% M#$%&'#(
A'&( %la (&
A
0 =
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@==.!? ==.!?
@=0.=8? =0.= 8?
@=.?!>
N4-e#s en !&s s#n -#-en'#s +alan(ea"#s le(7as = M > e*esen'an el a(ae# "e -#-en'# "es"e %n e3'e-# al #'# e3'e-# "e %n -&e-+#.
R%(3',)#( •
;#-en'#s en a'&(%la(&#nes, "e'e-&na"#s *# el -'#"# "e "&s'&+%(&/n "e -#-en'#s.
La (#nen(&/n "e s&!n#s %s%al en &n!en&ea es %sa"a a$%, &.e. L#s -#-en'#s *#s&'&#s (a%san el#n!a(&/n en la *a'e &ne&# "e %n ele-en'# "e &!a. @aa *#*/s&'#s "e (#-*aa(&/n, l#s s&!%&en'es s#n l#s es%l'a"#s !enea"#s, %san"# %n ;'#"# "e ;a'&8. N#'a $%e en el anl&s&s s%*e&#, el *#(es# &'ea'&# %e llea"# a T0.01 "e *e(&s&/n. El 7e(7# "e $%e el es%l'a"# "e anl&s&s "e -a'&8 ) el es%l'a"# "e anl&s&s "e "&s'&+%(&/n "e -#-en'#s &!%ale a 0.001 "e *e(&s&/n es -ea (#&n(&"en(&a. •
;#-en'#s en a'&(%la(&#nes "e'e-&na"#s *# el -'#"# "e -a'&8
L#s "&a!a-as (#-*le'#s "e (#'an'e ) -#-en'# le3'# s#n (#-# s&!%e. N#'a $%e el -'#"# "e "&s'&+%(&/n "e -#-en'#s s#l# "e'e-&na l#s -#-en'#s en las j%n'as. Desa#llan"# "&a!a-as "e -#-en'#s le3'#es (#-*le'#s e$%&ee "e (l(%l#s a"&(&#nales %san"# l#s -#-en'#s "e'e-&na"#s en las a'&(%la(&#nes ) el e$%&l&+&# &n'en# "e la se((&/n. •
D
CONCLUSIONES Una &!a (#n'&n%a es a$%ella &!a s#*#'a"a *# "#s # -s a*#)#s *aa l#!a %na -a)# &!&"e8, "e -#"# $%e se *%e"e (al(%la el ee('# $%e %na (a!a 'en"a s#+e &!as &n"&&"%ales "e &!%ales l%(es. El -'#"# "e "&s'&+%(&/n "e -#-en'#s (ae "en'# "e la (a'e!#a "el ;'#"# "e Des*la8a-&en'# "e anl&s&s es'%('%a. La en'aja "e 7a(e $%e las &!as sean (#n'&n4as a las &!as "e %n s#l# 'a-# es $%e "an -#-en'#s le('#es ) le(7as "e -en# -a!n&'%", l# $%e llea (#ns&!# &!as "e -en# se((&/n 'ansesal ) *# l# 'an'#, -as e(#n/ -&(as. El -'#"# "e l#s ?es ;#-en'#s es %n 'e-a $%e ne(es&'a "e %n a-*l&# es'%"&# "e+&"# a $%e se a*l&(an eje(&(&#s (#n s%+*#+le-as ) %n -s "&&(%l'#s# $%e l#s "e-s -'#"#s.
BIBLIOGRA"A
L-7+#( S&n!e , “Resistencia de los Materiales”
P1-&,( %7 http://patricia-vateccofic2008.blogspot.com/ http://marilycita.blogspot.com/2008/07/metodo-de-tres-momentos.html http://www2.ula.ve/dsiaportal/dmdocuments/Resistencia%20de%20ateriales %20!ema%20".pdf
