175606462 Slope Deflection Trabajo Final

UNIVERSIDAD PERUANA DE CIENCIAS APLICADAS FACULTAD FACULT AD DE INGENIE INGENIERÍA RÍA INGENIERIA CIVIL

TRABAJO FINAL: MÉTODO DE ANÁLISIS DE ESTRUCTURA SLOPE DEFLECTION

CURSO CURSO

: MECÁNIC MECÁNICA A DE MATER MATERIAL IALES ES

PROF PROFES ESOR OR : JAVI JAVIER ER MORE MORENO NO ALUMN ALUMNOS OS : JOSÉ JOSÉ CUY CUYA CALD CALDER ERÓN ÓN JOSÉ DE LA ROCA HENRY FLORES GAMARRA RAUL PACO AYUQUE FECHA

:

OCTUBRE DEL 2011

LIMA – PERÚ

TABLA TABLA DE CONTENIDO CONTEN IDO INTRODUCCION.........................................................................................................1 2.

“SLO “SLOPE PE - DEFL DEFLEC ECTI TION ON”” ..... ........ ...... ...... ...... ...... ...... ..... ..... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ...... ....... ....34 34

2.1 2.1

Intro Int roduc ducc! c!nn ...... .......... ........ ........ ........ ........ ........ ........ ....... ....... ........ ........ ........ ........ ........ ........ ........ ........ ........ ........ .......... ............ .......... .....3 .344

2.2 2.2

SUPOSI SUPOSICIO CIONES NES B"SICA B"SICAS S DEL #$TODO #$TODO%. %..... ........ ........ ........ ........ ......... ........... ............ ............ ........... .....3& 3&

2.2.1

Con'(ncon()% Con'(ncon()% ............................................. ........................... ..................................... ..................................... ........................ ...........3& .....3&

2.3 2.3

DEDU DEDUCC CCI* I*N N DE DE LAS LAS ECUA ECUACI CION ONES ES DEL DEL SLOP SLOPEE-DE DEFL FLEC ECTI TION ON%% ... ........ ......... ....3& 3&

2.3.1 2.3 .1

#o+(nto #o+(nto)) d( (+,otr (+,otr+(n +(nto to ..... .............. ................. ................. .................. .................. .................. ................. .............3 .....3

2.3.2 2.3 .2

#o+(nto) #o+(nto) /(n /(n(rd (rdo) o) ,or d(),0 d(),0+(n +(nto) to) .... ............. .................. .................................... ...........................41 41

2.4 2.4

PLAN PLANTE TEA# A#IE IENT NTO O ENE ENERA RALL DEL DEL #$TO #$TODO DO DEL DEL SLOP SLOPEE-DE DEFL FLEC ECTI TION ON%& %&33

2.4.1 2.4 .1

A,0cc! A,0cc!nn d(0 d(0 S0o,( S0o,( D(0( D(0(cto ctonn (n (n /) /) ......... ................. ................. .................. ............................&3 ...................&3

2.4.2 2.4 .2

A,0cc! A,0cc!nn d( S0o,( S0o,( D(0(ct D(0(cton on (n (n (0 n5 n50)) 0)) d( ,!rt ,!rtco). co)....... ............... ........................6 ...............644

2.& 2.&

E7ERCI E7ERCICIO CIOS S PROPUE PROPUESTO STOS% S% ....... ........... ........ ........ ........ ....... ....... ........ ........ ........ .......... ........... ........... ............ ......18 18&&

2. “SLOP “SLOPE E - DEFLEC DEFLECTI TION” ON”

2.1

Introducc!n

E0 ,ro()or (or/( A. #n(9 ,r()(nt! (n 1:1& (0 +;todo d(0 n50)) d( ()tructur)

S0o,(-d(0(cton (n un ,u<0cc!n )oo u( un (?t(n)!n d( ()tudo) nt(ror() c(rc d( ()u(ro) )(cundro) r(0do) ,or @(nrc #nd(r0 9 Otto #or (ntr( 0o) o) 18-1:88.

Su

,o,u0rdd )( +ntu'o (ntr( 0o) n/(n(ro) d( 0 ;,oc ,or c) 1& o) )t 0 ,rc!n d( (0 +;todo d( d)tr
t(n(r (n cu(nt u( ()t( +;todo )!0o () ,0c<0(  ()tructur) con nodo) r=/do) co+o co+o () (0 c)o d( 0) '/) contnu) contnu) 9 ,!rtco) ,!rtco) r=/do) r=/do) 9 u( no )( con)d(r con)d(r (0 ((cto d( d(or+con() d(or+con() ,or cr/ ?0 u( )on 0) u( )( ,roduc(n (n 0) c(rc).

@o9 d= ()t( +;todo (n )u or+ con'(ncon0 () ,oco Gt0 d(<do  0o) ,ro/r+) tn 'ndo) d( n50)) d( ()tructur) u( (?)t(n (n 0 ctu0dd. Sn (+<r/o )( )/u( con)d(rndo co+o uno d( 0o) +;todo) d( d(),0+(nto +5) +,ortnt( d( cu(rdo con 0) )/u(nt() con)d(rcon()%

•

Pr 0/uno) ))t(+) ()tructur0() )+,0() '/) +rco) r=/do) (0 +;todo

,u(d( ,r()(ntr un )o0uc!n r5,d 9 ,rctc. •

L) (cucon() und+(nt0() d(0 +;todo )r'(n d( <)( ,r (0 d()rro00o d(0

+;todo d( d)tr
34

L) (cucon() und+(nt0() d(0 +;todo ()t<0(c(n 0 <)( d( ntroducc!n d( +;todo) d( or+u0c!n +trc0.

•

2.2 SUPOSICIONES B"SICAS DEL #$TODO% - Todo) Todo) 0o) +(+
Lo) +o+(nt +o+(nto) o) (n 0o) nodo) nodo) (n )(ntd )(ntdoo d( 0) +n(c +n(c00) 00) d(0 d(0 r(0o> r(0o> )on )on n(/t' n(/t'o). o).

-

L) rotcon rotcon() () d( 0o) nodo) nodo) (n )(ntd )(ntdoo nt- nt-orr orro o )on )on ,o)t' ,o)t'o). o).

2.3 DEDUCCI*N DE LAS ECUACIONES DEL SLOPE-DEFLECTION% I+/n( un ,!rtco (0 cu0  )do )o+(tdo  un ))t(+ d( cr/) cu0u(r% ,()o ,ro,o cr/ '' u(r d( ))+o ))+o '(nto '(nto (tc. (tc. Co+o r()u0tdo d( 0 ,0cc!n d( ()t( ))t(+ d( cr/) 0 ()tructur )ur( d(or+ d(or+co con() n() rotco rotcon() n() 9 d(),0 d(),0+(n +(nto) to) (n 0o) nodo) 9  )u '( )( /(n(rn /(n(rn u(r) (n cd uno d( 0o) (0(+(nto) d( 0 ()tructur.

35

F/ur 6% P!rtco So+(tdo  un S)t(+ d( Cr/)

i

A

j

B

C

A

B

C

To+ndo (0 (0(+(nto i− j d(0 ))t(+ ,!rtco d(),u;) d( d(or+do )( t(n(% F/ur % An50)) d(0 E0(+(nto -> W

P

 j

θ j θ

Δ

RELATIVO

i

i

Pr (0 n50)) d( ()t( (0(+(nto )( c( u)o d( 0 (cuc!n <5)c u)d (n (0 n50)) +trc0 d( ()tructur)%

[F ] = ⎣F T

⎣

⎦+

F desplazamientos

⎦

Ecuc!n 2.1

empotramiento

Dond(% F T  Fu(r) n0() (n 0o) (?tr(+o) d( 0 <rr.

36

Fempotramiento  Fu(r) /(n(rd) (n 0o) (?tr(+o) d( 0 <rr

i− j

d(<do  0)

cr/) (?t(rn) ( P! " ctunt() )o
Fdesplazamientos  Fu(r) /(n(rd) (n (0 (0(+(nto d(<d)  0o) d(),0+(nto) d( 0 <rr. Co+o )( +(ncon! 0 nco d(0 ,r()(nt( c,tu0o (0 +;todo d(0 S0o,(-d(0(cton d(),r(c 0) d(or+con() d(<d)  0) u(r) ?0() 9 cortnt() (n 0o) (0(+(nto) t(n(ndo )o0+(nt( (n cu(nt 0) d(or+con() ,or 0(?!n +o+(nto).  A)= 0 (cuc!n <5)c (n (0 n50)) +trc0 ,0cd 0 +;todo d(0 S0o,(-d(0(cton u(d con'(rtd (n% [F

T#T$%&'

]=

⎣F ⎦ emp

 [

)

T#T$%&'

+ [F ] des

 ]=⎣)

emp



Ecuc!n 2.2

⎦+⎣)

desplazamiento

W

P

P

M θ j

θi

i

⎦

 j

W

M

 ji

TOTALES

 j i

Δ RELATIV O

=

M

M

ij

ji

)

RELATIVO

θ

F

i

M

ij DESPLAZAMIENTOS

ij TOTALES

T#T$%&'

Δ

F

M

[

θ j

+

j

]

 #o+(nto) tot0() /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr.

3*

 ji

DESPLAZAMIETOS

⎣ )

empotramientos

( P! "

⎦  #o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr ,or cr/) (?t(rn)

ctunt() )o
c(ro () d(cr 0 <rr ()t5 (+,otrd.

⎣ )

desplazamientos

⎦  Fu(r)

/(n(rd) (n (0(+(nto) d(<d)  0o) d(),0+(nto)

(θi θ  j  Δ relatio"

 A contnuc!n )( ()tudn 0o) +o+(nto) d( (+,otr+(nto 9 0o) +o+(nto) d(<do)  0o) d(),0+(nto). 2.3.1 #o+(nto) d( (+,otr+(nto

⎣ )

empotramientos

⎦ % Co+o 9 )( +(ncon! nt(ror+(nt( 0o) +o+(nto) d( (+,otr+(nto

)on 0o) u( )( /(n(rn (n 0o) nodo) d( 0 <rr d(<do  0) cr/) (

P! "

(?t(rn)

,0cd) )o
F

ij F

)

= #o+(nto /(n(rdo (n (0 nodo  j d( 0 <rr ij d(<do  ( P  ! " = #o+(nto /(n(rdo (n (0 nodo  j d( 0 <rr ij d(<do  ( P  ! "

 ji

Lo) +o+(nto) d( (+,otr+(nto )( c0cu0n% •

U)ndo un +;todo d( n50)) d( ()tructur) co+o% "r( +o+(nto c)t/0no cr/ untr '/ con>u/d (ntr( otro).

 A +n(r d( (>(+,0o )( c0cu0n 0o) +o+(nto) d( (+,otr+(nto d( 0 )/u(nt( '/ u)ndo (0 +;todo d( c)t/0no.

3,

F/ur :% / Cr/d

J  A

B

S( con)d(rn co+o r(dundnt() 0) r(ccon() d(0 (?tr(+o B

)

.

 /  . L) .

condcon() (n (0 nodo B )on t0() u(% 9

θ =-

Lu(/o%

.

⎡

%

θ

.

=

1

Δ

)

=.

0)

⎤ ⋅ ⎢ ⎥ d = ⎣ & 2 0) . ⎦

-

Ecuc!n 2.3

⋅

1

Δ. =

%

-

⎡ ) 0) ⎤ ⎢ ⋅ ⎥ d = ⎣ & 2 0 /  . ⎦

Ecuc!n 2.4

-

⋅

P0nt(ndo 0 (cuc!n d( +o+(nto) ,r 0 '/ )( t(n(%

4

-

= )(" − ). −

) ) (" = ) . +

! ⋅ 

! ⋅  

+ ( / .⋅ 3 " = -

− ( / .⋅ 3 "



Ecuc!n 2.&

D(r'ndo 0 (cuc!n d( +o+(nto con r(),(cto  cd un d( 0) r(dundnt()% 0)( "

0)(

=

"

0). •

=−3

0 /.

Pr (0 c)o d( θ . %

θ.=-=

1

% -

⎡

⎢ ⎣

) 0)

⋅

& ⋅2 0 )

% ⎡ ! ⋅  ⎤ ⎥ d = 1 - ⎢  ⎦ ⎣ .

+ ) . − / .⋅ 3

37

⎤ ⎥ ⎦

⋅ [ ] d

! ⋅

-=

3

/ .⋅ 3

+ ) .⋅ 3 −

6



/ .⋅ %

+ ) .⋅ % −

%

-



3

- = ! ⋅% 6



Ecuc!n 2.K



D(),(>ndo / . d( 0 nt(ror (cuc!n )( t(n(% 5 ). ! % = + % 3

/.

⋅

⋅

Ecuc!n 2.6

Pr (0 c)o d( Δ

•

⎡ 1 - ⎣⎢ %

Δ. = - =

/  3

-= -=

3

3

/ .⋅ %

4

3

−

⎤ ⎥ d = ⎦ . 

⎡ ! ⋅  1- ⎢−  % ⎣ %

− ) . + /.

⎤ ⋅ 3 ⎥ ⋅ [ − 3 ] d ⎦

⋅

−

,

! ⋅%

%

). 3

⋅

−

3

⋅

& ⋅2 0 / 

! 3

⋅

.

) 0)

.

4

−

,



) .⋅ %

-



Ecuc!n 2.



D(),(>ndo / . d( 0 nt(ror (cuc!n )( t(n(%

/ 

.

3 =

,

( ! ⋅% " +

3 



( )

⋅% .

Ecuc!n 2.:

"

I/u0ndo 0) (cucon() 2.6 9 2.: )( (ncu(ntr (0 '0or d(

)

.

%



3 ⋅ ! ⋅% + 3 ⋅ ) . ⋅ %  ⋅ ) . ! ⋅% = + ,  3 % ) . = ! ⋅%



Ecuc!n 2.18



4-

R((+,0ndo 0 (cuc!n 2.18 (n cu0u(r d( 0) (?,r()on() d(

/ .

=

! ⋅%

/ . )( o
Ecuc!n 2.11



Un '( conocdo) 0o) '0or() d( 0) r(ccon() (n (0 nodo )( ,0nt( 0 ()t5tc d( 0 '/ o
) $=

/$

=

! ⋅%



Ecuc!n 2.12



! ⋅%

Ecuc!n 2.13



 A)= 0o) +o+(nto) d( (+,otr+(nto ,r un '/ d( )(cc!n con)tnt( 9 con un cr/ con)tnt( )o
)ij

F

F

=−)

 j i

! ⋅ % = 



Ecuc!n 2.14

F/ur 18% #o+(nto) d( E+,otr+(nto 2

J

JL 12

•

2

JL 12

Por +(do d( t<0) d( c50cu0o d( +o+(nto) 0) cu0() +u()trn un /r5co d( 0 '/ con (0 ))t(+ d( cr/ 9 (0 '0or d( 0o) +o+(nto) (n unc!n d( 0) cr/) ( P! "

ctunt() (n 0 <rr 9 0 0on/tud d( 0 <rr.

2.3.2 #o+(nto) /(n(rdo) ,or d(),0+(nto)

4

 Aor )( ()tudn 0o) +o+(nto) d(<do)  d(),0+(nto) ⎣ )

desplazamientos

 ⎦

 .

F/ur 11% #o+(nto) d( E+,otr+(nto

M  ji

 j

θ j

DESPLAZAMIETOS

Δ

RELATIVO

θi i

M

ij

DESPLAZAMIENTOS

Co+o )( ,u(d( '(r (n 0 /ur nt(ror 0o) +o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr )( ,roduc(n ,or 3 d(),0+(nto).

. Rotc!n d(0 nodo i (θi " <. Rotc!n d(0 nodo  j (θ j " c. D(),0+(nto r(0t'o (ntr( nodo) (Δ relatio"  A contnuc!n )( n0 cd d(),0+(nto ,or )(,rdo. 2.3.2.1. #o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr i− j ,or un rotc!n (n (0 nodo i (θ i " Pr (ncontrr 0 r(0c!n (?)t(nt( (ntr( 0o) +o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr 9 0 rotc!n ocurrd (n (0 nodo

i (θ i "  con)d;r()( un <rr -> )+,0(+(nt(

,o9d (n )u (?tr(+o  9 (+,otrd (n )u (?tr(+o > 0 cu0  )urdo un rotc!n (n )u (?tr(+o  co+o )( +u()tr  contnuc!n%

4

F/ur 12% Rotc!n (n (0 nodo  d( 0 <rr ->

θ

i

Mij )ij−θ i

−θ

M −θ

i

 ji

j

 #o+(nto /(n(rdo (n (0 nodo i d(<do  un rotc!n (n (0 nodo i (θ i "

) ji − θ i

 #o+(nto /(n(rdo (n (0 nodo  j d(<do  un rotc!n (n (0 nodo

i( θ i"  A contnuc!n )( c( u)o d(0 t(or(+ d( 0 '/ con>u/d ,r c0cu0r (0 '0or d( 0o) +o+(nto) )i j − θ

i

9

) ji − θ

i

(n unc!n d( 0 rotc!n ( θ i "

Pr no )turr (0 do 9 ('tr ,o)<0() conu)on() )( u) 0 )/u(nt( con'(nc!n% )

i j −θ i

)

)  ji −θ i

)

$

.

43

F/ur 13% D/r+ d( #o+(nto) d( 0 / Cr/d M

MB

A

= +

M

A

M

A

M

B

M

B

+

Diagramas de Momentos

M ! " #L$ EI

 

A

  M    A

EI

%iga Cargada

θ

M EI B

! #L$ MB

 

"

EI

P0nt(ndo un )u+tor d( +o+(nto) (n (0 nodo  )( t(n(%

 ) =i

.

⋅%⎞⎛ ⎞ ⎛

⎛ ) ⎜ ⋅ ⎟⋅⎜ %⎟− ⎜ ⋅ ⎝  &2 ⎠ ⎝ 3 ⎠ ⎝ 

)

$

⋅%⎞⎛ ⎞

&2

⎟⋅⎜ %⎟= ⎠⎝3 ⎠

) =) $

.

 Aor c(ndo un )u+tor d( u(r) (n 9 u)ndo 0 nt(ror r(0c!n%

4 F8 = -

44

⎛

−⎜

⋅

%⎞ ⎛ ) %⎞ . ⎟+⎜ ⋅ ⎟ + θ i= &2 ⎠ ⎝  &2 ⎠

)

⎝

$

⎛  ) % ⎞ ⎛  ( ) $ " % ⎞ −⎜ ⋅ ⎟+⎜ ⋅ ⎟ + θ i= $ ⎝  &2 ⎠ ⎝  ⎠ &2 ⎛

−⎜

⎝4

⋅

) $% ⎞ &2

⎟ + θ i= ⎠

D(),(>ndo ) $  )( o
)$ =

%

θi

Ecuc!n 2.1&

 ,or con)/u(nt(%  θi &2

) .=

Ecuc!n 2.1K

% Lu(/o )ij−θ i

)  ji − θ i

 

4&2

% &2

%

θi

Ecuc!n 2.16

θi

Ecuc!n 2.1

2.3.2.2 #o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr i− j ,or un rotc!n (n (0 nodo  j (θ j "

45

Pr (ncontrr 0 r(0c!n (?)t(nt( (ntr( 0o) +o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr 9 0 rotc!n ocurrd (n (0 nodo  j ( θ  j " )( con)d(r un <rr -> (+,otrd (n )u (?tr(+o  9 )+,0(+(nt( ,o9d (n )u (?tr(+o > 0 cu0  )urdo un rotc!n (n )u (?tr(+o > co+o )( +u()tr  contnuc!n% F/ur 14% Rotc!n (n (0 nodo > d( 0 <rr ->

M −θ ij

) i j −θ

 j

θ  j

i

M −θ  ji

 j

 #o+(nto /(n(rdo (n (0 nodo  d(<do  un rotc!n (n (0 nodo  j (θ j "

)

 ji −θ  j

 #o+(nto /(n(rdo (n (0 nodo > d(<do  un rotc!n (n (0 nodo

 j (θ  j "

R(0ndo (0 +)+o n50)) u( )( o ,r (0 c)o d( 0 rotc!n (n (0 nodo  )( o
) i j − θ j )  ji − θ  j

 

 &2

% 4 &2

θ  j

Ecuc!n 2.1:

θ  j

Ecuc!n 2.28

%

2.3.2.3. #o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr i− j ,or un d(),0+(nto r(0t'o ( Δ/  "

46

Pr (ncontrr 0 r(0c!n (?)t(nt( (ntr( 0o) +o+(nto) /(n(rdo) (n 0o) (?tr(+o) d( 0 <rr 9 (0 d(),0+(nto r(0t'o

( Δ/ "

(ntr( nodo) )( con)d(r un <rr ->

(+,otrd (n )u) do) (?tr(+o) 0 cu0  )urdo un d()c(n)o '(rtc0 (n uno d( )u) ,o9o) co+o )( +u()tr  contnuc!n% . F/ur 1&% D(),0+(nto R(0t'o (ntr( nodo) d( 0 <rr ->

i

M −Δ ij

Δ

RELATIVO

&  j

M −Δ  ji

) i j −Δ / 

&

 #o+(nto /(n(rdo (n (0 nodo i d(<do 0 d(),0+(nto

r(0t'o ( Δ/  " ) ji −Δ / 

 #o+(nto /(n(rdo (n (0 nodo  j d(<do 0 d(),0+(nto

r(0t'o ( Δ/  " Nu('+(nt( )( r5 u)o d(0 +;todo d( 0 '/ con>u/d ,r c0cu0r (0 +o+(nto (n unc!n d(0 d(),0+(nto r(0t'o (ntr( 0o) (?tr(+o) d(0 (0(+(nto. Pr no )turr (0 do 9 ('tr ,o)<0() conu)on() )( u) 0 )/u(nt( con'(nc!n% ) i j −Δ /   ) )  ji −Δ /   )

$

.

4*

F/ur 1K% C50cu0o d( #o+(nto) d(<do) 0 d(),0+(nto R(0t'o (ntr( nodo) i

M

ij

−Δ &  j

=

M

 ji

−Δ

&

+

M

A

M

B

M

B

Diagramas de Momentos

M

A

+

%iga Conj'gada ! #L$ AM

 

" M EI

EI

A

Δ& MB EI !

 

"

@c(ndo )u+tor d( u(r) (n 9%

4F

8

=-

 ⎛) ⋅% ⎞  ⎛ ) ⋅% ⎞ $ . − ⎜ ⎟+ ⎜ ⎟ =⎝ &2 ⎠  ⎝ &2 ⎠

4,

#L$ M EI

B

)$=)

Ecuc!n 2.21

.

 Aor ,0nt(ndo un )u+tor d( +o+(nto) con r(),(cto 0 nodo >%

4)

=-

 j

⎛ ) $ ⋅ % ⎞ ⎛  ⎞⎛ . ) ⋅ % ⎞ ⎛  ⎞ ⎜ ⎝  &2 ⎟⎠ ⎜⎝ 3 % ⎠⎟ −⎝⎜  &2 ⎟⎠ ⎜⎝ 3 % ⎠⎟ − Δ / = 



) $⋅ % − ) $⋅ % = Δ /  3 &2 6 &2 

) $⋅ % = Δ / 6 &2

D(),(>ndo ) $ % )

6&2 $

=



⋅ Δ

Ecuc!n 2.22

/

%

 ,or con)/u(nt(% ).

6 &2 =



⋅ Δ

Ecuc!n 2.23 /

%

Lu(/o%

)

6

i j −Δ / 

)  ji −Δ / 

 &2 %  6

Ecuc!n 2.24

Δ

⋅

/ 

⋅

Ecuc!n 2.2&

Δ

&2

%



/ 

47

I+,ortnt(% E0 )/no d(0 +o+(nto 'r= d(,(nd(ndo ) (0 d(),0+(nto r(0t'o c( /rr 0 <rr (n )(ntdo orro ! nt-orro.

En (0 d()rro00o d( ()t( c,tu0o )( ,0nt( 0 )/u(nt( con'(nc!n ,r d(t(r+nr (0 )/no d(0 +o+(nto d(<do 0 Δ/  C)o A. S (0 d(),0+(nto r(0t'o

Δ/  c( u( 0 <rr t(nd  rotr (n )(ntdo d( 0)

+n(c00) d(0 r(0o> 0o) +o+(nto) (n 0o) (?tr(+o) )on ,o)t'o). F/ur 16% Rotc!n d( 0 <rr (n )(ntdo d( 0) +n(c00) d(0 r(0o>

Δ RELATIVO

Mij −Δ&

M ji −Δ& )

i j −Δ / 



6 &2

)  ji −Δ

/ 





Ecuc!n 2.2K

Δ/

%

C)o B. S (0 d(),0+(nto r(0t'o

Δ/  c( u(

0 <rr t(nd  rotr (n )(ntdo nt-

orro 0o) +o+(nto) (n 0o) (?tr(+o) )on n(/t'o). F/ur 1% Rotc!n d( 0 <rr (n )(ntdo contrro  0) +n(c00) d(0 r(0o>

M −Δ  ji

M ij −Δ

Δ RELATIVO

&

5-

&

)

i j −Δ / 



6 &2

)  ji −Δ

 − %

/ 

Ecuc!n 2.26

Δ /

Su+ndo 0o) 3 ((cto) n0do) )( ,u(d( d(cr u(% E0 +o+(nto /(n(rdo (n (0 nodo i  d(<do  0o) d(),0+(nto) θi θ j 9

•

Δ/ 

() /u0 %

)

 (

)

"

ij − θ M

ij

i

i j desplazamientos

) i j desplazamientos =

()

4 &2

%

"

 &2

θi +

%

M

−θ  j

()

i j −Δ

" / 

6 &2

θ j 9

Δ



%

Ecuc!n 2.2

/ 

E0 +o+(nto /(n(rdo (n (0 nodo  j  d(<do  0o) d(),0+(nto) θi θ j 9

•

() /u0 %

)

)

 (  j i − θ

i

 ji desplazamientos

)  ji

desplazamientos

"

=  &2 

θi +

M

()

 j i − θ

"

M

 j

4 &2 &2

%

6

θ  j 9

()

 j i −Δ

" / 

Δ / 

Ecuc!n 2.2:



%

#o+(nto) tot0() F/ur 1:% C50cu0o d( +o+(nto) tot0() W

P

M  j

 j i

TOTALES

θ  j Δ

θi i

M

ij

TOTALES

5

RELATIVO

Δ/  

R(to+ndo 0 (cuc!n

) T#T$%&' = ) empotramiento + ) desplazamientos  ,0c5ndo0 0 nodo i d( 0 <rr -> )( t(n(%

)

= ) i j Total

F

ij

+)

i j desplazamientos

F

) Total

)



ij

θ

)

ij

F

) ij = )ij +

"

()

M ( ij − i M

4 &2

%

θi +

 &2

%

i j −θ

"

M

 j

()

i j −Δ

" / 

6 &2

θ  j 9

Δ / 



%

Ecuc!n 2.38

 0o +)+o )( c( (n (0 nodo  j  o
F

) Total

)



ij

 j i

θ

F

)  ji

M (

)

 &2

=)

%

i

()

M

"  j i − θ

 j

M

()

"  j i − Δ

/ 

6 &2

4 &2

θ +

+

 ji

" ji− i

θ 9

%

j



Δ

%

Ecuc!n 2.31

/ 

S(ndo ()t) 0) (cucon() <5)c) u)d) (n (0 +;todo d(0 S0o,(-d(0(cton ,r un <rr -> d( )(cc!n con)tnt(%

4  &2 6 &2 ) ij= )ij F + θ i+ θ  j 9  Δ /  &2

% )

F

=)  j i

ji

%

%

4 &2

6 &2

+ θ + &2

%

i

θ 9

%

j

Ecuc!n 2.38

Ecuc!n 2.31

Δ 

%

/ 

5

2.4 PLANTEA#IENTO ENERAL DEL #$TODO DEL SLOPE-DEFLECTION% Pr )o0uconr ()tructur) c(ndo u)o d(0 S0o,(-d(0(cton )( )/u(n 0o) )/u(nt() ,)o)% . A,0cr 0) (cucon() <5)c) d( +o+(nto d(0 S0o,(  cd un d( 0) <rr) d( 0 ()tructur. E)to) +o+(nto) u(dn (n unc!n d( 0) rotcon() (n 0o) (?tr(+o) 9 d( 0o) d(),0+(nto) r(0t'o) (ntr( 0o) (?tr(+o) d( cd <rr. <. P0nt(r un (cuc!n d( (u0
()tructur. En 0/un) ()tructur) () n(c()ro

,0nt(r (cucon() d( (u0
0/uno) (0(+(nto) o (n tod 0 ()tructur co+o )( '(r5 +) d(0nt( (n 0o) (>(rcco) t,o.  A0 ()t<0(c(r tod) 0) (cucon() d( (u0
Δ/   ()to) )( )u)ttu9(n (n 0) (cucon() d(

+o+(nto) n0() ,0nt(d) (n (0 ,)o .

d. C0cu0do) 0o) +o+(nto) )( ,u(d(n o
E>(+,0o 2.1

53

C0cu0r 0o) +o+(nto) (n 0o) (?tr(+o) d( 0) <rr) A-B 9 B-C d( 0 )/u(nt( '/ +(dnt( 0) (cucon() d(0 S0o,(-d(0(cton.

Pr dr )o0uc!n  0 '/ ,ro,u()t )( ,0cn 0) (cucon() <5)c) d(0 S0o,( 

cd un d( 0) <rr) d( 0 ()tructur. Co+o 0 ()tructur no ,r()(nt d(),0+(nto) r(0t'o) (n nn/uno d( 0o) ,o9o) (0 G0t+o t;r+no d( 0) (cucon() no )( t(n( (n cu(nt.

L) (cucon() ,r cd <rr )on%

)

F

=) ij

)

+

ij

 j i

ji

%

 &2

θ+ i

&2 +

4 &2

θ+

%

θ

%



F

=)

4 &2

i

%

j

θ j

C50cu0o d( #o+(nto) d( (+,otr+(nto

•

Con)d(rndo todo) 0o) nodo) d( 0 ()tructur (+,otrdo) )( c0cu0n 0o) +o+(nto) d( (+,otr+(nto ,r cd un d( 0) <rr)%

) $.





P a  :

F

⋅

,-  4  5

⋅

= %



⋅

⋅

= 6



= 35;55 N - +.

54

) .$ = − F

P ⋅a

% F

F

) .

=−)

.



⋅ :



=−

! ⋅% = 

,-⋅ 4 



6 =

5

⋅

= − *; N - +.





5- < 4 

= 66;6* N - +.

R((+,0ndo 0o) '0or() (n 0) (cucon() )( t(n(%

) $.

4 < ---= 35;55 + θ

⋅

 < ---$

⋅ θ.

+

6 6 4 < --- < ---⋅θ + ⋅ θ. ) . $ = − *; + $ 6 6 4 < --- < ---) = 66;6* + ⋅θ + ⋅ θ . . 4 4  < ---4 < ---) = − 66;6* ⋅θ + ⋅θ + . .  4 4

E'0undo cd uno d( 0o) '0or() d( 0) (cucon() )( o
) $.

= 35;55 + 6666;6* ⋅ θ + 3333;33 ⋅ θ $

.

= − *; + 3333;33 ⋅ θ + 6666;6* ⋅ θ

) .$

) .

) .

$

= 66;6* + ----⋅ θ + 5---⋅ θ .

.



= −66;6* + 5---⋅ θ + ----⋅ θ .



L) condcon() d( ,o9o d( 0 ()tructur )on t0() u(%

$

 d()conocdo

θ.

 d()conocdo

θ



θ

 8 (+,otr+(nto d(0 nodo

55

T(n(ndo (n cu(nt 0) nt(ror() con)d(rcon() 0) (cucon() u(dn con'(rtd) (n%

) $.

)

= 35;55 + 6666;6* ⋅ θ

+ 3333;33 ⋅ θ

(6"

θ + 6666;6* ⋅ θ

("

$

.

= − *; + 3333;33

⋅

$

.$

) .

= 66;6* + ---- ⋅ θ

.

(3" .

= − 66;6* + 5--- ⋅ θ

(4"

) .

.

@)t ()t( ,unto )( t(n(n (n (0 ))t(+

 θ θ  ) $

.

$.

 )  )  ) . .$

.

4 (cucon() 9

.

P0nt(ndo 0 condc!n d( (u0
 ) i = -  (n 0o) nodo) A 9 B )( t(n(%

Nodo A%

4)

$

= - = )$.

)$ . = -

Lu(/o 0 (cuc!n 1 u(d con'(rtd (n% - = 35;55 + 6666;6* ⋅θ

$

+ 3333;33 ⋅ θ

.

Nodo B%

4

.

= - =

)

+

) . = -

) .$

) .$

+

K nc!/nt)

) .

(6"

56

 A)= )u+ndo 0) (cucon() 2 9 3 )( o
$

+ 6666;6* ⋅ θ

(5" .

S )( to+n 0) (cucon() 1 9 & )( t(n( un ))t(+ d( do) (cucon() 9 do)

nc!/nt)  θ



%

θ

$

.

- = 35;55 + 6666;6* ⋅ θ - = − 4;44 + 3333;33 ⋅ θ

$

$

+ 3333;33 ⋅ θ

.

+ 6666;6* ⋅ θ

.

R()o0'(ndo (0 ))t(+ )( o
−3

< - rad −3

θ. = ;4,<-

rad

R((+,0ndo 0o) '0or() d( 0) rotcon() (n 0) (cucon() nc0() d( +o+(nto) d(0 (>(rcco )( (ncu(ntrn 0o) '0or() d( 0o) +o+(nto) (?tr(+o)% −3 ) $ . = 35;55 + 6666;6* ⋅ (− 6;-* < -− 3" + 3333;33 ⋅ ( ;4, < -" = - HN - +. −3

−3

) . $ = − *; + 3333;33 ⋅ ( − 6;-* < " + 6666;6* ⋅ ( ;4, < -" = − ,;4, HN - +. ) . ,;4, HN - +. −3 = 66;6* + ---- ⋅ ( ;4, < - " = −3

)  . = − 66;6* + 5--- ( ;4, < - " = − 57;* HN - +.

⋅

Co+,ro<c!n%

•

4)

$

=

) $.

5*

= -



4)

•

.

 

= ) . $ + ) . = − ,;4, + ,;4, = -

E)to )( c( con (0 n d( '(rcr u( 0 )o0uc!n d(0 ))t(+ d( (cucon() )( o d( +n(r corr(ct E)t5tc d( 0 ()tructur%

Conocdo) 0o) +o+(nto) r()u0tnt() (n 0o) (?tr(+o) d( cd (0(+(nto )( ,u(d(n c0cu0r 0o) cortnt() ,r cd <rr +(dnt( 0) (cucon() <5)c) d( (u0
 4 ) = -   F = -  )= co+o 0) r(ccon() (n 0o) nodo).

C

i

 AB

BA

BC

CB

8 n -1.4 n-+

1.4 n-+

&8 n+

-&:.26 n-+

-&:.26 n-+

8

 AB

R A

BA

BC

RB

Brr AB% 8 n 8

-1.4 n-+

 AB

•

4)

$

BA

= −,;4, − (,-<4" + 6 ⋅>. $ = -

>. $ = 66;7 HN.

5,

CB

RC

•

F

?

= − ,- + 66;7 + > $. = -

> $ . = 3;-7 HN.

Brr BC% &8 n+

1.4 n-+

-&:.26 n-+

BC

•

4

CB

) = ,;4, − 57;* − ⎡( 5- < 4  "  ⎤ + 4 ⋅ >

⎣



⎦

.

>  = 74;45 HN. .

• +

F

?

= − ( 5- < 4 " 74;45 + > . = 

> . = -5;55 HN.

C0cu0o d( r(ccon()% Nodo A

  AB 8

R A

•

4F

8

= / $ − >$ . = -

57

=-

/ = > $

= 3;-7

HN.

$.

Nodo B BA

BC

-1.4 n-+

1.4 n-+

RB

•

4

F 8 = / . − > . $− > .  = -

/

= 66;7 + -5;55 = *;46 HN.

.

Nodo C

CB

&:.26 n-+

-&:.26 n-+

RC

•

F 8= / − > .= -

4 /  

•

= 74;45 HN.

=>

4)

.



=

)

.

− )@ = -

6-

)@ = )  . = − 57;*

HN - +.

Eu0
4 ) = − ( ,-⋅ 4 " + (6 ⋅ /  " − ⎣ ( 5-⋅ 4 "⋅ , ⎦ + (- ⋅ /  " + ) 4 ) = − ( 3- " + (-34;*6" − ( 6-- " + (744;5" − ( 57;* " = 4 F = − ,- − (5-⋅ 4" + / + / + / 4 F = − ,- − -- + 3;-7 + *;46 + 74;45 = $

.





$

•

$

8

.

 



8



D/r+) d( 0 ()tructur%

 A contnuc!n )( ,r()(ntn 0o) d/r+) d( cortnt( 9 +o+(nto ,r 0 nt(ror '/. F/ur 28% D/r+) d( Cortnt( 9 d( #o+(nto d( 0 E)tructur 18&.&& n 13.8:n

K:.:1 n

162.4Kn :4.4& n

D/r+ d( Cortnt(

-1.43 n-+ &:.2K n-+

2:.K3 n-+ &2.3& n-+ D/r+ d( #o+(nto

6

E>(+,0o 2.2

C0cu0r 0o) +o+(nto) (n 0o) (?tr(+o) d( 0) <rr) A-B 9 B-C d( 0 )/u(nt( '/. T(n/ (n cu(nt u( )( ,r()(nt un )(nt+(nto  Δ  (n (0 ,o9o B con un '0or d( 2 c(nt=+(tro). U)( (0 +;todo d(0 S0o,(-D(0(cton.

E)t( (>(rcco () )+0r 0 nt(ror ,(ro )( d(<( con)d(rr (0 G0t+o t;r+no d( 0

(cuc!n 9 u( (n ()t( c)o ocurr( un d(),0+(nto r(0t'o (ntr( 0o) nodo) A 9 B )= co+o (ntr( 0o) nodo) B 9 C. E) +,ortnt( t(n(r (n cu(nt (0 )(ntdo d(0 d(),0+(nto ,r ,od(r d(t(r+nr (0 6

)/no u( t(ndr5 (0 t;r+no &2 Δ /  

%

 A

B

6

C

 An0ndo 0 ()tructur d(),u;) d( ocurrdo (0 )(nt+(nto con'(nc!n ,ro,u()t (n 0 )(cc!n 2.3.2.3

9 c(ndo u)o d( 0

,r d(t(r+nr (0 )/no d(0 t;r+no d(0

d(),0+(nto r(0t'o )( '( u(%

L <rr AB t(nd(  /rr (n )(ntdo orro 0u(/o (0 t;r+no

•

6 &2

Δ / 

)(r5

6 Δ /  &2

)(r5



% ,o)t'o (n 0) do) (cucon() d( +o+(nto) d( 0 <rr. L <rr BC t(nd(  /rr (n )(ntdo ntorro 0u(/o (0 t;r+no

•

n(/t'o (n 0) do) (cucon() d( +o+(nto) d( 0 <rr.  Aor )( ,u(d( (ncontrr 0) (cucon() ,r cd <rr% 4 &2

F

)

ij

)

=)

ij

=)

 j i

+

F

ji

 &2 θ +

&2 +



%

i

θ+

%

6 &2 θ 9 j

%

4 &2

i



%

/

%

θ 9 j

Δ

6 &2

 

Δ



%

/ 

C50cu0o d( #o+(nto) d( (+,otr+(nto

•

Son 0o) +)+o) '0or() (ncontrdo) ,r (0 (>(+,0o nt(ror.

) F

$.

= 35;55 N - +.

F

) .$ = − *; N - +. )

F .

F

=−)

.

= 66;6* N - +.

R((+,0ndo 0o) '0or() t(n(+o)%

) $. ) .$

4 < ---= 35;55 + θ

⋅

 < ---$

+

⋅θ

6 < ---.

+



⋅ ( -;- "

6 6 6 4 < ---6 < --- < ---- θ + ⋅θ + ⋅ ( -;- " ⋅ = − *; + .  $ 6 6 6



%

63

)

= 66;6* + .

) +

4 < ----

 < ----

6 < ----

⋅ θ. + ⋅ θ − ⋅ ( -;- "  4 4 4  < ---4 < ---6 < ---= − 66;6* ⋅θ + ⋅θ − ⋅( -;- "

.

.

4



4

4



6

O<);r'()( u( (0 )/no d(0 t;r+no &2 Δ /  d(,(nd( d(0 )(ntdo d( rotc!n d( 0 <rr 

% 9 no d( u( Δ )( ,o)t'o o n(/t'o.

E'0undo cd uno d( 0o) '0or() d( 0) (cucon() )( o
) $.

= 6666;6* ⋅ θ + 3333;33 ⋅ θ + 6,;,, $

.

= 3333;33 ⋅ θ + 6666;6* ⋅ θ − 3*;*,

) .$

) .

) .

$

= ----⋅ θ + 5---⋅ θ .

= 5---⋅ θ + ----⋅ θ .

.

− ,;33 

− 4;6* 

L) condcon() d( ,o9o d( 0 ()tructur )on t0() u( % θ$

 d()conocdo

θ.

 d()conocdo

θ

 8 (+,otr+(nto d(0 nodo

 A)= 0) (cucon() u(dn con'(rtd) (n%

) $.

)

= 6666;6* ⋅ θ + 3333;33 ⋅ θ + 6,;,, $

= 3333;33 ⋅ θ + 6666;6* ⋅ θ − 3*;*, .$

) ). 

(6"

.

("

$ .

= --- ⋅ θ − ,;33 . = 5--- ⋅ θ − 4;6*

.

(3" (4"

.

64

Nu('+(nt( )( ,r()(nt un ))t(+ d( 4 (cucon() 9 0) +)+) K nc!/nt) d(0

(>(+,0o nt(ror  θ

$

 θ  )  )  )  ) . .

$.

.$

P0nt(ndo 0 condc!n d( (u0
.

.

 ) i = -  (n 0o) nodo) A 9 B )( t(n(%

Nodo A%

4

) =-=)

$.

$

) $. = -

Lu(/o 0 (cuc!n 1 u(d con'(rtd (n% - = 6666;6* ⋅ θ

$

+ 3333;33 ⋅ θ + 6,;,,

(6"

.

Nodo B%

4

.

)

+)

= - = ) .$ + ) . =.

)

.$

Su+ndo 0) (cucon() 2 9 3 )( o
$

+ 6666;6* ⋅ θ − 46;

(5"

.

S )( to+n 0) (cucon() 1 9 & )( o
nc!/nt)  θ

 $

%

θ .

65

- = 6666;6* ⋅ θ

+ 3333;33 ⋅θ

$

- = 3333;33 ⋅ θ

+ 6,;,, .

+ 6666;6* ⋅ θ − 46;

$

.

R()o0'(ndo (0 ))t(+ )( o
−3

< - rad −3

θ. = 5;3* <-

rad

R((+,0ndo 0o) '0or() d( 0) rotcon() (n 0) (cucon() nc0() d(0 (>(rcco )( (ncu(ntrn 0o) '0or() d( 0o) +o+(nto)% ) $ . = -;

HN - +.

) . $ = − 45;3 ) .  = 45;36

HN - +.

HN - +.

) . = −4;,

HN - +.

Co+o )( o<)(r' (0 nodo A 9 (0 nodo B ,r()(ntn (rror() d( c(rr( ,(ro ()to) )on c(,t<0() co+,rdo) con 0 +/ntud d( 0o) '0or() d( 0o) +o+(nto). •

E)t5tc d( 0 ()tructur%

E0 c0cu0o d( 0o) cortnt() d( 0) <rr) 9 d( 0) r(ccon() d( 0 ()tructur )( c( d( 0 +)+ or+ co+o )( d()rro00o (n (0 (>(rcco nt(ror. Lo) r()u0tdo) o
>$ . = 7; HN.

66

>. $ = 6-;,7 HN. >.  = 4-;-5 HN. > . = 57;76 HN.

R(ccon()%

/ $ =

3;-7

/ . =

--;73

/ 

= 57;76

HN. HN.

HN.

)@ = − 4;,

HN - +.

E>(+,0o 2.3. C0cu0r 0o) +o+(nto) (n 0o) (?tr(+o) d( 0) <rr) A-B 9 B-C d( 0 )/u(nt( '/. L ()tructur ,r()(nt un )(nt+(nto

  Δ  (n (0 ,o9o A d(

3 c(nt=+(tro) 9 un

)(nt+(nto (n (0 ,o9o B d( 1 c(nt=+(tro. U)( (0 +;todo d(0 S0o,(-D(0(cton.

 A0 n0r 0 ()tructur d(),u;) d( ocurrdo) 0o) )(nt+(nto) (n 0o) nodo) )( ,u(d( '(r u( (0 d(),0+(nto r(0t'o (ntr( (0 nodo A 9 B () /u0 0 )(nt+(nto

6*

u( )ur( (0 nodo A +(no) (0 )(nt+(nto u( )ur( (0 nodo B. E)crto (n or+ d( (cuc!n )( t(n(%

Δ

= Δ

$ −.

− Δ $

.

Δ $ − .  3 c+. - 1 c+. Δ $ − .  2 c+.  8.82 +.

En (0 c)o d( 0 <rr BC (0 nodo B )( )(nt 1 c+. +(ntr) u( (0 nodo C ,(r+n(c( (n )u ,o)c!n nc0.

Δ .−  = Δ

.

− Δ

Δ .−



 1 c+. - 8.

Δ .−



 1 c+.  8.81 +.

Co+o )( ,u(d( '(r 0) do) <rr) t(nd(n  /rr (n )(ntdo ntorro 0u(/o (0 6

t;r+no &2 Δ /  )(r5 n(/t'o (n 0) (cucon() d( cd <rr. 

%

E0 ,roc(d+(nto d( d()rro00o () /u0 0 ,0nt(do (n 0o) do) (>(rcco) nt(ror().

P0nt(ndo 0) (cucon() ,r cd <rr%

•

4 <---)

$.

) .$ ) .

= 35;55 + θ

<----

⋅

$

+

6<----

⋅ θ.

−



⋅ ( -;- "

6 6 6 4 < ---6 < --- < ---⋅θ + ⋅ θ. − ⋅ ( -;- " = − *; +  $ 6 6 6 4 < ---6 < ---⋅ . − ⋅ ( -;- " = 66;6* +  θ 4 4

6,

).

 < ---= − 66;6* +

4

⋅

6 < ---θ

.

− 4



⋅ ( -;- "

R(cordndoθ = - ,or (0 (+,otr+(nto.

E'0undo 0o) t;r+no) d( cd (cuc!n )( o
•

) $.

)

= 6666;6* ⋅ θ + 3333;33 ⋅ θ + ;

(6"

= 3333;33 ⋅ θ + 6666;6* ⋅ θ − -4;44

("

$

$

.$

) .

.

.

= ---- ⋅ θ − 7;*

(3"

= 5--- ⋅ θ − -4;*

(4"

.

) .

•

4

4

.

P0nt(ndo (0 (u0
=$.

= ) . $+ ) .  = -

)

 A)= )( o
$

+ 3333;33 ⋅ θ + ; .

$

+ 6666;6* ⋅ θ − 33;6 .

L )o0uc!n d( 0) (cucon() nt(ror() ()% θ $= − 4;,

−3

< - rad

θ. = ,;7,<-

•

−3

rad

R((+,0ndo ()to) '0or() (n 0) (cucon() nc0() d( +o+(nto) )( o
67

) $. = -

HN - +.

) . $ = − 6-;64

HN - +.

HN - +.

) .  = 6-;64

) . = − 57;*

HN - +.

R(ccon()%

/ $ =

6;56

HN.

/ . =

63;*,

HN.

/ 

= 77;66

HN.

)@ = − 57;*

HN - +.

E>(+,0o 2.4. C0cu0r 0o) +o+(nto) (n 0o) (?tr(+o) d( 0) <rr) A-B 9 B-C d( 0 )/u(nt( '/. E0 ,o9o A ()t r(,r()(ntdo ,or un r()ort( con un r/d( d(

A  = 5--- AB C m

En 0/un) oc)on() 0) ()tructur) c'0() no ,r()(ntn ,o9o) ,(r(ct+(nt( r=/do) co+o )( ,u(d( ,r()(ntr 0 t(n(r co+o ,o9o un )u(0o co+,r()<0(. E)t( t,o

*-

d( )u(0o ,u(d( ,r()(ntr )(nt+(nto) d(r(nc0() 0 )(r )o+(tdo  cr/). S )( conoc(n 0) ,ro,(dd() d(0 )u(0o )u r/d( ,u(d( )(r r(,r()(ntd con (n r()ort(

d(

/u0 r/d( () d(cr 0o) r()ort() )on d(0con() u( (n (0 n50)) ()tructur0 )( ,u(d(n (+,0(r ,r r(,r()(ntr r/d(c(). Co+o )( ,udo '(r (n (0 (>(rcco 2 9 3 0o) )(nt+(nto) u( )ur(ron 0o) ,o9o)

(rn

conocdo). En ()t( (>(rcco (0 )(nt+(nto u( ocurr( (n (0 ,o9o A () d()conocdo (0 nodo B ,(r+n(c( (n )u ,o)c!n nc0 0u(/o (0 d(),0+(nto r(0t'o (ntr( 0o) do) nodo) )(r5  Δ .

P0nt(ndo 0) (cucon() d( +o+(nto ,r cd <rr%

•

4 <---)

$.

) .$ ) .

).

= 35;55 + θ

 <----

⋅

$

+

6 <----

⋅ θ.

−

Δ

⋅



6 6 6 4 < ---6 < --- < ---⋅θ + ⋅ θ. − = − *; + Δ  $ 6 6 6 4 < ---θ ⋅ . = 66;6* + 4  < ---⋅ θ. = − 66;6* + 4 ⋅

R(cordndoθ = - ,or (0 (+,otr+(nto.  E'0undo 0o) t;r+no) d( cd (cuc!n )( o
•

) $.

) .$

•

$

+ 3333;33 ⋅ θ − 666;6*

$

.

= 66;6* + ---- ⋅ θ

.

= − 66;6* + 5--- ⋅ θ

(6"

Δ

⋅

.

= − *; + 3333;33 ⋅ θ + 6666;6* ⋅ θ − 666;6*

)

)

= 35;55 + 6666;6* ⋅ θ

Δ

⋅

("

.

(3" .

(4" .

P0nt(ndo (0 (u0
*

 ) $= )

$.

=-

- = 35;55 + 6666;6* ⋅ θ $ + 3333;33 ⋅ θ − 666;6*

(6"

Δ

⋅

.

)

.

= ) . $+ ) . = -



⋅θ

+ 6666;6* ⋅ θ − 666;6*

Δ

⋅

(5"

- = − 4;44 + 3333;33 $

.

O
(6"

Δ

⋅

.

- = − 4;44 + 3333;33 ⋅ θ

$

+ 6666;6* ⋅ θ − 666;6*

Δ

⋅

(5"

.

Co+o )( ,u(d( '(r )( t(n(n 2 (cucon() 9 3 nc!/nt). Pr (ncontrr 0 t(rc(r (cuc!n d(0 ))t(+ )(,r+o) (0 tr+o AB d( 0 ()tructur 9 r(0+o) )u d/r+ d( cu(r,o 0
 A0 c(r un cort( (n 0o) (?tr(+o) d( cu0u(r <rr (0 +o+(nto nt(rno u( ,r(c( )(r5 ,o)t'o () d(cr )(ntdo nt-orro. E)to )( d(<(  0 con'(nc!n u)d (n 0 d(ducc!n d( 0) (cucon() <5)c) d(0 )0o,(. @c(ndo un )u+tor d( +o+(nto) (n (0 ,unto B )( o
*

4)

= − ( / $ ⋅6" + ,-⋅  + ) . $ = -

.

D(),(>ndo ).$ =

). $ %

/  $⋅6

− 6-

R(corddo u( 0 u(r d( 0o) r()ort() ()t dd ,or 0 (?,r()!n  F= A ⋅ Δ  0 r(cc!n d(0 ,o9o A  / $  )( ,u(d( ()cr<r (n unc!n d(0 )(nt+(nto d(0 ,o9o

/ $ = 5---⋅ Δ . •

R((+,0ndo ()t (?,r()!n (n 0 (cuc!n d( +o+(nto

) . $ = (5--- Δ"⋅6 ⋅

) . $ = 3----

⋅

Δ−

). $  )( o
− 6-

6-

(*"

E)t (cuc!n r(0con (0 +o+(nto

). $

con (0 d(),0+(nto r(0t'o d( 0 <rr AB.

S )( /u0 ()t (cuc!n 6 con 0 (cuc!n 2 )( o
⋅

Δ − 6-

= − *; + 3333;33 ⋅ θ

- = ,,;,7 + 3333;33 ⋅ θ

$

$

+ 6666;6* ⋅ θ

+ 6666;6* ⋅ θ − 666;6* − 3666;6*

(,"

Δ

⋅

.

E0 ))t(+ d( (cucon()  )o0uconr ()% - = 35;55 + 6666;6* ⋅ θ $ + 3333;33 ⋅ θ − 666;6*

(6"

Δ

⋅

.

- = − 4;44 + 3333;33 ⋅ θ

$

Δ

⋅

.

+ 6666;6* ⋅ θ − 666;6* .

*3

Δ

⋅

(5"

- = ,,;,7 + 3333;33 ⋅ θ

$

+ 6666;6* ⋅ θ − 3666;6*

Δ

⋅

(,"

.

So0uconndo (0 ))t(+ d( (cucon() d( 3?3 )( o
Δ = ;5* < -− 3m $

Con ()to) '0or() d( d(),0+(nto) (ncontr+o) (0 '0or d( 0o) +o+(nto)% ) $. = -

HN - +.

) . $ = − ,;73

HN - +.

HN - +.

) .  = ,;,*

) . = − 5,;5*

HN - +.

R(ccon()%

/ $ =

;,5

/ . =

*3;3

/ 

= 73;73

HN. HN. HN.

)@ = − 5,;5*

HN - +.

*4

BIBLIORAFIA

C@A7ES A0(?nd(r. 1::8. Structur0 An50)). N(Q 7(r)9. Edtor0 Pr(ntc( @00 @EAO NORRES Cr0() JILBUR BENSON 7on UTU S(no0

1:6K. E0(+(ntr9

Structur0 An09)) Edtor0 #c rQ @00

S@EPLE Erc.1:&8. Contnu) B(+ Structur() A (/r(( o F?t9 #;tod nd t( #(tod o +o+(nt D)tr
TI#OS@ENO S P 1:K&. T(or9 o )tructur() 7,n. #crQ @00 o/Hu) Ltd #c Cor+n 7cH C.1:6&. Structur0 An09)). N(Q orH Int(?t Educton0 Pu<0)tor) IA JAN Cu. 1:&3 Sttc009 Ind(t(r+nt( Structur() 7,n #c rQ @00odHu) Ltd PLU##ER L. Fr(d. 1:44. Fund+(nt0 o Ind(t(r+nt( Structur(). N(Q orH Pt+n Pu<0)n/ Cor,orton. ONALE CUEAS O)cr #. 2882 An50)) E)tructur0. #;?co D.F. L+u) Nor(/ (dtor() #ARS@AL J. T. NELSON @. #. 1:66 E)turctur() 2( Londr() Pt+n. Pu<0)n/ L+t(d

ALLECILLA B. Cr0o) R+ro. 2883. Fu(r) S=)+c) ,rnc,o) 9 ,0ccon() NSR : Bo/ot5 Co0o+<. +,r() Ltd.

*5