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A#0$B%, 9E,!2%,K ∂σx ∂τyx ∂τzx + + =0 ∂x ∂y ∂z ∂τxy ∂σy ∂τzy + + =0 ∂x ∂y ∂z ∂τzx ∂τzy ∂σz + + =0 ∂x ∂y ∂z S H&'UK !" #$%&'#'$%(" &( )$*&( "$)*( ! "+,(*-#'( !.(*! &( ! ,'(/! τxy νy + τxz νz = 0 σy νy + τyz νz = 0 τyz νy + σz νz = 0 5 !" #$%&'#'$%(" &( #$%.$*%$ (% $" (6.*(7$" x=0 5 01234 x = L8 σx = 0 01294 τxy : τxz =!" #$%".!%.(" 0 νy : νz , "$% (;+'<! (%.(" ! .$*"$* $" #$"(%$" &'*(#.$*(" &( Mx ! %$*7! "! '(%.( ! ! "+,(*-#'( !.(*! νx = 042 > ,!*.'* &( 012?4 "( $).'(%(% !" .(%"'$%(" +"!%&$ τxy = θG ! ū (y, z) "!.'"D!#( (5 &( @$$A( ∂ ū ∂ ū −z : τxz = θG +y ∂y ∂z τxy = σx = σy = σz = 0 =! "+".'.+#'B% &( (".$" <! $*(" (% "( "!.'"D!#(% "' ! ! (#+!#'B% 012C4 7+(".*!% ;+( !" (#+!#'$%(" &( (;+' ')*'$ ! (#+!#'B% ∂ 2 ū ∂ 2 ū + =0 ∂y 2 ∂z 2 (% .$&$" $" ,+%.$" &( 01214 ! "(##'B% .*!%"<(*"! &( 012E4 #' '%&*$2 F!*! "!.'"D!#(* !" #$%&'#'$%(" &( )$*&( ∂ ū ∂ ū − z cos (ν, y) + + y cos (ν, z) = 0 ∂y ∂z 012G4 01234: &()( .(%(*"( (% C: &$%&( C (" ( #$%.$*%$ &( ! "(##'B% .*!%"<(*"! R 0<(* H'I+*! 12J42 F(*$ ∂ ū ∂ ū ∂ ū cos (ν, y) + cos (ν, z) ≡ ∂y ∂z ∂ν (% #$%"(#+(%#'!: ! #$%&'#'B% &( #$%.$*%$ 012G4 ,+(&( ("#*')'*"( #$7$ ∂ ū = z cos (ν, y) − y cos (ν, z) ∂ν (% 012KL4 012KK4 C2 =!" #$%&'#'$%(" &( #$%.$*%$ 01294 "( "!.'"D!#(% "'8 ˆ ˆ ˆ ˆ τxy dy dz = 0: R τxz dy dz = 0 R ! "#$%&'!()*+ "# (%&,# γ -# "#.+# #+ /%- (0&-%- "# #/!-,)()"!" 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"- A!+*)+!-@/!0$ ,! τxy < τxz ," !+"-.!E0$- 0)- '$, 0)/ )-"-.", +" ("')0!+$+% Q2-72" ψ 2"+" ,"* $*4!.*$*!$ "- ') 0)-0"*-!"-." $' "72!'!4*!)6 "' ,!,."/$ +" ."-,!)-", 89%=P; +"4" ,$.!,K$0"* '$, 0)-+!0!)-", +" 0)-.)*-) 89%R; < 89%S;6 < 0)-+!0!)-", +" 0)/ $.!4!'!+$+% &- $2,"-0!$ +" K2"*#$, /@,!0$, '$, 0)-+!0!)-", +" 0)/ $.!4!'!+$+ *"72!"*"- 72" T ∇2 τxy = 0 ∇2 τxz = 0 !" #$%&'#'$%(" &( #$)*!+','-'&!& ./(0 #1!-21'(0 +(3+$ &( (-!"+'#'&!&4 0("1-+!% &( (3'5'0 '51!-&!& &( &(0'/!&!" S ! " #$% ∂ 2 τxy ∂ 2 τxy + =0 ∂y 2 ∂z 2 ∂ 2 τxz ∂ 2 τxz + =0 ∂y 2 ∂z 2 & #'&! #( &#$) ) *)%+$% " ,-./01 ∇ τxz = ∇2 τxy = 2 ∂ 2 τxz ∂ 2 τxz + ∂ 2y ∂z 2 ∂ 2 ∂ψ ∂ ∂ 2 ∂ψ − 2 =− =− 2 ∂y ∂y ∂z ∂y ∂y ∂ 2ψ ∂ 2ψ + 2 ∂y 2 ∂z =− ∂ 2 ∇ ψ=0 ∂y ,-.231 2 2 ∂ τxy ∂ τxy + ∂ 2y ∂z 2 2 = 2 ∂ ∂ψ ∂ ∂ ∂ψ + 2 = 2 ∂y ∂z ∂z ∂z ∂z #'& 4' #()4 2 2 ∂ ψ ∂ ψ + 2 ∂y 2 ∂z = ∂ 2 ∇ ψ=0 ∂z ∇2 ψ = #+ . ,-.2/1 5 4)! #'&"$#$'& ! 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'+1+*$E " -%'&+9)/ ) !" <$'&+9) &+,4!+58"*" $&-" &$ $&8'+#$ 8%,% GIt / 0" -"#!" *$ 4$'5!$& 4'%<$$ !%& <"!%'$& 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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o 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 $+6#/1%. B+. 55C) u1 = 4,96 × 10−3 u2 = 5,33 × 10−3 u3 = 5,17 × 10−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φ = M Lθ 2 2 ,/ 2%/26 H#, 1% ,/,$"3% 4, 4,56$0%-!./ %10%-,/%4% ;, ,;-$!<, ,/ 5#/-!./ 4, 1% 2,/;!./ 4, -6$2, 71% K/!-% -60=6/,/2, 4, 2,/;!./ /6 /#1%: ? 1% 4!;26$;!./ %;6-!%4% γ 1 89 = 2 ˆ τ τ γ dV V $,,0=1%D%/46 1% $,1%-!./ -6/;2!2#2!A% γ = Gτ ,/ L;2% K12!0% ? 4,;-60=6/!,/46 1% !/2,"$%1 4, A61#0,/ ,/ 1% !/2,"$%1 % 16 1%$"6 4, 1% A!"% xF 1% !/2,"$%1 ,/ 1% 13/,% 0,4!% 4, 1% ;,--!./ s ? ,/ 1% 4!$,--!./ 4,1 ,;=,;6$ e * 1 89 = 2 ˆ V 1 τ2 dV = G 2 ˆ ˛ ˆ L s e τ2 de ds dx G ;!,/46 1% A!"% 4, 0%2,$!%1 E606"L/,6 ,1 0.4#16 4, -6$2, G ;, =#,4, ;%-%$ 5#,$% 4, 264%; 1% !/2,"$%1,;F #;%/46 1% E!=.2,;!; 4, H#, 1%; 2,/;!6/,; τ ;6/ -6/;2%/2,; ,/ ,1 ,;=,;6$F 1% !/2,"$%1 ,/ (M !"#$ !%&""!'( &) !%&"*$+ ,-& $( ./ 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