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" 4)! L$*8+ !$! $&$#$)4 !. K)! '+%)! #'&"$#$'& ! !+M& " N&$")! & ,-./21 O ,-./B1. PL'%)
ˆ ˆ
σxy dx dy =
R
ˆ ˆ
R
∂ψ
dx dy
∂z
E'% 4 + '% :) " Q)(!!G !+) ! C ψνy ds O ! )&(4) " )#( %"' #'& ,-.2H1. C& ;'%:) !$:$4)%
! )&(4) 4) % !(4+)&+ " ;( %R)! & 4) "$% ##$8& z . E'% 4' +)&+' 4)! #()#$'& ! ,-./21 ! )&(4)&.
>$&)4: &+ 4) #()#$8& ,-./B1 % @($ % @( A
´
Mx = −
ˆ ˆ R
∂ψ
∂ψ
y
+z
∂y
∂z
@( (!)&"' 4 + '% :) " Q)(!! *( " +%)&!;'%:)%!
dy dz
&A
ˆ ˆ ∂
∂
(yψ) +
(zψ) − 2ψ dy dz
Mx = −
∂y
∂z
R
ˆ ˆ
ˆ
2ψ dy dz
= − {yψ cos (ν, y) + zψ cos (ν, z)} ds +
C
,-.271
R
!"#$%& '(#)&%& %! '*+,*$!$-! %! %!.*(+&'/*$!
∂ 2 γxy
,*( !0!+,1* ∂y∂z
.#$'/5$ %! 1* %! ,1&)&+/!$-* ! -&61!'!( '*$%/'/*$! %! 1& .*(+&
εij,kl + εkl,ij − εik,jl − εjl,ik = 0
7#! # &$%* 1& 1!3 %! 8**9! ,#!%!$ ! '(/6/( ! !$ .#$'/5$ %! -!$ /*$!
-
=
∂ 2 γxy
∂z∂y
2 3 & ,&(-/( %! # %!4$/'/5$ !$
! R "# $%& '"(!)% #!*+,"*"%-" ./%"0&1 ,& !%-"('&, 2" ,3%"& #" &%$,& 2"4!2/ & ,& ./%2!.!)% 56789:7
;% ./%#".$"%.!&
ˆ ˆ
Mx =
2ψ dy dz
R
;%-/%."#1 -/2&# ,&# ".$&.!/%"# 2!<"'"%.!&,"# = ,&# ./%2!.!/%"# 2" ./%-/'%/ '"<"'!2&# & -"%#!/%"#
#" #&-!#<&."% #! ψ /4"2"." ,&# ".$&.!/%"# 5678>:1 56789: = 5678?:7 @"'/ +"'*&%"." !%2"-"'*!%&2& ,&
./%#-&%-" 2" ,& "0+'"#!)% 5678>:7 ;#-& ./%#-&%-" -!"%" A$" #"' 2"-"'*!%&2& +/' ,&# ./%2!.!/%"# 2"
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∂u
τxy
=
+ θz
∂y
G
∂u
τxz
=
− θy
∂z
G
5678E:
2"'!F&%2/ ./% '"#+".-/ & z = & y '"#+".-!F&*"%-"1 = G&.!"%2/ ,& 2!<"'"%.!&1 #" /4-!"%"
1
G
$#&%2/ 567>H: 2&
∂τxy ∂τxz
−
∂z
∂y
= −2θ
∂ 2ψ ∂ 2ψ
+ 2 = −2Gθ
∂y 2
∂z
56786:
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!" #$%&#'()* +,-+ '#%.#/# !!#0"& " %"&1(& /#2
∂ ū
∂ ū
− ∂ψ
∂y − z 3
∂y = τxz = θG ∂z + y
/#&(4"*/5 !" %&(6#&" &#'%#715 /# z !" '#0.*/" &#'%#715 /# y 8 &#'1"*/5 6(#69&5 " 6(#69&5
∂ψ
∂z
= τxy = θG
∂2ψ ∂2ψ
+
∂z 2
∂y 2
∂2ψ ∂2ψ
+
∂z 2
∂y 2
!
=
=
∂
∂ ū
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∂
−z
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∂z
∂y
∂y
∂z
2
∂ u
∂2u
∂y
∂z
θG
−
+
−
= −2Gθ
∂z∂y ∂z∂y
∂z
∂y
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T=
σ 0
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∂
∂
∂u
∂u
σe
+
σe
+p=0
∂y
∂y
∂z
∂z
5*$.# σ 2#$(3&$3*- ?& *D",*(56$ &$3*,5#, ,*(!?3&
p
∂ 2u ∂ 2u
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∂y
∂z
σe
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τxz = −
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∂z
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∂u
τy =
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(2Gθ) =
∂z p
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∂u σe
∂u
τz = −
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∂y p
∂y
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∂u σe
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"!"#$"*" 3 .) ,"4$% *$! ,+&,% "&( 8%,% !%& <"!%'$& *$ ",#"& 8%,4%)$)-$& *$ -$)&+%)$& *$ 8%'-$/
0" ,"3%'$& -$)&+%)$& *$ 8%'-$ &$ 8%)8$)-'") $) !"& "!"&: *%)*$ F"3 .)" <"'+"8+9) 8."&+ !+)$"! $)-'$
!" 4"'-$ &.4$'+%' $ +)R$'+%' *$ !" ,+&,"/ &-% $& 4"'-+8.!"',$)-$ 8+$'-% $) $! 4$'5! *$ "!"& ")8F"&
3" =.$ $! *$ "!"& $&-'$8F"& )% -+$)$ $&4$&%' .)+R%',$/
1
1.4E+05
1.0E+05
6.0E+04
2.0E+04
-2.0E+04
-6.0E+04
-1.0E+05
-1.4E+05
u
0.8
4.20
3.40
2.60
1.80
1.00
0.20
0.6
Z
0.4
0.2
00
τxy
0.4
0.2
Y
1.2E+05
1.0E+05
8.0E+04
6.0E+04
4.0E+04
2.0E+04
0.0E+00
τxz
;+1.'" 6/QS -%'&+9) *$ .)" &$88+9) MNOGBB
T$! ,%*$!% ).,U'+8% 4.$*$ %#-$)$'&$ -",#+U) !" '+1+*$E " -%'&+9)/ ) !" <$'&+9) &+,4!+58"*"
$&-" &$ $&8'+#$ 8%,% GIt / 0" -"#!" *$ 4$'5!$& 4'%<$$ !%& <"!%'$& ItIP N = 14,6 3 ItIP B = 63,4 $)
-")-% =.$ !%& <"!%'$& $=.+<"!$)-$& 4'%<+&-%& 4%' $! ,%*$!% ).,U'+8% &%) ItIP N = 12,8 3 ItIP B = 60,2
!"!2! $%&'()* +, -* 3-0% /-.+&.+%
&-$ -$'8$' $I$,4!% 8%''$&4%)*$ " .) -.#% 8."*'"*% ?0"*% B = 200,, 3 $&4$&%' t = 10,,D
*%)*$ F"#+-."!,$)-$ &$ .-+!+E" !" R9',.!" *$ P'$*-/ ) !" ;+1.'" 6/6 &$ ,.$&-'") !%& '$&.!-"*%&
%#-$)+*% 4%' $!$,$)-%& 5)+-%&/ ) $&-$ 8"&% &$ F" ",4!+"*% !%& '$&.!-"*%& $) !"& $&=.+)"& *%)*$
@G
180000
140000
100000
60000
20000
-20000
-60000
-100000
-140000
-180000
1.5
u
1
Z
8.50
6.50
4.50
2.50
0.50
0.5
00
0.2
0.6
0.4
0.8
1
Y
τxy
180000
160000
140000
120000
100000
80000
60000
40000
20000
0
τxz
!"#$% &'() *+$,!-. /0 #.% ,011!-. 234566
7%, 80.,!+.0, .+ ,+. #.!9+$:0, 0. 07 0,;0,+$ < 7% 9-$:#7% /0 4$0/8 .+ %;$+=!:% 1+$$018%:0.80 7%,
80.,!+.0, /0 1+$80' >. 7% !"#$% &'&'1? ;#0/0 @0$,0 A#0 07 B#C+ /0 7%, 9#0$D%, /0 1+$80 0, E%,8%.80
#.!9+$:0 % ;+1+ A#0 .+, %70C0:+, /0 7% 0,A#!.%? ;0$+ %77F G%< #.% :%<+$ 80.,!-. 0. 7% ;%$80
!.80$.%' >. 7% !"#$% &'&'/ ,0 G% "$%H1%/+ 07 :-/#7+ /0 7% 80.,!-. /0 1+$80 ;%$% %10.8#%$ 7+
0=;$0,%/+ %.80,'
I% $!"!/0D % 8+$,!-. A#0 $0,#78% /0 7% 5/% 9-$:#7% /0 4$0/8 J,#;+.!0./+ #.% ,011!-. 1+.
0,A#!.%, % 90o K 0,
(It G)Bredt
equiv
(2Sm )2
= G ´ ds = G × 6859 cm4
0. 8%.8+ A#0 /07 :+/07+ .#:L$!1+ $0,#78%
t
4
um
=
G
×
7067
cm
(It G)N
equiv
M0"N. 7% ;$!:0$% 9-$:#7% /0 4$0/8? 07 @%7+$ /0 7% 80.,!-. /0 1+$80 J#.!9+$:0K @%70)
τ=
)
´ m
G (2S
ds
2
q
Mt
G (2Sm )
θG (B − t)
t
=
=θ
=θ
=
= θG × 9,5 [cm]
t
2Sm t
2Sm t
perim
2
O. %.P7!,!, /08%77%/+ 0. 0,80 0C0:;7+ :#0,8$% A#0 0. 7%, ;%$80, $018%, 7%, 80.,!+.0, /0 1+$80 0. 7%
;%$0/ 0=80$.% ,+. 56 Q :%<+$0, A#0 0. 7% ;%$0/ !.80$.% 0. 8%.8+ A#0 7% :P=!:% 80.,!-. /0 1+$80
0. 7% ;%$80 1#$@% 0=10/0 0. #. R6 Q 07 @%7+$ /0 7% 80.,!-. :0/!% /0 1+$80 τmedia = 9,81θG
τexterna = 10,81θG
τinterna = 8,81θG
τmáxima = 12,85θG
!"! #$%&'()* +*,-.) (, ,-,()/0, .% (, &%&12,-,
!"!#! $%&&'() &'*&+,-*
S0%:+, 07 0C0:;7+ :P, ,0.1!77+ /07 1#%7 80.0:+, #.% ,+7#1!-. 0=%18%) 7% ,011!-. 1!$1#7%$'
T0,+7@%:+, 07 ;$+E70:% 0. 9+$:% .#:L$!1% 1+. 7+, ,!"#!0.80, ;%$P:08$+,) /!P:08$+ D = 151:?
p = 13% < σe = 100UV:' M#;+."%:+, A#0 07 :-/#7+ /0 07%,8!1!/%/ /07 :%80$!%7 0, G = 8×104 W3%
< A#0 07 "!$+ ;+$ #.!/%/ /0 7+."!8#/ 0, θ = 0,01$%/V:' >7 9%18+$ /0 0,1%7% $0,#78%
α=
σe
p
(2Gθ) = 160000 [W;%]
XR
100
600.000
450.000
300.000
150.000
0.000
-150.000
-300.000
-450.000
-600.000
80
Z
60
40
20
Z
Y
0
(a)
(b)
20
40
60
80
100
120
140
Y
1.3E+08
1.2E+08
1.2E+08
1.1E+08
1.1E+08
1.1E+08
1.0E+08
9.8E+07
9.4E+07
9.0E+07
8.6E+07
8.2E+07
(c)
(d)
!"#$% &'&( )*$+!,- ./ #- )#0* 1#%.$%.*
2- /+)/ 1%+* 3%+ )/-+!*-/+ ./ 1*$)/ +/ 1%31#3%-(
τ =−
∂u
∂u
α=−
× 160000 [45%]
∂s
∂s
3% )/-+!,- ./ 1*$)/ /6%1)% 7%3/ 5%$% #- $%.!* 1#%38#!/$%
r
τ = Gθr
2- 3% !"#$% &'9 +/ *0+/$7%- /- 3% 5%$)/ !:8#!/$.% ;%< 3%+ 3=-/%+ ./ -!7/3 ./3 ./+53%:%>!/-)*
)$%-+7/$+%3 +*0$/ #- 1#%$)* ./ 3% +/11!,- ? % 3% ./$/1@% ;0< 3%+ )/-+!*-/+ ./ 1*$)/ % 3* 3%$"* ./
1#%38#!/$ 3!-/% $%.!%3' 2- /+)A B3)!>% 3% 3=-/% 1*-)!-#% /+ /3 7%3*$ /6%1)* /- )%-)* 8#/ 3*+ 1=$1#3*+
!-.!1%- /3 7%3*$ *0)/-!.* -#>C$!1%>/-)/
!"!#! $%&&'() *)+,*- &%--*.*
D- +/"#-.* 1%+* ./3 8#/ +/ 1*-*1/ 3% +*3#1!,- /6%1)% /+ /3 ./ #- )#0* ./ +/11!,- 1!$1#3%$'
E/+*37%>*+ /3 1%+* rm = 101> ? t = 11> 1*- 3%+ >!+>*+ 7%3*$/+ ./ pF σeF G ? θ ./3 /G/>53*
%-)/$!*$'
H%$% $/+*37/$ /+)/ 5$*03/>% ./0/ -*)%$+/ 8#/ /3 1*-)*$-* ./ 3% +/11!,- /+)A I*$>%.* 5*$
.*+ 3=-/%+( 3% ./+1$!5)% 5*$ /3 $%.!* /6)/$-* ? 3% ./+1$!5)% 5*$ /3 $%.!* !-)/$-*' J%+ 1*-.!1!*-/+
./ 1*-)*$-* ./3 5$*03/>% $/8#!/$/- 8#/ u +/% 1*-+)%-)/ /- /3 1*-)*$-*' K#%-.* @%? #- B-!1*
1*-)*$-* 0%+)% 1*- +#5*-/$ 8#/ %33= u 7%3/ 0' K#%-.* @%? >A+ ./ #- 1*-)*$-*F /- /+)/ /G/>53* ?
/- "/-/$%3 /- +/11!*-/+ >#3)!1/3#3%$/+F +/ +#5*-/ 8#/ u = 0 +*0$/ /3 1*-)*$-* /6)/$-*F /- )%-)*
LM
80
60
Exacta
Numerica
50
60
τ[MPa]
r[mm]
40
40
30
20
20
10
0
0
20
40
0
0
60
20
r [mm]
40
60
80
r[mm]
!"
#"
$%&'(! )*+, -.//%01 /%(/'2!( 3!/%4!
5'. .1 267 68(67 /6186(167 9.#. !7.&'(!(7. '1 :!26( /6178!18. .1 /!9! '16 9. .2267; /'<6 :!26(
9.#. 9.8.(3%1!(7. /636 =!(8. 9. 2! 762'/%01*
>6( 68(6 2!96 2! =(.7%01 16(3!2
p
9.#. !=2%/!(7. 76#(. 8696 2! 7'=.(?/%. 9.2%3%8!9! =6( .2
/6186(16 .@8.(16* A7 9./%( 5'. 2! =(.7%01 16 7626 7. !=2%/! 76#(. 2! 3.3#(!1! 7%16 8!3#%B1 76#(.
2!7 461!7 9.2%3%8!9!7 =6( 267 /6186(167 %18.(%6(.7*
>!(! .2 .C.3=26 7.1/%226 5'. 167 6/'=! .1861/.7,
-6#(. .2 /6186(16 .@8.(16
-6#(. .2 /6186(16 %18.(16
re = 10533"
7. %3=61. 2! /619%/%01
u=0
ri = 9533" 7. %3=61. 2! /619%/%01 u = /8.* /'<6 :!26( 7!29(D 9.2
!1D2%7%7
2! =(.7%01 2!8.(!2 7. !=2%/! 76#(. 2! 3.3#(!1! < 76#(. 2! 7'=.(?/%. /%(/'2!( %18.(1! 9. (!9%6
ri
E.#%96 ! 2! 7%3.8(F! 9.2 =(6#2.3!; .2 (.7'28!96 .7 %9B18%/6 ! 26 2!(&6 9. /'!25'%.( (!9%6; 26 /'!2
=.(3%8. (.762:.( .2 =(6#2.3! /636 '1%9%3.17%61!2
A1 2! $%&'(! )*G! 7. 3'.78(! .2 9.7=2!4!3%.186 9. 2! 3.3#(!1! .18(. '1 :!26( 3D@%36 .1 .2
#6(9. %18.(16 H!78! .2 :!26( 1'26 %3='.786 .1 .2 #6(9. .@8.(16* A1 2! $%&'(! )*G# 7. /63=!(!1 2!7
762'/%61.7 .@!/8! 2%1.!2" < 1'3B(%/!*
!"!#! $%&&'() *%&+,)-./,* 01234
I617%9.(.367 !H6(! '1! 7.//%01 (./8!1&'2!( 9. 9%3.17%61.7
3%7367 :!26(.7 9.
p; σe; G
<
θ
b = 5/3
<
h = 20/3;
/61 2!7
9.2 .C.3=26 !18.(%6(* J8%2%4!196 2!7 /619%/%61.7 9. 96#2. 7%3.8(F!;
1'3B(%/!3.18. !2/!14! /61 !1!2%4!( '1 /'!(86 9. 2! 7.//%01
A1 2! =!(8. 7'=.(%6( 9.(./H! 9. 2! $%&'(! )*KL 7. :.1 2!7 2F1.!7 9. 1%:.2 9. 267 9.7=2!4!3%.1867
9. 2! 3.3#(!1!* A1 2! =!(8. %1M.(%6( 9. 2! $%&'(! )*KL 7. H! &(!?/!96 2!7 8.17%61.7 9. /6(8. ! 26
KN
0.005
84
Exacta
Numerica
0.004
82
u [mm]
τ [Mpa]
0.003
0.002
80
78
0.001
0
9.5
10
76
9.5
10.5
10
10.5
r[cm]
r[cm]
!"
#"
$%&'(! )*+, -.//%01 !1'2!( /.((!3!
2!(&4 3. 2!5 261.!5 3. 5%7.8(6! 94(%:418!2 ; .1 2! <!(8. 5'<.(%4( 3.(./9! 3. 2! =&'(! ! 24 2!(&4 3.2
.>. ?.(8%/!2* @'.3. 4#5.(?!(5. A'.,
.1 2! 7.14( 3%(.//%01 3. 2! 5.//%01 2! 8.15%01 3. /4(8.
τxz
/(./.1 .1 B4(7! <(C/8%/!7.18.
2%1.!2 .18(. .2 /.18(4 ; .2 #4(3. /474 %13%/!(6! 2! 8.4(6! 3. D4'247#"
.1 2! 3%(.//%01 7!;4( 2!5 8.15%41.5 3. /4(8. τyz .5 <(C/8%/!7.18. 1'2! .1 843! 2! <!(8. /.18(!2
; 2'.&4 /(./. .1 B4(7! <!(!#02%/!
τxz ! 2! 7%8!3 3.2 2!34 7C5 2!(&4 .5
8.15%01 τyz ! 24 2!(&4 3.2 2!34 7C5 /4(84
2! 8.15%01 3. /4(8. 7CE%7!
7!;4( A'. 2! 7CE%7!
!<(4E%7!3!7.18. '1 FG H
2! 8.15%01 3. /4(8. 5. !1'2! .1 .2 /.18(4 ; 8!7#%I1 .1 2!5 .5A'%1!5
J2 747.184 84(54( 5. <'.3. /!2/'2!( /474 345 ?./.5 .2 ?42'7.1 #!>4 2! 7.7#(!1! 3.B4(7!3!
7'28%<2%/!34 <4( .2 B!/84(
α
Mt = 2 × V ol ×
σe
2Gθ = V ol × 32 × 1010
p
.1 .58. /!54 3.#. 8.1.(5. .1 /'.18! A'. 3.#%34 ! 2! 5%7.8(6! 5. 9! 8(!#!>!34 /41 '1 /'!(84 3. 2!
5.//%01K <4( 24 /'!2 9!; A'. %1/2'%( '1 B!/84( L 5% .2 /C2/'24 3.2 ?42'7.1 5. (.!2%:! 54#(. .2 /'!(84
3. 5.//%01 '8%2%:!34* J?!2'!134 1'7I(%/!7.18. .2 ?42'7.1 (.5'28!
M = 16823N m
/41 24 /'!2 2! (%&%3.: .A'%?!2.18. .5
GIt =
M
= 1, 6823M N m2
θ
It = 2103cm4
.58. M28%74 ?!24( 5. <'.3. /47<!(!( /41 .2 747.184 3. %1.(/%! <42!( 3. 2! 5.//%01
3542cm3 ;
1
/41 2! !<(4E%7!/%01 ! 5.//%41.5 3.2&!3!5 !#%.(8!5 3 hb3
OP
= 833cm4 *
J=
bh
12
(b2 + h2 ) =
N! <(%7.(! (.5'28!
2
2
1.5
1.5
Y
2.5
Y[cm]
2.5
1
1
0.5
0.5
00
10
20
τ [MPa]
30
0
40
0
2
4
0
2
4
X
6
8
10
6
8
10
40
τ [MPa]
30
20
10
0
X[cm]
!"#$% &'()* +,--!./ $,-0%/"#1%$
23$ 4#2#,403 564 %10% 7 1% 4,"#/8% 564 9%:% ;1% 4,--!./ /3 -%1!<-% -353 8, 2%$,8 8,1"%8%=' >%
$,1%-!./ ,/0$, ,1 ?%13$ 8, It -%1-#1%83 7 ,1 535,/03 231%$ 8, !/,$-!% ,4
It
= 0,594
J
ρ=
!"!#! $%&&'() *)+,*- *.'%-/*
@3/4!8,$,534 1% 5!45% ",35,0$A% 8, 1% 4,--!./ %/#1%$ -,$$%8% ?!40% %/0,4B 2,$3 %C3$% %9!,$0%B
,4 8,-!$ -3$0%8% % 13 1%$"3 8, #/ 5,$!8!%/3' >% ,D2$,4!./ E#, 4, #0!1!F% 2%$% 8,0,$5!/%$ 1%4
0,/4!3/,4 8, -3$0, ,/ ,1 -%43 8, 4,--!3/,4 %9!,$0%4 8, 2,E#,G3 ,42,43$ ,4 1% 4!"#!,/0,*
Mx = GJρ θ
Jρ =
ˆ
s
e3
ds = πDe3
3
τ = 2Gθy
83/8, ,/ 1% H10!5% ,D2$,4!./ y ,4 1% 8!40%/-!% 8,1 2#/03 83/8, 4, ,?%1H% 1% 0,/4!./ 8, -3$0, %1
2#/03 5,8!3 8,1 ,42,43$'
I/ 1% !"#$% &'(( 4, 5#,40$%/ 134 $,4#10%834 390,/!834' J3$ #/ 1%83 4, !/-1#7,/ 134 8,421%F%K
5!,/034 8, 1% 5,59$%/% ,/ 038% 1% 4,--!./' I/ ,40, -%43 0383 ,1 -3/03$/3 8, 1% 4,--!./ ;,D0,$!3$B
!/0,$!3$ 7 ,1 -3$0,= 0!,/, 8,421%F%5!,/03 /#13 2%$% 1% 5,59$%/%' >% 5,59$%/% ,/03/-,4 %1-%/F%
4# 56D!53 8,421%F%5!,/03 %2$3D!5%8%5,/0, ,/ 1% 1A/,% 5,8!% 8, 1% 2%$,8' I4034 8,421%F%5!,/034
43/ 2$6-0!-%5,/0, #/!L3$5, ,/ 0383 ,1 8,4%$$3113 8, 1% 4,--!./ 4%1?3 /%0#$%15,/0, ,/ 1% F3/% 8,1
-3$0,'
M 134 ,L,-034 8, ?!4#%1!F%$ 134 8,0%11,4 4, 5#,40$%/ %521!%8%4 1%4 1A/,%4 8, /!?,1 %43-!%8%4 %
1% F3/% 83/8, 1% 4,--!./ ,406 %9!,$0% ;8,$,-C%= 7 ,1 2#/03 32#,403 8, 1% 4,--!./ ;!FE#!,$8%=' >%
%4!5,0$A% E#, 2#,8, /30%$4, 4, 8,9, % 1% %2$3D!5%-!./ /#5N$!-% 7 % 1% L3$5% ,/ E#, ,1 "$%<-%83$
!/0,$2$,0% 134 $,4#10%834 2,$3 /3 ,4 #/% -#,40!./ LA4!-%'
!/%15,/0, ,/ ,1 -#%8$3 4#2,$!3$ 8,1 "$6<-3 4, 5#,40$%/ 1%4 0,/4!3/,4 8, -3$0, ,/ ,1 ,42,43$ 8,
1% 2%$,8 #0!1!F%/83 1% ,D2$,4!./ %2$3D!5%8% 7 1% 390,/!8% /#5N$!-%5,/0, ,/ 9%4, % 1% %/%13"A%'
J#,8, /30%$4, E#, 134 $,4#10%834 43/ 2$6-0!-%5,/0, -3!/-!8,/0,4'
!"!0! $%&&'() 1+,/'&%,+,*I/ 1% !"#$% &'(O 4, ?, #/% 4,--!./ -#%8$%8% -3/ 834 2%$,8,4 !/0,$/%4' >%4 2%$,8,4 ,D0,$/%4
43/ 8, ,42,43$ #/!L3$5,B ,/ 0%/03 E#, 1% !/0,$/% ?,$0!-%1 0!,/, ,1 8391, E#, 1%4 ,D0,$/%4 7 1% !/0,$/%
(&
8
6
Aproximada
Numerica
4
τ [MPa]
2
0
-2
-4
-6
-8
9.6
9.8
10
r [cm]
10.2
10.4
!"#$% &'(() *+,,!-. %.#/%$ %0!+$1%
23$!43.1%/ +/ 5!653 +67+63$ 8#+ /%6 +91+$.%6' :. /% 63/#,!-. .#5;$!,% <+ /% 5+50$%.% 13<3 +/
,3.13$.3 +61= !57+<!<3 <+ <+67/%4%$6+> +. 1%.13 8#+ ,%<% #.3 <+ /36 ,3.13$.36 !.1+$.36 1+.<$= #.
?%/3$ @.!,3 <+ <+67/%4%5!+.13 8#+ $+6#/1% <+/ %.=/!6!6'
A36 <+67/%4%5!+.136 <+ ,%<% #.3 <+ /36 ,3.13$.36 !.1+$.36 $+6#/1%. B+. 55C)
u1 = 4,96 × 10−3
u2 = 5,33 × 10−3
u3 = 5,17 × 10−3
/36 ,#%/+6 63. $+/%1!?%5+.1+ 6!5!/%$+6 ,3. #.% <!D+$+.,!% 5=9!5% 5+.3$ %/ E F' B+.1$+ /%6 ,+/<%6 (
G HC' I+0+ .31%$6+ 8#+ /%6 1+.6!3.+6 <+ ,3$1+ 63. 7$373$,!3.%/+6 %/ "$%<!+.1+ <+/ <+67/%4%5!+.13>
73$ /3 8#+ 6! <36 ,+/<%6 %<G%,+.1+6 6+ +/+?%. ?%/3$+6 6!5!/%$+6 +/ "$%<!+.1+ 7$35+<!3 +. /% 7%$+<
,35@. 6+$= 5#G 0%J3' :. /% K"#$% 6+ 2%. !.,/#!<3 /%6 ,#$?%6 <+ .!?+/ <+ /36 <+67/%4%5!+.136'
L/%$%5+.1+ +. /%6 7%$+<+6 !.1+$.%6 +61=. 5#G +67%,!%<%6 +. 1%.13 8#+ +. /%6 7%$+<+6 +91+$.%6
+61=. 5#G %7$+1%<%6 G +. D3$5% 7$=,1!,%5+.1+ #.!D3$5+6 6%/?3 +. /%6 +68#!.%6 <3.<+ %7%$+,+.
6!."#/%$!<%<+6'
M%$% +61% 6+,,!-. +. 7%$1!,#/%$ 73<$N% <+,!$6+ 8#+ /%6 7%$+<+6 !.1+$.%6 1!+.+. #.% ,3.1$!0#,!-.
<+67$+,!%0/+ % /% 13$6!-. G 8#+ 73<$N%. %.%/!4%$+ /% 7!+4% ,35% #. 1#03 ,#%<$%<3 6!. 7%$+<+6
!.1+$.%6 #6%.<3 /%6 +97$+6!3.+6 <+ O$+<1'
! ! "#$%& '( )&*+,-( ./%0+'( -& 1%&-2
A% 6+"#.<% D-$5#/% <+ O$+<1 6+ 7#+<+ 301+.+$ +. 0%6+ %/ !"#$" "% &' $%#('!)*$"+# &' ,*
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1
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1
89 =
2
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τ
τ γ dV
V
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4!$,--!./ 4,1 ,;=,;6$ e *
1
89 =
2
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V
1
τ2
dV =
G
2
ˆ ˛ ˆ
L
s
e
τ2
de ds dx
G
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(M
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1
01 =
2G
ˆ ˛
L
eτ 2 ds dx
s
%&&234$5$( . $#.%$ 4$) *&()!.(&) &( 6-("!'( &4 7-8. & ".%*& τ =
3-& & )$"$% 6-&%$ & 4$ !(*&:%$4 & !(*&:%$% &( x/
1
01 =
2G
ˆ ˛
L
s
e
q 2
e
q2
ds dx =
2G
ˆ ˛
L
s
˛
s
1
ds
e
%&&234$5$( . 4$ 3%!2&%$ 6'%2-4$ & =%& * > q = M/2Sm ?
M 2L
M Lθ =
2
4GSm
˛
s
1
ds
e
)& 3-& & &)3&8$% &4 @(:-4. &)3&"AB". & *.%)!'(
M
θ=
2
4GSm
˛
s
1
ds
e
9 &B(!% 4$ %!:! &5 &,-!C$4&(*& $ *.%)!'(/
2
4Sm
¸
GIt = G 1
ds
s e
2
4Sm
¸
It = 1
ds
s e
DE
9 )!&( . q ".()*$(*& )& 4.
1 q2L
1
ds dx =
e
2 G
0( ;$)& $4 3.)*-4$ . 2&("!.($ .+ !:-$4$( . <0 9 01 )& *!&(&
1
1 q2L
M Lθ =
2
2 G
q
e
˛
s
1
ds
e