INGENIERIA CIVIL
UNIVERSIDAD NACIONAL DE SAN ANTONIO ABAD DEL CUSCO “FACUL “FACULT TAD DE INGENI ING ENIERÍ ERÍA A CIVI CIVIL L” ESCUELA PROFESIONAL: INGENIERÍA CIVIL
LINEA DE ESTRUCTURAS
TORSIÓN
CURSO
: RESISTENCIA DE MATERIALES
PRESENTADO POR GREIDY
: FLORES OMONTE JHON
CODIGO
: 141581
DOCENTE PALOMINO
: Ingeniero DANNY NIETO
CUSCO, 28 DE NOVIEMBRE DE 2015 RES R ESIS IST TEN ENCI CIA A DE MA MATE TER RIAL ALES ES
Pg gin in! !1
INGENIERIA CIVIL
DEDICATORIA E"#e E"#e #r!$ #r!$!% !%o o e" &e&i &e&i'! '!&o &o ! #o&! #o&!"" (!" (!" )er" )er"on on!" !" en gene gener! r!(* (* )er )ero )ri+or&i!( ! (o" e"#,&i!n#e" &e Ingenier-! Ci.i( /,e e"#n in#ere"!&o" en !)ren&er +" "o$re (! r!+! &e e"#r,'#,r!"
RES R ESIS IST TEN ENCI CIA A DE MA MATE TER RIAL ALES ES
Pg gin in! !0
INGENIERIA CIVIL
DEDICATORIA E"#e E"#e #r!$ #r!$!% !%o o e" &e&i &e&i'! '!&o &o ! #o&! #o&!"" (!" (!" )er" )er"on on!" !" en gene gener! r!(* (* )er )ero )ri+or&i!( ! (o" e"#,&i!n#e" &e Ingenier-! Ci.i( /,e e"#n in#ere"!&o" en !)ren&er +" "o$re (! r!+! &e e"#r,'#,r!"
RES R ESIS IST TEN ENCI CIA A DE MA MATE TER RIAL ALES ES
Pg gin in! !0
INGENIERIA CIVIL
ÍNDICE 12 PROLOGO
Pgin! 4
02 INTROD3CCIN
Pgin! 5
2 TORSIN
Pgin! 6
42 CONCL3SIONES
Pgin! 8
52 7I7LIOGRAFA
RES R ESIS IST TEN ENCI CIA A DE MA MATE TER RIAL ALES ES
P!gin! 9
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INGENIERIA CIVIL
PROLOGO E( +o#i.o )or e( ',!( &e"!rro((!+o" e"#e #e+! &e TORSIN* e" )!r! !)ren&er +" "o$re '+o !'#;! e"#! <,er=! en (!" e"#r,'#,r!" &e #o&o #i)o &e 'on"#r,''ione"* )!r! !"- e"#!r )re)!r!&o" en ',!(/,ier "i#,!'in* e" )or e"o /,e +e +o#i.o re!(i=!r e"#e #e+! &e TORSIN )!r! !"- 'o+)!r#ir(o 'on #o&o" (o" e"#,&i!n#e" &e Ingenier-! Ci.i( > !"- e"#? e" #?n n )re) )re)!r !r!& !&o" o" )!r! )!r! "," "," <,#, <,#,r! r!"" o$r! o$r!"* "* (o" (o" )ro$ )ro$(e (e+! +!"" /,e /,e "e )re"en#!ron !( re!(i=!r (! +onogr!<-! no <,eron +,'@o" >! /,e 'on (! e( ,"o &e( &e( in#e in#errne# "e ),& ),&o re' e'o o)i(! )i(!rr in !"- #er+in!r (! +onogr!<-! 'on ei#o
RES R ESIS IST TEN ENCI CIA A DE MA MATE TER RIAL ALES ES
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INGENIERIA CIVIL
INTRODUCCIÓN En (! '!rrer! &e Ingenier-! Ci.i(* en (!" 'on"#r,''ione" &on&e ei"#en e"#r,'#,r!" > +eg! e"#r,'#,r!" 'o+o en (o" e&iB'io"* r!"'!'ie(o"* @i&roe(?'#ri'!"* re)re"!"* ),en#e" > en ',!(/,ier #i)o &e 'on"#r,''ione"* "ie+)re #ienen e"#r,'#,r!" in#ern!" @e'@o" &e !'ero* 'o+;n+en#e ((!+!&o e"/,e(e#o e"#e e"/,e(e#o @e'@o &e !'ero #iene /,e !g,!n#!" '!rg!"* &e 'on're#o > o#ro" !''e"orio" /,e "e &e$e &e )oner en (!" 'on"#r,''ione"* e"#!" '!rg!" /,e !g,!n#! gener!n #o&o #i)o &e <,er=!"* e" en &on&e en'on#r!+o" e( #i)o &e <,er=! /,e no" in#ere"! +,'@o ((!+!&! TORSIN* &on&e &e"!rro((!re+o" e( #e+! en e"#! +onogr!<-! &on&e .ere+o" ',n i+)or#!n#e e" e"#! <,er=!
TORSIÓN RESISTENCIA DE MATERIALES
Pgin! 5
INGENIERIA CIVIL DEFINICIÓN.- TORSIN* 'on origen e#i+o(gi'o en e( (!#-n #or"io* e" ,n #?r+ino /,e !(,&e e( !'#o > e( re",(#!&o &e #or'er E( 'on'e)#o ",e(e re &e (o" e"<,er=o" > &e "," !)(i'!'ione"* )ero "(o en e( '!"o &e r$o(e" &e "e''in 'ir',(!r* o &e #,$o" &e )!re& &e(g!&! L! #or"in &e r$o(e" &e "e''in !r$i#r!ri! e" ,n )ro$(e+! 'o+)(e%o &e( /,e "(o "e e)onen (!" <r+,(!" &e !)(i'!'in Con (! #or"in "e ini'i!* )or o#r! )!r#e* e( e"#,&io &e (o" )ro$(e+!" en (o" /,e e( e"<,er=o no "e &i"#ri$,>e ,ni ,n! .e= &e&,'i&!" (!" <r+,(!"* no @!> +" /,e ","#i#,ir en e((!" (o" .!(ore" &e (o" &!#o" > n!&! +" E( )ro'e&i+ien#o gener!( /,e "e "ig,e en #o&o" (o" '!"o" &e &i"#ri$,'in no ,ni
RESISTENCIA DE MATERIALES
Pgin! 6
INGENIERIA CIVIL 1 De( e!+en &e (!" &e (! !)(i'!'in &e (! (e> &e Hooe* "e &e#er+in!n ,n!" re(!'ione" en#re (o" e"<,er=o" en (o" &i"#in#o" ),n#o" &e (! "e''in* &e +!ner! /,e "e!n 'o+)!#i$(e" 'on (!" &e (!" <,er=!" re"i"#en#e" in#eriore" en ,n! "e''in* "e ((!+! e',!'ione" &e e/,i(i$rio Co+)ro$!'in &e /,e (! "o(,'in &e( "i"#e+! &e e',!'ione" &e (o" ),n#o" 1 > 0 "!#i" !(g,n!" &e e((!"* 'o+)ro$!r"e e)eri+en#!(+en#e L!" &o" )ri+er!" 'orre")on&en ! "e''ione" 'ir',(!re" 1 L!" "e''ione" 'ir',(!re" )er+!ne'en 'ir',(!re" &e"),?" &e (! #or"in 0 L!" "e''ione" )(!n!" )er+!ne'en )(!n!" > no "e !(!$e!n &e"),?" &e (! #or"in L! )ro>e''in "o$re ,n! "e''in #r!n".er"!( &e ,n! (-ne! r!&i!( &e ,n! "e''in )er+!ne'e r!&i!( &e"),?" &e (! #or"in 4 E( r$o( e"# "o+e#i&o ! (! !''in &e )!re" #or"ore" o #or"ion!n#e" /,? !'#;!n en )(!no" )er)en&i',(!re" ! ", e%e 5 Lo" e"<,er=o" no "o$re)!"!n e( (-+i#e &e )ro)or'ion!(i&!& Re'or&!n&o /,e en (o" '!)-#,(o" !n#eriore" "e e"#,&i '+o '!(',(!r (o" e"<,er=o" > (!" &e )!r#e" &e +!/,in!ri! /,e "e en',en#r!n en #or"in M" e")e'-B'!+en#e* "e e"#,&i!rn (o" e"<,er=o" > (!" &e "en#i&o" o),e"#o" Son '!n#i&!&e" .e'#ori!(e"
RESISTENCIA DE MATERIALES
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INGENIERIA CIVIL /,e ),e&en re)re"en#!r"e +e&i!n#e e'@!" ',r.!"* o )or .e'#ore" &e )!r Lo" e(e+en#o" "o+e#i&o" ! #or"in "e en',en#r!n en +,'@!" "i#,!'ione" &e ingenier-! L! !)(i'!'in +" 'o+;n (! re)re"en#!n (o" e%e" &e #r!n"+i"in* /,e "e e+)(e!n )!r! #r!n"+i#ir )o#en'i! &e ,n ),n#o ! o#ro
Geeralidades Con"i&ere+o" ,n! $!rr! "in ning,n! (i+i#!'in !( +o.i+ien#o &e "," ),n#o"* 'o+o (! &e (! Bg,r! 51* /,e e"# "o+e#i&! ! +o+en#o" #or"ore" en "," e#re+o"* no ei"#ien&o ning,n! o#r! "o(i'i#!'in En e"#!" 'on&i'ione"* (!" "e''ione" &e (! $!rr! e)eri+en#!n ,n giro re(!#i.o &e &ire''in * /,e "er 'on"#!n#e )or ,ni&!& &e (ongi#,& &e (! $!rr! En e( Te+! * En"!>o &e Tor"in >! @e+o" &eno#!&o 'o+o ! e"#e ng,(o )or ,ni&!& &e (ongi#,& S!(.o en e( '!"o e")e'i!( en /,e (! $!rr! "e! &e "e''in 'ir',(!r 2@,e'! o +!'i=!2* o',rrir /,e (!" "e''ione" '#e ini'i!(+en#e )(!n!" &e (! $!rr!* &e%!rn &e "er )(!n!" Se &eno+in! ! e"#e
En ',!n#o ! (!" #en"ione"* e( +o+en#o #or"or gener!r en (!" "e''ione" ,n 'ier#o "i"#e+! &e #en"ione" #!ngen'i!(e" e"##i'!+en#e e/,i.!(en#e ! ?(* no ei"#ien&o #en"ione" nor+!(e" E( +o&o &e #r!$!%o &e (! $!rr! &e"'ri#o @!"#! !/,- "e 'ono'e 'o+o #or"in ),r!* o #or"in ,ni
RESISTENCIA DE MATERIALES
Pgin! 8
INGENIERIA CIVIL S,)ong!+o" !@or! /,e (! $!rr! &e (! Bg,r! 51 #,.ie"e i+)e&i&o" (o" e")(!=!+ien#o" &e !(!$eo (o" &e &ire''in * /,e "e +,e"#r!n orien#!#i.!+en#e en ,no &e "," e#re+o"* )or e%e+)(o )or e +o+en#o n,(o" en (! "e''in &e( e+)o#r!+ien#o* >! /,e no @!> o#r!" <,er=!" &e &ire''in o +o+en#o" e'#ore" en e( )ro$(e+! 'on (!" /,e ),e&!n e/,i(i$r!r"e E" #!+$i?n e.i&en#e* )or 'on&i'in &e 'on#orno &e( )ro$(e+!* /,e en #o&o" (o" ),n#o" &e (! "e''in o),e"#! !( e+)o#r!+ien#o e" K Por #!n#o* en ',!(/,ier (-ne! )!r!(e(! !( e%e &e (! $!rr!* (! #en"in e.o(,'ion!r &e"&e ,n .!(or )o"i$(e+en#e &i"#in#o &e 'ero en e( e+)o#r!+ien#o* @!"#! ,n .!(or n,(o en e( e#re+o o),e"#o Se 'ono'e 'o+o #or"in no ,ni o#r!" 'ir',n"#!n'i!" /,e ),e&en )ro.o'!r #or"in no ,ni! !r+oni=!'in @!r !)!re'er (!" #en"ione" nor+!(e" )ro)i!" &e (! #or"in no ,ni o#ro" #i)o" &e "e''in ',>! #en&en'i! !( !(!$eo e" )e/,e!* > gener!(+en#e ),e&en !n!(i=!r"e 'on ",B'ien#e !)roi+!'in $!%o (! @i)#e"i" &e #or"in ,ni
Cir',(!re"* #!n#o +!'i=!" 'o+o @,e'!" &e )!re& &e(g!&! o no M!'i=!"* 'o+o (!" re'#!ng,(!re"* ',!&r!&!"* e(-)#i'!"* e#'
RESISTENCIA DE MATERIALES
Pgin! 9
INGENIERIA CIVIL
Cerr!&!" &e )!re& &e(g!&!* 'o+o (!" "e''ione" en '!%n > "i+i(!re" Se''ione" (!" "e''ione" en T* &e )!re& &e(g!&!
L! $re.-"i+! in#ro&,''in ! (! #or"in /,e 'on#ienen (o" e)-gr!
TORSIÓN EN BARRAS DE SECCIÓN CIRCULAR Se!"# !$%&'$ (%e' )e *'$e) )e&+')' L! #or"in en ,n! $!rr! &e )erB( 'ir',(!r @,e'o &e )e/,eo e")e"or >! <,e )re"en#!&! en e( #e+! !( e"#,&i!r e( en"!>o &e #or"in "o$re (! 'onBg,r!'in &e (! Bg,r! 50! Ree(!$or!+o" "eg,i&!+en#e (o" re",(#!&o" o$#eni&o" !((- )!r! o$#ener ,n!
T =2 πe R σ xθ
51
L! <r+,(! !n#erior 'on#iene !(g,n!" !)roi+!'ione" !&i'ion!(e": )or ,n! )!r#e e( re! &e (! 'oron! 'ir',(!r "e !)roi+! )or ", &e"!rro((o en ,n re'#ng,(o e"#re'@o* > )or o#r! (! re",(#!n#e &e (!" #en"ione" en ,n "e'#or &i 'on"i&er!r /,e (! re",(#!n#e &e (!" #en"ione" )!"! )or e( r!&io +e&io &e (! 'oron!* 'o+o "e in&i'! en (! Bg,r! 50$ E" &e'ir: 2
2
T =π ( Re − Ri ) σ xθ
Re + Ri 2
50
No o$"#!n#e* (! e)re"in 50 !n#erior "ig,e 'on#enien&o (! !)roi+!'in &e /,e e" 'on"#!n#e en e( e")e"or* )or (o /,e "(o &e$e ,"!r"e )!r! e")e"ore" &e(g!&o" > en e"#o" '!"o" (! +e%or! &e )re'i"in re")e'#o &e (! <r+,(! +" "en'i((! 51 no "er gr!n&e
RESISTENCIA DE MATERIALES
Pgin! 1K
INGENIERIA CIVIL
Por o#r! )!r#e* &e 8 > 9 #ene+o" /,e: σ xθ= γ xθ .G =θ . G . R
5
Co+$in!n&o e"#! e',!'in 'on (! 51* o $ien 'on (! 50* "e ),e&e o$#ener (! re(!'in en#re e( ng,(o gir!&o )or ,ni&!& &e (ongi#,& &e $!rr! > e( +o+en#o #or"or T En )!r#i',(!r 'o+$in!n&o 51 > 5 #ene+o":
Don&e J0eR e" e( +o+en#o )o(!r &e iner'i! &e (! "e''in re")e'#o &e( 'en#ro &e re!" &e (! +i"+! /,e 'oin'i&e 'on e( 'en#ro &e "i+e#r-! geo+?#ri'o &e (! "e''in Pen"e+o" !@or! en e( )ro$(e+! &e #or"in &e ,n! $!rr! "i+i(!r !( !n!(i=!&o @!"#! !@or!* )ero en e( /,e e( e")e"or no ),e&! ",)oner"e &e(g!&o Bg,r! 5!* o $ien ',>! "e''in "e! &e
Pgin! 11
INGENIERIA CIVIL Con"i&ere+o" e( )ro$(e+! >! "e! e( &e )!re& gr,e"! o e( &e( 'i(in&ro +!'i=o &i.i&i&o en $!rr!" ',>!" "e''ione" "on 'oron!" 'ir',(!re" 'on'?n#ri'!" 'o+o in'i&! (! Bg,r! 5' E( /,e "e!n rrrrK i+)(i'! /,e e"!" $!rr!" no "e #r!n"+i#en ning,n! #en"in en#re e((!" ! #r!.?" &e (! ",)erB'ie /,e (!" "e)!r! En &eBni#i.!* '!&! ,n! e"# "o+e#i&! "o(!+en#e ! (!" #en"ione" * #r!$!%!n&o &e ,n! +!ner! !n(og! ! (! &e (! Bg,r! 50!
Por #!n#o e" e( r!&io R )or e( .!(or r &e e"! 'oron! )!r#i',(!r E( +o+en#o #or"or 'orre")on&ien#e "er ,n &i
S,"#i#,>en&o r )or ", .!(or &!&o )or 5 n,e.!+en#e )!r! e( r!&io r "e #iene:
RESISTENCIA DE MATERIALES
Pgin! 10
INGENIERIA CIVIL Don&e* 'o+o &i%i+o"* @! &e "er 'on"#!n#e en r )or ",),e"#o (o e" en Seg,i&!+en#e !)(i'!re+o" e"#e re",(#!&o ! (o" '!"o" &e (! Bg,r! 5 ! > $
Secció circ+lar h+eca de 'ared ,r+esa In#egr!n&o (! e',!'in 55 en#re (o" r!&io" in#erior > e#erior in&i'!&o" en (! Bg,r! 5! ! > $ re")e'#i.!+en#e "e ((eg! in+e&i!#!+en#e !:
Tenien&o en ',en#! /,e e( +o+en#o )o(!r &e iner'i! &e (! 'oron! 'ir',(!r .!(e J$42!40* (! <r+,(! !n#erior !&+i#e ,n! e)re"in !n(og! ! (! 54:
Con e( .!(or &e( giro )or ,ni&!& &e (ongi#,& !n#erior* (!" #en"ione" "e o$#ienen in+e&i!#!+en#e &e 5* "in +" /,e ree+)(!=!r R )or e( .!(or &e( r!&io:
Co+o "e !)re'i!* (!" #en"ione" "on )ro)or'ion!(e" ! (! &i"#!n'i! r !( 'en#ro &e re!" &e( )erB(* #!( 'o+o "e )o&-! !n#i'i)!r ! (! .i"#! &e 5
Perl circ+lar %acio En e"#e '!"o* @e+o" &e in#egr!r 55 en#re rK > e( r!&io e#erior* r$ O "i+)(e+en#e )!r#i',(!ri=!r 56 )!r! !K Se o$#iene in+e&i!#!+en#e /,e:
N,e.!+en#e* &!&o /,e e( +o+en#o )o(!r &e iner'i! &e( '-r',(o .!(e J$40* )o&e+o" e"'ri$ir (! e',!'in !n#erior &e (!
RESISTENCIA DE MATERIALES
Pgin! 1
INGENIERIA CIVIL
Y ","#i#,>en&o ?"#! en 5 &e
N#e"e '+o (o" ),n#o" 'en#r!(e" &e (! $!rr!* rK* no "o)or#!n #en"in C!$e &e'ir /,e e( +!#eri!( 'er'!no ! e"e ),n#o e"# &e"!)ro.e'@!&o* "ien&o &e"&e e"#e ),n#o &e .i"#! +" eB'ien#e e( )erB( @,e'o Sin e+$!rgo e( @e'@o &e /,e ,n! $!rr! +!'i=! e" +" <'i( &e +!#eri!(i=!r 'on#r!rre"#! en +,'@o" '!"o" ", &e".en#!%! en ',!n#o ! !)ro.e'@!+ien#o &e( +!#eri!(
N!#e- -.$e &' /$-!"# e# .'$$'- )e -e!"# # !$%&'$ En ',!n#o ! (! !)#i#,& )!r! re"i"#ir #or"in* en#en&i&! 'o+o !)!ri'in &e #en"ione" &e .!(or +o&er!&o* $!%o (! !''in &e ,n +o+en#o #or"or* ),e&e &e'ir"e /,e (!" "e''ione" +" i&ne!" "on (!" 'err!&!"* &e )!re& &e(g!&! C,!n&o "e 'o+)!r! ,n! "e''in &e e"#e #i)o 'on ,n! "e''in +!'i=! &e( +i"+o re!* en'on#r!+o" /,e e( )erB( @,e'o #iene +!>or rigi&e= ! #or"in* > &e"!rro((! +enore" #en"ione" +i+!" (! #en"in #!ngen'i!( e" )r'#i'!+en#e (! +i"+! en #o&o" (o" ),n#o" En )!r#i',(!r (! "e''in 'ir',(!r @,e'! ),e&e "er e")e'i!(+en#e 'on.enien#e &e$i&o #!n#o ! /,e no !(!$e!* 'o+o ! /,e e" )#i+! en 'ier#o "en#i&o* 'o+o "e .er +" #!r&e L!" "e''ione" @,e'!" &e )!re& gr,e"! )re"en#!rn ,n! +!>or .!ri!'in &e (! #en"in en &ire''in r!&i!(* 'on (o /,e )re"en#!rn +!>ore" #en"ione" +i+!" /,e ,n )erB( &e )!re& &e(g!&! Pero ! '!+$io !(e%!n e( )e(igro &e )o'! #en"in en 'o+)!r!'in 'on (o" e#eriore" Por e( 'on#r!rio* (!" "e''ione" !$ier#!" &e )!re& &e(g!&! "on +,> )o'o !)ro)i!&!" )!r! "o)or#!r +o+en#o #or"or* &e$i&o ! /,e &e$en RESISTENCIA DE MATERIALES
Pgin! 14
INGENIERIA CIVIL gener!r gr!n&e" #en"ione" E" e( '!"o &e (!" "e''ione" en L > en T !,n/,e #eng!n )o'! )ro)en"in ! (! #or"in no ,ni 'o+,ne" en e"#r,'#,r! +e#(i'!* &e$en &i"e!r"e (!" 'on&i'ione" &e !)o>o > &e+" ,n e%e " /,e e" #!ngen#e ! &i'@! (-ne! +e&i! en '!&! ),n#o* > /,e #en&r e( "en#i&o 're'ien#e &e (! 'oor&en!&! " E( e%e n e" )er)en&i',(!r !( " en '!&! ),n#o* > #iene e( "en#i&o "!(ien#e en e( )erB(* 'o+o "e in&i'! E( e")e"or &e (! )!re& en '!&! ),n#o "er e"* ),&ien&o .!ri!r 'on " E( e%e * 'o+o e" @!$i#,!(* #iene (! &ire''in (ongi#,&in!( &e (! $!rr!* > )or #!n#o e" )er)en&i',(!r !( &i$,%o* 'o+o "e +,e"#r! en (! Bg,r! 54$ En )ri+er (,g!r* no#!re+o" /,e e( )ro&,'#o "e" &e$e "er 'on"#!n#e ! (o (!rgo &e (! 'oor&en!&! " &e( )erB( En e 7* /,e en gener!( ),e&en #ener #en"ione" &i "7 re")e'#i.!+en#e* (! re'i)ro'i&!& &e (!" #en"ione" #!ngen'i!(e" eige /,e ?"#!" 'oin'i&!n en .!(or 'on (!" #en"ione" "en (!" re")e'#i.!" e"/,in!" &e ,n e(e+en#o &e( )erB( o$#eni&o )or &o" 'or#e" "'#e > &o" 'or#e" '#e* !+$o" &e &i+en"in Bni#!* 'o+o e( +o"#r!&o en (! Bg,r! 54$ E"#!" #en"ione" " "ern 'on"#!n#e" en )or (! n!#,r!(e=! &e( )ro$(e+! en #or"in ,nie /,e
RESISTENCIA DE MATERIALES
Pgin! 15
INGENIERIA CIVIL
Se 'ono'e 'o+o ,%o &e #en"ione" !( )ro&,'#o ""e"* /,e )er+!ne'e 'on"#!n#e en " 3n! 'on"e',en'i! &e e"#! )ro)ie&!& &e( ,%o &e #en"ione" e"* )or e%e+)(o* /,e en "e''ione" &e e")e"or e 'on"#!n#e* " "er #!+$i?n 'on"#!n#e C!(',(!re+o" !@or! e( .!(or &e( +o+en#o #or"or T )ro&,'i&o )or (!" #en"ione" " L! <,er=! "o$re ,n &i #iene (! &ire''in &e( e%e " en e"e ),n#o En (! e)re"in !n#erior ","#i#,ire+o" &" )or e( .e'#or &" * /,e #en&r e( "en#i&o &e "* )!r! &!r '!r'#er .e'#ori!( ! (! 'i#!&! <,er=! E( +o+en#o &e e"! <,er=! &i
Co+o "!$e+o"* e( &o$(e &e( re! &e ,n #ring,(o ),e&e '!(',(!r"e 'o+o e( +&,(o &e( )ro&,'#o .e'#ori!( &e &o" .e'#ore" /,e 'oin'i&!n 'on &o" &e "," (!&o" Por #!n#o e( )ro&,'#o r&" #iene 'o+o +&,(o e( &o$(e &e( re! &e( #ring,(o r!>!&o en (! Bg,r! 55$ Si in#egr!+o" 511 ! (o (!rgo &e #o&! (! (-ne! 'err!&! /,e re'orre (! 'oor&en!&! "* #ene+o" e( +o+en#o #or"or #o#!(:
En &on&e e( )ro&,'#o ""e" @! "!(i&o &e (! in#egr!( )or "er 'on"#!n#e L! in#egr!( ! (o (!rgo &e (! (-ne! +e&i! &e( )erB( &e r&" re)re"en#! e( &o$(e &e( re! &e #o&o" (o" #ring,(o" &i!&o en (! Bg,r! 55$* > )or #!n#o re)re"en#! e( &o$(e RESISTENCIA DE MATERIALES
Pgin! 16
INGENIERIA CIVIL &e( re! in#erior en'err!&! )or (! (-ne! +e&i! &e( )erB( L(!+!n&o ?"#e re!* (! in#egr!( #en&r e( .!(or 0
!
En &eBni#i.!* )re"'in&ien&o &e( '!r'#er .e'#ori!( &e !+$o" +ie+$ro" &e (! e',!'in* )o&e+o" e"'ri$ir e( +o+en#o #or"or #o#!( 'o+o:
E" &e"#!'!$(e (! 'oin'i&en'i! en#re 510 > 51 E"#! ;(#i+! e)re"! /,e )!r! ,n! "e''in 'ir',(!r &e )!re& &e(g!&! e" T0R0e * )ero e( re! "e'#ori!( en e"#e )erB( e" R0* )or (o /,e &i'@! e)re"in re.ier#e ! (! ! "e!n &e )!re& &e(g!&!* gr,e"!* o "e''in 'ir',(!r +!'i=! En e"! e)re"in J e" e( +o+en#o )o(!r &e iner'i! P,e&e &e+o"#r!r"e /,e )!r! "e''ione" 'err!&!" &e )!re& &e(g!&! e( giro )or ,ni&!& &e (ongi#,& &e $!rr! !&+i#e ,n! e)re"in "i+i(!r:
Don&e IT e" e( +&,(o &e #or"in* /,e %,eg! e( +i"+o )!)e( /,e e( +o+en#o )o(!r &e iner'i! en (! "e''in 'ir',(!r &e )!re& &e(g!&! RESISTENCIA DE MATERIALES
Pgin! 1
INGENIERIA CIVIL #iene ig,!(+en#e &i+en"ione" &e (ongi#,& e(e.!&! ! (! 4Q )o#en'i!* )ero /,e no 'oin'i&e n,+?ri'!+en#e 'on e( +o+en#o )o(!r &e iner'i! )!r! ),e&en 'on",(#!r"e en (! $i$(iogr!<-! 1*14 S, .!(or* .(i&o e")e'-B'!+en#e )!r! )erB(e" 'err!&o" &e )!re& &e(g!&!* e":
Sien&o LS (! (ongi#,& &e (! (-ne! +e&i! &e( )erB( 'err!&o* /,e re'orre (! 'oor&en!&! " E"#e re",(#!&o e" &eno+in!&o
Lo" .!(ore" &e( +&,(o &e #or"in IT 'orre")on&ien#e" ! )erB(e" 'o+er'i!(e" nor+!(i=!&o" #!n#o !$ier#o" 'o+o 'err!&o"* ),e&en en'on#r!r"e en (!" #!$(!" !( ,"o* %,n#o ! (o" &e o#r!" +!gni#,&e" e"##i'!" &e (! "e''in* 'o+o e( +o+en#o &e iner'i!* e( +&,(o re"i"#en#e* e#' Lo" 'on#eni&o" !'er'! &e (! #or"in )re"en#!&o" en e"#e #e+! "on +,> ",'in#o"* >! /,e e((o e" (o )re#en&i&o )!r! e( ',r"o P!r! <,#,r! !+)(i!'in ),e&en 'on",(#!r"e ',!(/,ier! &e (!" re 8
E#-' )e T$-!"# En e"#e +o+en#o &e( ',r"o "o(!+en#e ne'e"i#!+o" +o"#r!r e( e* "o$re ,n e(e+en#o RESISTENCIA DE MATERIALES
Pgin! 18
INGENIERIA CIVIL &io re!( /,e #-)i'!+en#e )er+i#e !)(i'!r ! ,n e(e+en#o &io &e #or"in E( en"!>o &e #or"in "e re!(i=! ,",!(+en#e "o$re ,n! )ro$e#! +!'i=! &e geo+e#r-! 'i(-n&ri'!* !)(i'!n&o ,n +o+en#o 'o(ine!( 'on (! &ire'#ri= &e (! $!rr!* /,e ((!+!+o" +o+en#o #or"or > &eno#!+o" 'o+o T* > ',>o eo Teri'!+en#e (o" re",(#!&o" &e$ier!n "er !n(ogo" )!r! !+$!" geo+e#r-!"* )ero &e$e #ener"e no#i'i! &e /,e en (! )r'#i'!* en e( '!"o &e "e''in &e )!re& &e(g!&!* ),e&en !)!re'er &e(g!&! en 'o+)!r!'in 'on e( &i+e#ro* &i'@o" o ),e&e 'on"i&er!r"e &e )!re& &e(g!&! ! (o" #,$o" 'on re(!'in De +!>or /,e 0K* orien#!#i.!+en#e A",+i+o" /,e n,e"#ro en"!>o e"#!r re!(i=!&o "o$re ,n! )ro$e#! ',>! re(!'in De e"# en#re e"o" .!(ore"* &e
L! Bg,r! 5! +,e"#r! ,n #,$o &e )!re& &e(g!&! "o+e#i&o ! ,n +o+en#o #or"or T L! &i"#ri$,'in &e #en"ione" /,e "e gener! en ,n! "e''in &e (! $!rr! )er)en&i',(!r ! ", e%e* 'on"#! &e ,n "i"#e+! &e #en"ione" #!ngen'i!(e" /,e #ienen (! &ire''in 'ir',n RESISTENCIA DE MATERIALES
Pgin! 19
INGENIERIA CIVIL #eng! (! &ire''in 'ir',n
L!" Bg,r!" 5$ > 5' +,e"#r!n (! %,"#iB'!'in &e /,e (! #en"in #!ngen'i!( &e$e #ener !)roi+!&!+en#e (! &ire''in #!ngen#e ! (! (-ne! +e&i! &e( )erB(* e" &e'ir (! &ire''in L! Bg,r! 5$ +,e"#r! !+)(i!&o e( e(e+en#o &i !$!r'! e( )e/,eo e")e"or &e( #,$o en (! &ire''in r Co+o "e in&i'!* (! #en"in r e" e.i&en#e+en#e n,(! en (!" )!re&e" in#erior > e#erior &e( #,$o* >! /,e no ei"#e ning,n! !''in !)(i'!&! "o$re &i'@!" ",)erB'ie" /,e "on ",)erB'ie" e#eriore" &e( "(i&o E"#o" &o" ),n#o" e"#n +,> )ri+o" en e( "(i&o* >! /,e e( e")e"or e" )e/,eo D!&! (! e.o(,'in 'on#in,! /,e "e e")er! )!r! (!" .!ri!$(e" &e( )ro$(e+!* e" r!=on!$(e !",+ir /,e en#re e"o" &o" ),n#o" )ri+o" &e .!(or n,(o* r no ),e&e 're'er "igniB'!#i.!+en#e* > ),e&e &e")re'i!r"e
En#re (!" o$"er.!'ione" e)eri+en#!(e" &e( en"!>o '!$e &e"#!'!r /,e: 1 L! $!rr! no e)eri+en#! .!ri!'ione" &e (ongi#,& !)re'i!$(e"* > (!" "e''ione" '#e &e (! $!rr! )er+!ne'en )(!n!" > "in '!+$io" &e &i+en"in !)re'i!$(e" RESISTENCIA DE MATERIALES
Pgin! 0K
INGENIERIA CIVIL 0 En e( #r!+o (ine!(* "e o$"er.! )ro)or'ion!(i&!& en#re e( )!r #or"or !)(i'!&o > e( ng,(o gir!&o en#re &o" "e''ione" /,e "e en',en#r!n ! ,n! 'ier#! &i"#!n'i! L! Bg,r! 6$ +,e"#r! e( e(e+en#o &i '!+$io &e ng,(o"
L! "eg,n&! o$"er.!'in e)eri+en#!( !n#erior 'on&,'e ! /,e ei"#e )ro)or'ion!(i&!& en#re (! #en"in > e( ng,(o E"#o e" &e$i&o ! /,e e" )ro)or'ion!( ! T* > e( ng,(o gir!&o en#re &o" "e''ione" e" )ro)or'ion!( A 'on#in,!'in %,"#iB'!+o" (!" +en'ion!&!" )ro)or'ion!(i&!&e" Si 'on"i&er!+o" ,n e(e+en#o &i e( e")e"or e en "en#i&o r!&i!(* 'o+o +,e"#r! (! Bg,r! !* .e+o" /,e e( +o+en#o &e (! <,er=! /,e !'#;! "o$re e( &i e( +o+en#o #o#!( "er:
,e e)re"! (! )ro)or'ion!(i&!& T Por o#r! )!r#e* 'on"i&er!n&o ,n! (ongi#,& &e $!rr! &* > ((!+!n&o !( ng,(o gir!&o en#re "e''ione" )or ,ni&!& &e (ongi#,& &e $!rr!* ng,(o /,e "e +i&e en e( en"!>o* (! Bg,r! $ )one &e +!niBe"#o /,e e( &e")(!=!+ien#o 'ir',n
Pgin! 01
INGENIERIA CIVIL &e ,n ),n#o e#erior &e (! "e''in &ere'@!* ",)onien&o "in )?r&i&! &e gener!(i&!& /,e (! "e''in i=/,ier&! no gir!* "er )or ,n! )!r#e &* > )or o#r! )!r#e &R Lo" ng,(o" &* "e e)re"!n en r!&i!ne" Por #!n#o:
,e e)re"! (! )ro)or'ion!(i&!& > e( ng,(o &e giro en#re "e''ione" )or ,ni&!& &e (ongi#,& &e $!rr! /,e )re#en&-!+o" %,"#iB'!r
Vo(.ien&o ! (! "eg,n&! o$"er.!'in e)eri+en#!( !n#erior* (! )ro)or'ion!(i&!& en#re e( )!r #or"or !)(i'!&o T > e( ng,(o gir!&o )or ,ni&!& &e (ongi#,&* i+)(i'! (! )ro)or'ion!(i&!& en#re (! #en"in > e( ng,(o* >! /,e &? > 8 "e #iene /,e
L! 'on"#!n#e &e )ro)or'ion!(i&!& en#re e"#!" +!gni#,&e" e" ,n! '!r!'#er-"#i'! &e( +!#eri!(* /,e "e ((!+! M&,(o &e Cor#!&,r! > e" &eno#!&! 'o+o G:
Se ),e&e &e+o"#r!r /,e (! 'on"#!n#e G no e" in&e)en&ien#e &e (!" o#r!" 'on"#!n#e" e("#i'!" &e( +!#eri!( S, .!(or en <,n'in &e (!" >! 'ono'i&!" E* e":
RESISTENCIA DE MATERIALES
Pgin! 00
INGENIERIA CIVIL
A,n/,e !/,- o+i#i+o" e"! &e+o"#r!'in )or $re.e&!&* #?ng!"e no#i'i! &e /,e e( 'o+)or#!+ien#o &e( +!#eri!( i"#ro)o /,e&! 'o+)(e#!+en#e &e"'ri#o )or &o" 'on"#!n#e" e("#i'!" Se &eBnen en (! (i#er!#,r! &i.er"!" 'on"#!n#e" e("#i'!" )or 'on.enien'i!* en#re e((!" G o#r!" )o"i$(e" "on e( +&,(o &e L!+? > e( +&,(o g(o$!(* ! #-#,(o in
T$-!"# M'- /e$' En e"#e '!)i#,(o "e !n!(i=! e( '!(',(o )or #or"ion* )!r! '!rg!" e"#!#i'!" E( '!(',(o )or #or"ion* )!r! '!rg!" e"#!#i'!" E( '!(',(o &e (o" !r$o(e" ',!n&o !'#,!n +o+en#o" #or"ore" / .!ri!n en <,n'ion &e( #ie+)o '!(',(o )or (o" +o+en#o" #or"ore" in#eriore"* (o" re)re"en#!re+o" )or ,n! (ini! 'on &o" 'ir',(o" En ,no &e e((o" )on&re+o" ,n ),n#o* /,e in&i'! e( 'o+ien=o &e (! "!e#!@!'i! e( (e'#or* > en o#ro* ,n! 'r,= /,e re)re"en#! e( Bn!( &e (! "!e#! /,e "!(e &e( (e'#or Bg5 P!r! '!(',(!r (o" +o+en#o" #or"ore" M #or /,e ",rgen en (!" "e''ione" &e (! $!rr!* $!%o (! !''ion &e +o+en#o" #or"ore" e#eriore" o &e ,n! '!rg! #r!n".er"!(* e+)(e!+o" e( +e#o&o &e (!" "e''ione" Tr!ere+o" +en#!(+en#e ,n! "e''ion )!r! (! $!rr! Bg 5* )or !X! * )or e%e+)(o > !)!r#!+o" ,n! &e (!" )!r#e"* en e"#e '!"o (! i=/,ier&!* > .e!+o" e( e/,i(i$rio &e (! )!r#e &er'@!
RESISTENCIA DE MATERIALES
Pgin! 0
INGENIERIA CIVIL
L! in#er!''in en#re (!" )!r#e" &e (! $!rr! (! ","#i#,i+o" )or ,n +o+en#o #or"or M#or /,e e/,i(i$r! !( +o+en#o #or"or P!r! e( e/,i(i$rio &e (! )!r#e "e)!r!&! e" ne'e"!rio /,e (! ",+! !(ge$r!i'! &e #o&o" (o" +o+en#o" /,e !'#;!n "o$re e((!* "e! ig,!( ! 'ero De !/,-* en e"#e '!"o "e o$#iene M#or Si "o$re (! )!r#e "e)!r!&! !'#;!n .!rio" +o+en#o" e#eriore"* en#on'e" )or e( +i"+o )ro'e&i+ien#o* no" 'on.en'e+o" &e /,e e( +o+en#o #or"or en (! "e''in e" n,+?ri'!+en#e ig,!( ! (! ",+! !(ge$r!i'! &e (o" +o+en#o" #or"ore" e#eriore" /,e !'#;!n ! ,n! &e (!" )!r#e" &e (! "e''in
P!r! i(,"#r!r e( '!r'#er &e (! &i"#ri$,'in > (! +!gni#,& &e (o" +o+en#o" #or"ore" ! (o (!rgo &e (! $!rr!* "e 'on"#r,>e e( &i!gr!+! gr!B'o &e e"#o" +o+en#o" S, 'on"#r,''in e" !n(og! ! (! &e (o" gr!B'o" &e (!" <,er=!" !i!(e" en e( '!"o &e #r!''in o 'o+)re"in P!r! 'on#r,ir (o" gr!B'o"* e" ne'e"!rio )oner"e &e !',er&o re")e'#o ! (o" "igno"* /,e "e! !&+i#i&o )or #o&o" P,e&e "er !&+i#i&! en ',!(/,ier reg(! Lo ;ni'o /,e i+)or#! e" !#ener"e ! (! +i"+! reg(! en #o&o e( gr!B'o No"o#ro" "eg,ire+o" (! reg(! "ig,ien#e Bg 54 E( +o+en#o #or"or en (! "e''in !X! "e 'on"i&er! )o"i#i.o* ',!n&o e( +o+en#o e#erior gir! (! )!r#e "e)!r!&! en &ire''in 'on#r!ri! ! (!" &e (!" +!ne'i((!" &e( re(o%* "i "e o$"er.! e"#! )!r#e &e"&e (! "e''in E( +o+en#o #or"or en (! "e''in "e 'on"i&er! neg!#i.o* "i e( +o+en#o e#erior gir! (! )!r#e "e)!r!&! en (! &ire''in &e (!" +!ne'i((!" &e( re(o% "i "e (! o$"er.! &e"&e (! "e''in I(,"#r!re+o" 'on e( e%e+)(o /,e "ig,e (! 'on"#r,''in &e( gr!B'o &e (o" +o+en#o" #or"ore" Bg 55 .e!+o" e( r$o( CD* !)o>!&o en (o" 'o%ine#e" A > 7 /,e "e en',en#r!n en e/,i(i$rio $!%o (! !''in &e (o" +o+en#o" !)(i'!&o" en (!" "e''ione" E* > L #r!'e+o" (! "e''in ! X! en ,n (,g!r !r$i#r!rio &e( #r!+o" DL An!(i=!n&o e( e/,i(i$rio &e (! )!r#e "e)!r!&! &ere'@!* "e &e+,e"#r! /,e MZ #or K RESISTENCIA DE MATERIALES
Pgin! 04
INGENIERIA CIVIL Si #r!=!+o" &e"),?" (! "e''in $X$ en ',!(/,ier (,g!r &e( #r!+o L > !n!(i=!+o" e( e/,i(i$rio &e (! )!r#e /,e "e en',en#r! !(! &ere'@! &e (! "e''in* en#on'e" re",(#! /,e M TOR 0 De !',er&o 'on (! 'on.en'in !n#erior "o$re (o" "igno" &e (o" +o+en#o" #or"ore"* e"#e +o+en#o "e 'on"i&er! )o"i#i.o Tr!=!&o
L! "e''in ' ' en e( #r!+o E* &e( e/,i(i$rio &e (! )!r#e &ere'@! "e o$#en&r 0 M TOR K De &on&e "e &e&,'e /,e* M TOR 21 #+ E( gr!B'o o$#eni&o #iene (! !r* /,e en (o" (,g!re" &on&e "e !)(i'!n (o" +o+en#o"
E#eriore"* (!" or&en!&!" &e( gr!B'o .!r-!n ";$i#!+en#e en ,n! +!gni#,& ig,!( ! (! &e( +o+en#o e#erior !)(i'!&o RESISTENCIA DE MATERIALES
Pgin! 05
INGENIERIA CIVIL Si e"#n &!&!" (!" '!rg!" #r!n".er"!(e" /,e origin!n (! #or"in &e (! $!rr! Bg 50 en#on'e" "e '!(',(!n )re.i!+en#e (o" +o+en#o" #or"ore" e#eriore" origin!&o" )or e"#!" '!rg!" En e( '!"o re)re"en#!&o en (! Bg,r! 50* e( +o+en#o #or"or e#erior &e$i&o ! (! <,er=! P 3n! .e= '!(',(!&o" (o" +o+en#o" e#eriore" "e &e#er+in! (o" +o+en#o" #or"ore" in#eriore" > "e 'on"#r,>en "," grB'o"* 'o+o "e in&i' !n#erior+en#e Lo" +o+en#o" #or"ore"* &e /,e "e @!$( en e( )!rgr!
Ve!+o"* !n#e (o" re",(#!&o" &e (o" en"!>o" Si "o$re (! ",)erB'ie &e ,n! $!rr! &e "e''in 'ir',(!r #r!=!+o" ,n! re& re'#!ng,(!r* en#on'e" &e"),?" &e o',rrir (! &e II* )or e%e+)(o* no .!r-!n T!+)o'o .!r-!n (! (ongi#,& &e (! $!rr! > ", &i+e#ro E"#o &e+,e"#r! (! !,"en'i! &e #en"ione" nor+!(e" en (!" "e''ione" #r!n".er"!(e" > (ongi#,&in!(e" &e (! $!rr! A"-* ),e" &,r!n#e (! #or"in* en (!" "e''ione" #r!n".er"!(e" > (ongi#,&in!(e" !'#;!n ;ni'!+en#e #en"ione" #!ngen'i!(e"* e" &e'ir e( e"#!&o #en"ion!( &e (o" ),n#o" &e ,n! $!rr! #or"ion!&! e" &e"(i=!+ien#o ),ro RESISTENCIA DE MATERIALES
Pgin! 06
INGENIERIA CIVIL e( &i+e#ro A7 &e (! "e''in e#re+! Bg 5 gir!r! 'ier#o !ng,(o [ re")e'#o ! ", )o"i'in ini'i!(* )er+!ne'ien&o re'#o E" (gi'o ",)oner* /,e (o" r!&io" &e( re"#o &e (!" "e''ione" #r!n".er"!(e" #!+$i?n gir!r!n* )er+!ne'ien&o re'#o" L!" <r+,(!" /,e "e o$#ienen
So$re (! $!"e e"#! ",)o"i'in* "on 'onBr+!&!" )or (o" en"!>o" E( ),n#o A "e &e")(!=! )or e( !r'o AAZ > e( ),n#o C* )or e( !r'o +enor CCZZZ P!r! e"#!$(e'er (! (e> &e &i"#ri$,'in &e (!" #en"ione" #!ngen'i!(e" en (! "e''in #r!n".er"!( &e (! $!rr! #or"ion!&!* !n!(i'e+o" 'on +!" &e#!((e ", &e 58 En (! Bg,r! 58 e"#! re)re"en#!&!* ! +!>or e"'!(! (! )!r#e &e (! $!rr! /,e "e en',en#r! en#re (!" "e''ione" I > II* !"i 'o+o e( (!&o N &e( e(e+en#o LMN En ng,(o &e &i"#or"in &e( e(e+en#o LMN /,e "e en',en#r! "o$re (! ",)erB'ie &e (! $!rr! Bg 56* e" ig,!( !( 'o'ien#e &e( "eg+en#o NNZ Y (ongi#,& &e( e(e+en#o &= Bg 58
Se)!r!n&o +en#!(+en#e &e e"#! )!r#e &e (! $!rr! ,n 'i(in&ro &e r!&io \ !r$i#r!rio > re)i#ien&o (o" +i"+o" r!=on!+ien#o"* o$#en&re+o" e( ng,(o &e &i"#or"in 'orre")on&ien#e !( e(e+en#o "i#,!&o ! ,n! &i"#!n'i! \ &e( e%e &e (! $!rr!
Seg;n (! (e> &e Hooe )!r! e( &e"(i=!+ien#o o$#ene+o"
RESISTENCIA DE MATERIALES
Pgin! 0
INGENIERIA CIVIL Co+o .e+o"* en (! #or"in (! &e (!" #en"ione" #!ngen'i!(e" "on )ro)or'ion!(e" ! (! &i"#!n'i! !( 'en#ro &e gr!.e&!& E( gr!B'o &e (!" #en"ione" #!ngen'i!(e" en (! "e''in #r!n".er"!( &e (! $!rr! e"# re)re"en#!&! en (! Bg,r! 5* ! (! &ere'@! En e( 'en#ro &e gr!.e&!& &e ,n! "e''in 'ir',(!r* (!" #en"ione" #!ngen'i!(e" +i+!" !)!re'en en (o" ),n#o" &e (! "e''in /,e "e en',en#r! en (! ",)erB'ie &e (! $!rr!
Co'ien&o (! (e> &e &i"#ri$,'in &e (!" #en"ione" #!ngen'i!(e" e" <'i( >! &e#er+in!r ", +!gni#,&* )!r#ien&o &e (! 'on&i'in &e e/,i(i$rio An!(i'e+o" e( e/,i(i$rio &e (! )!r#e &e (! $!rr! ! (! &ere'@! &e (! "e''in !X! Bg 59 L! ",+! !(ge$r!i'! &e (o" +o+en#o" re")e'#o !( e%e &e (! $!rr!* /,e !'#;!n "o$re (! )!r#e "e)!r!&!* e" ig,!( ! 'ero * e" &e'ir
Sien&o \]&F e( +o+en#o #or"or e(e+en#!( &e (!" <,er=!" in#eriore" /,e !'#;!n "o$re e( re! &F
* e( +o+en#o #or"or en (! "e''in en ',e"#in In#ro&,'ien&o e"#! <r+,(! e( .!(or &e (! #en"in &e 5 o$#en&re+o"
Tenien&o en ',en#! /,e
RESISTENCIA DE MATERIALES
Pgin! 08
INGENIERIA CIVIL
Don&e JP e" e( +o+en#o )o(!r &e iner'i! &e (! "e''in @!((!re+o"
In#ro&,'ien&o en (!
Si "o$re (! $!rr! "e)!r!&! !'#,!"en .!rio" +o+en#o" e#eriore" en (,g!r &e ,no* en#on'e" en (! en (!" "ig,ien#e" "e in#ro&,'irn (! ",+! !(ge$r!i'! &e e"#o" +o+en#o" e#erno" /,e e" &e +!gni#,& ig,!( !( +o+en#o #or"or in#erior + #or A "i ),e" (! <r+,(! &eBni#i.! )!r! (! &e#er+in!'in &e (!" #en"ione" #!ngen'i!(e" e" (! #or"in e"
De e"#! <r+,(! "e &e&,'e en (o" ),n#o" e/,i&i"#!n#e" &e( 'en#ro &e (! "e''in (! #en"ione" ] "on ig,!(e" L!" #en"ione" +i+!" en (o" ),n#o" "i#,!&o" en $or&e &e (! "e''in "on
L! '!r!'#er-"#i'! geo+?#ri'! ^) "e &eno+in! +o&,(o )o(!r &e (! "e''in o +&,(o &e (! "e''in en (! #or"in En e( '!"o &e ,n! "e''in 'ir',(!r +!'i=!
> e( &e ,n! "e''in !n,(!r
RESISTENCIA DE MATERIALES
Pgin! 09
INGENIERIA CIVIL
L! 'on&i'in &e re"i"#en'i! e"#!#i'! &e ,n r$o( "o+e#i&o ! ,n! #or"in "e e"'ri$e !"i
A/,- ] e" (! #en"in #!ngen'i!( !&+i"i$(e C,!n&o "e #r!#e &e '!rg!" e"##i'!" "e 'on"i&er!
A)!r#e &e 'o+)ro$!r (! re"i"#en'i!* e"#!
Por e"#! <r+,(! "e '!(',(! e( &i+e#ro &e( r$o(* )!r#ien&o &e (! 'on&i'in &e re"i"#en'i! E( +o+en#o !&+i"i$(e )or (! 'on&i'in &e re"i"#en'i!* "e o$#iene )or (!
L!" #en"ione" #!ngen'i!(e" no "o(o !'#;!n en (!" "e''ione" #r!n".er"!(e" "ino /,e 'o+o "e &e")ren&e &e (! (e> &e re'i)ro'i&!& &e (!" #en"ione" #!ngen'i!(e"* #!+$i?n en (!" "e''ione" (ongi#,&in!(e" Bg 51K
RESISTENCIA DE MATERIALES
Pgin! K
INGENIERIA CIVIL
En (!" "e''ione" in'(in!&!" ",rgen #!n#o #en"ione" nor+!(e" 'o+o #!ngen'i!(e" )rin'i)!(e" "on (!" /,e )re"en#!n e( +i+o in#er?" E"#!" "e o$#ienen )or (! ` `++in 2] L! )ri+er! &e e"#!" #en"ione" e" &e #r!''in > (! "eg,n&! &e 'o+)re"in De (! bZZK 15K Bg 511 * ! Lo" en"!>o" &e+,e"#r!n /,e (o" +!#eri!(e" en* &,r!n#e (! #or"in )or ,n )(!no +" )re'i"!+en#e* )or ,n! ",)erB'e @e(i'oi&!( in'(in!&o re")e'#o !( e%e &e( r$o( 45K 511 * $ e" &e'ir )or (o" )(!no" &on&e !'#;!n (!" #en"ione" &e #r!''in +i+!" A "i* ),e" en (! #or"in* en #o&o" (o" ),n#o" &e (! $!rr!* ee')#o (o" ,$i'!&o" en ", e%e &on&e (!" #en"ione" "on ig,!(e" ! 'ero ",rge e( e"#!&o #en"ion!( $i!i!( &e &e"(i=!+ien#o ),ro L! )!r#e &e (! $!rr!* )ri+! ! (! ",)erB'ie* re",(#! +!" #en"!&! /,e (! /,e "e en',en#r! 'er'! &e ", e%e E" &e'ir* e( e"#!&o #en"ion!( 'orre")on&ien#e ! (! #or"in no e" @o+og?neo Si "e #or"ion! ,n #,$o &e )!re&e" &e(g!&!"* en#on'e" "e ),e&e 'on"i&er!r /,e )r'#i'!+en#e #o&o" (o" ),n#o" &e (! )!re& #ienen (!" +i"+!" #en"ione"* (o /,e "igniB'! /,e e( e"#!&o #en"ion!( e" en e"#e '!"o @o+og?neo Lo" en"!>o" ! (! #or"in &e e"#o" #,$o" "e e+)(e!n* gener!(+en#e* )!r! e( e"#,&io &e( &e"(i=!+ien#o ),ro > en )!r#i',(!r )!r! &e#er+in!r e( (-+i#e &e ,en'i! )or &e"(i=!+ien#o
Re&'!"# e#/$e e& M3e#/ 4%e -e T$'#-3!/e ' %# $.&, L' P/e#!' &' Ve&!)') A#+%&'$ Gener!(+en#e* en (o" )ro$(e+!" #?'ni'o" "e &!n )or 'ono'i&o" (! )o#en'i! /,e #r!n"+i#e e( r$o( > ", .e(o'i&!& !ng,(!r* e)re"!&! en re.o(,'ione" )or +in,#o Con e"#o" &!#o" "e '!(',(! e( +o+en#o RESISTENCIA DE MATERIALES
Pgin! 1
INGENIERIA CIVIL #or"or #r!n"+i#i&o )or e( r$o( Co+o "e "!$e* (! )o#en'i! en '!$!((o" &e .!)or e" ig,!( ! 1
Sien&o P e( e"<,er=o 'ir',n n * e( n;+ero &e re.o(,'ione" )or +in,#o &e( r$o( ),e"#o /,e Pr &e 51 "e o$#iene
De6$3'!#e- De-*&'7'3!e#/ 4%e A3*'' &' T$-!"# )e B'$$'- -e Se!"# C!$%&'$ P!r! '!(',(!r (!" &e
E( ng,(o &e #or"in 'o+)(e#o [ 'orre")on&ien#e ! (! (ongi#,& = +e&i&! &e"&e (! "e''in &e e+)o#r!+ien#o* "e o$#iene in#egr!n&o (! e)re"in !n#erior en#re K > =
Si e( +o+en#o #or"or > (! +!gni#,& GJ )* "e &eno+in! rigi&e= &e (! "e''in ! (! #or"in* "on 'on"#!n#e" en #o&o e( in#er.!(o &e in#egr!'in* en#on'e"
Co+o "e .e &e e"#! <r+,(!* ! +e&i&! /,e no" !(e%!+o" &e e+)o#r!+ien#o "e''in in+.i( e( ng,(o &e #or"in !,+en#! (ine!(+en#e E( ng,(o &e #or"in +i+o o',rre en e( e#re+o &e (! $!rr! ',!n&o dL
RESISTENCIA DE MATERIALES
Pgin! 0
INGENIERIA CIVIL
Por ", e"#r,'#,r!* e"#! <r+,(! e" !n(og! ! (!
P!r! 'on"eg,ir (! rigi&e= re/eri&! &e ,n !r$o(* e" ne'e"!rio /,e e( !ng,(o &e #or"ion ,ni#!rio +!i+o no "on ",)erior !( !&+i"i$(e
E"#! <r+,(! e)re"! (! 'on&i'in &e rigi&e= &e ,n! $!rr! &e "e''in 'ir',(!r ! (! #or"in En e((! e" e( ng,(o &e #or"in ,ni#!rio !&+i"i$(e e)re"!&o en r!&i!ne" )or ,ni&!& &e (ongi#,& &e( r$o( En (! +!>or-! &e (o" '!"o" (! +!gni#,& &e ng,(o &e #or"in ,ni#!rio !&+i"i$(e "e &! en gr!&o" )or +e#ro" &e (ongi#,& Sien&o !"- e( (,g!r &e (!
L! +!gni#,& "e e"'oge en <,n'in &e &e"#ino &e( r$o( > &e "," &i+en"ione" P!r! !r$o(e" &e &i+en"ione" +e&i&!" en e( M!n,!( &e 'on"#r,'#or &e +/,in!" "e re'o+ien&! ,n ng,(o &e #or"in !&+i"i$(e &e o*5 )or +e#ro &e (ongi#,& De !',er&o 505 "e ),e&e e( &i+e#ro &e r$o(* "i e"#! &!&! ", rigi&e= #enien&o en ',en#! /,e JP K1 &4 @!((!+o" Sien&o* N (! )o#en'i! /,e #r!n"+i#e e( r$o(* e)re"!&! en '!$!((o" &e .!)orc n (! .e(o'i&!& !ng,(!r &e( r$o(* en re.o(,'ione" )or +in,#o Lo" 'oe 7 "e &e#er+in!n en <,n'in &e )or (! #!$(! 51 )!r! G 8f1K 5 g<'+0 e" e( !ng,(o &e #or"in ,ni#!rio !&+i"i$(e en gr!&o" + e( &i+e#ro & "e o$#iene en '+
C#-/$%!"# )e &- G$9- )e &- #+%&- )e T$-!"# P!r! i(,"#r!r e( '!r'#er &e (! .!ri!'in &e (o" ng,(o" &e #or"in ! (o (!rgo &e (! $!rr!* ! .e'e" "e 'on"#r,>e (o" grB'o" &e (o" ng,(o" &e RESISTENCIA DE MATERIALES
Pgin!
INGENIERIA CIVIL #or"in grB'o" #r!n".er"!(e"
&e
(o" ng,(o" &e
giro &e
(!" "e''ione"
E( +?#o&o &e 'on"#r,''in &e e"#o" grB'o" (o e)(i'!re+o" en e( e%e+)(o !n!(i=!&o !n#erior+en#e en (! Bg,r! 510*! 'on"#r,ire+o" )re.i!+en#e e( gr!B'o &e (o" +o+en#o" #or"ore" Bg 510 $ P,e"#o /,e e( r$o( gir!* no @!> e"#e '!"o "e''ione" in+.i(e" A no"o#ro" no" in#ere"!n "o(!+en#e (o" ng,(o" &e giro &e (!" "e''ione" /,e "on 'on"e',en'i! &e (!" &e
Sien&o (ongi#,& &e( #r!+o A7 A&+i#i+o" (! reg(! "ig,ien#e )!r! (o" "igno" &e (o" ng,(o" &e #or"in: e( ng,(o [ "e 'on"i&er! )o"i#i.o* "i +ir!n&o &e i=/,ier&! ! &ere'@!* e( giro &e (! "e''in re",(#! orien#!&o en (! &ire''in &e (!" +!ne'i((!" &e( re(o% En n,e"#ro '!"o [ 7A "er )o"i#i.o A ,n! e"'!(! &e#er+in!&! 'o(o'!+o" (! or&en!&! [ 7A Bg 510*' E(
P,n#o o$#enien&o (o ,ni+o" )or ,n! re'#! 'on e( ),n#o E* ),e"#o /,e en e( #r!+o A7 (o" ng,(o" &e giro .!r-!n (ine!(+en#e RESISTENCIA DE MATERIALES
Pgin! 4
INGENIERIA CIVIL C!(',(e+o" !@or! e( ng,(o &e giro &e (! "e''in C re")e'#o ! (! "e''in 7 Tenien&o en ',en#! (! reg(! !&+i#i&! )!r! (o" "igno" &e (o" ng,(o" &e #or"in* o$#ene+o"
P,e"#o /,e (! "e''in 7 no e" in+.i(* e( ng,(o &e giro &e "e''in C re")e'#o ! (! "e''in A "er
E( ng,(o &e #or"in [ CA ),e&e re",(#!r )o"i#i.o* neg!#i.o o 'o+o '!"o )!r#i',(!r* "er ig,!( ! 'ero S,)ong!+o" /,e en e"#e '!"o e( !ng,(o [ CA re",(#o "er )o"i#i.o En#on'e" "i#,!n&o e"#! +!gni#,&* ! (! e"'!(! !&+i#i&!* &e"&e e( e%e &e( gr!B'o @!'i! !rri$!* o$#en&re+o" e( ),n#o M ,nien&o (o" ),n#o" M > "e o$#iene e( gr!
E#e$+' P/e#!'& )e &' T$-!"# D,r!n#e (! #or"in (o" +o+en#o" e#eriore" re!(i=!n ,n #r!$!%o 'o+o 'on"e',en'i! &e( giro &e (!" "e''ione" ! (!" /,e e"#n !)(i'!&o" E"#e #r!$!%o "e g!"#! en 're!r n,+?ri'!+en#e e" ig,!( !( #r!$!%o &e (!" <,er=!" in#eriore" Se ),e&e &e+o"#r!r &e ,n! +!ner! !n(og! ! 'o+o "e @i=o en e( '!"o &e #r!''in /,e e( #r!$!%o &e ,n +o+en#o #or"or e#erior* !)(i'!&o e"##i'!+en#e* e" ig,!( ! (! +i#!& &e( )ro&,'#o &e( .!(or &eBni#i.o &e( +o+en#o )or e( &e( ng,(o &e #or"in &eBni#i.o
E( #r!$!%o &e (!" <,er=!" in#eriore"* /,e e" e( "igno neg!#i.o > (! energ-! )o#en'i!( &e (! #or"in* e" n,+?ri'!+en#e ig,!( ! e"#e "e '!(',(! &e (! +i"+! +!ner! /,e "e '!(',(!$! e( #r!$!%o &e (!" <,er=!" in#eriore" en e( '!"o &e #r!''in 'o+)re"in E( #r!$!%o e(e+en#!( &e (!" <,er=!" in#eriore" RESISTENCIA DE MATERIALES
Pgin! 5
INGENIERIA CIVIL
Sien&o M TOR e( +o+en#o #or"or in#eriorc &[ e( ng,(o &e #or"in &e( e(e+en#o &e (ongi#,& &= Pero &e (!
E( #r!$!%o 'o+)(e#o &e (!" <,er=!" in#eriore" "e o$#iene in#egr!n&o (! e)re"in &e( #r!$!%o e(e+en#!( en #o&! (! (ongi#,& ( &e (! $!rr!
P!r! '!(',(!r (! energ-! )o#en'i!( /,e e" +!gni#,& ig,!( !( #r!$!%o &e (!" <,er=!" in#eriore" )ero &e "igno 'on#r!rio* #en&re+o"
Si e( +o+en#o #or"or > (! rigi&e= &e (! $!rr! no .!ri!n ! (o (!rgo &e( e%e* en#on'e"
E"#!
Re-%&/')- P$!#!*'&e- )e &' Te$' )e &' T$-!"# )e B'$$'- )e Se!"# # C!$%&'$ En e"#e #i)o &e $!rr!"* &,r!n#e (! #or"ion* (!" "e''ione" no )er+!ne'en )(!n!"* "ino /,e "e ',r.!n !(!$e!n Bg 51*! "i e( !(!$eo no e" re"#ringi&o* en#on'e" en (!" "e''ione" #r!n".er"!(e" no !)!re'en #en"ione" nor+!(e" E"#! #or"ion "e &eno+in! #or"ion ),r! o (i$re
RESISTENCIA DE MATERIALES
Pgin! 6
INGENIERIA CIVIL
L! #or"ion (i$re e" )o"i$(e "o(!+en#e "i e( +o+en#o #or"or > (! "e''ion #r!n".er"!( "on 'on"#!n#e" > (o" e#re+o" &e (! $!rr! no e"#!n e+)o#r!&o" E( '!(',(o &e (!" #en"ione" #!ngen'i!(e" en (!" $!rr!" &e "e''ion no 'ir',(!r re)re"en#! ,n)ro$(e+! $!"#!n#e 'o+)(i'!&o* /,e "e re",e(.! )or (o" +e#o&o" &e (! #eori! &e (! E(!"#i'i&!& E)one+o" !/,- (o" re",(#!&o" <,n&!+en#!(e" )!r! $!rr!" &e "e''ion re'#!ng,(!r ',!n&o !$ Bg 51*$ L!" #en"ione" #!ngen'i!(e" +!i+!" ",rgen en (o" ),n#o" 1 > 0 e" &e'ir* en e( 'en#ro &e (o" (!&o" (!rgo" > "on ig,!(e" !
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INGENIERIA CIVIL
Co+o "e .e &e (! #!$(!* )!r! (!" "e''ione" re'#!ng,(!re" e"#re'@!" &e re(!'ion !$ h 1K* "e ),e&e !&+i#ir b a 1 L! &i"#ri$,'ion &e (!" #en"ione" #!ngen'i!(e" en e( )eri+e#ro &e (! "e''ion &e ,n! $!rr!* ! (o (!rgo &e "," e%e" > &i!gon!(e"* e"#! re)re"en#!&! en (! o#ro" (!" Bno" !$ h 1K (! '!r!'#eri"#i'! geo+e#ri'! J TOR "e ),e&e '!(',(!r )or (!
L! #en"ion #!ngen'i!( +!i!+! #iene (,g!r en e( re'#!ng,(o &e +!>or !n'@,r! > "e @!((! )or (!
Sien&o $MA e( .!(or +!i+o &e( (!&o +enor &e( re'#!ng,(o E( !ng,(o &e #or"ion "e &e#er+in! )or (!
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INGENIERIA CIVIL )!re&e" &e(g!&!" &e "e''ion !$ier#! <,e e(!$or!&! )or V S V(!"o. E( &e+o"#ro /,e en e( '!"o &e (! #or"ion re"#ringi&! !)!r#e &e (!" #en"ione" #!ngen'i!(e" &e$i&!" !(! #or"ion ),r!* /,e "e &e#er+in! )or (!"
T$-!"# )e B'$$'- )e D'$e)e- Pe&+')'- )e Se!"# Ce$$')' Lo" )erB(e" &e )!re&e" &e(!g!&!" &e "e''ion 'err!&! "on +,'@o +!" rigi&o" ! (! #or"ion > )or (o #!n#o +!" 'on.enien#e" .e!+o" (! $!rr! 'i(in&ri'! ',>! "e''ion #r!n".er"!( e"#! re)re"en#!&! en (! Bg,r! 514 e( e")e"or &e (! )!re& j "e 'on"i&er! ",!.e+en#e .!ri!$(e ! (o (!rgo &e (! (ini! &e( 'on#orno &e #!( +!ner! /,e "e ),e&e )re"in&ir &e (!" 'on'en#r!'ion &e (!" #en"ione" P,e"#o /,e e( e")e"or &e (! )!re& e" in"igniB'!n#e* "e ),e&e !&+i#ir /,e (!" #en"ione" #!ngen'i!(e" origin!&!" )or (! #or"ion "e &i"#ri$,>en ,ni / )!"! )or e( ),n#o O "e! ig,!( ! 'ero
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INGENIERIA CIVIL E( )ro&,'#o ]j 'o+o +!gni#,& 'on"#!n#e* "e "!'! <,er! &e (! in#egr!(* /,e&!n&o (! e)re"in ∫ ❑ dw
C,!n&o !'#;!n +o+en#o" e#eriore"* e"#! <r+,(! &e$er in'(,ir (! ",+! &e (o" +o+en#o" e#eriore" /,e !'#;!n "o$re (! )!r#e "e)!r!&!* &e +!gni#,& ig,!( !( +o+en#o #or"or in#erior M #or en (! "e''in Tenien&o e"#o en ',en#!* o$#ene+o" &e 56
L! #en"in +i+! ",rge en e( (,g!r &e +enor e")e"or &e (! )!re&
E( ng,(o &e #or"in [ &e ,n! $!rr! &e (ongi#,& ( "e '!(',(! &e (! 'on&i'in &e /,e e( #r!$!%o &e( +o+en#o #or"or e#erior e" ig,!( !( &e (!" <,er=!" in#eriore" E( #r!$!%o &e( +o+en#o e#erior M !)(i'!&o e"##i'!+en#e* en e( &e")(!=!+ien#o !ng,(!r [ e"
C!(',(!+o" !@or! (! energ-! )o#en'i!( &e (! &e
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L! energ-! )o#en'i!( 'orre")on&ien#e !( e(e+en#o &e .o(,+en &( &" e"
Sien&o ( (! (ongi#,& &e (! $!rr! L! $!rr! )o#en'i!( #o#!( )!r! #o&! (! $!rr! "er*
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Tenien&o !@or! en ',en#! /,e (! energ-! )o#en'i!( e" n,+?ri'!+en#e ig,!( !( #r!$!%o A &e( +o+en#o e#erior* o$#ene+o"
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INGENIERIA CIVIL
CONCLUSIONES A( #er+in!r (! ',(+in!'in &e( #e+! &e TORSIN ((eg!+o" ! (! 'on'(,"in &e ',n i+)or#!n#e e" re!(i=!r e"#e #e+! /,e no" !>,&! en &i en e")e'i!( (! !>,&! /,e no" $rin&! en n,e"#r! '!rrer! &e Ingenier-! Ci.i( en (! r!+! &e E"#r,'#,r!"* /,e no" !>,&! ! !n!(i=!r (!" '!rg!" /,e #iene /,e !g,!n#!r ,n! e"#r,'#,r! >! /,e gener!n <,er=!" 'o+o e( +o+en#o , o#r!"* > )!r! e.i#!r /,e (!" e"#r,'#,r!" &e #o)o #i)o &e 'on"#r,''ione" "e &err,+$en )or e( )e"o e" ne'e"!rio !)ren&er &e TORSIN
RESISTENCIA DE MATERIALES
Pgin! 40