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zdrs/downrev.xmlPKykplB   P?Line 5"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK![a d$drs/shapexml.xmlUn0 }:tq.ڠnA}-ˉ] IΥ_?Jr,mQyxtsSluvfpL)2+J]4tfFuNWףNDZWྚk\U.;k)eٰ@hzf^lff(s.hPu*4'z$g@S֮ųA@@/|5_k :FdCؤ4%<vU qLU]N'z F| ހK ljwn@BJ~<l·StL8MsÐmNWa;#E.d2[2)%R^eᗪP>2 dRǦTgsB}pySCli>Te~/ vPC cN#x`B(DZ@1\('oS`a,He[ pe `hố2RkR-ח-bznfů]-dgt;%jaKWдh&xfQ*`J #O|KMI-<[M@ :Q4O-ы40‚?' 8.g`?u2E5M8EH͌&Ŋ5f-J܆ox5;uTx?kkvmfK*IAӤUaf_EƋf=֨.YAiZ;26p8E I{\KcCHG)9HHT?PK!GJtdrs/downrev.xmlD]K0C8w.]Eٺ1Z7؜aMR\^_>c !^AԟӷD!GHD/PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-![a d$*drs/shapexml.xmlPK-!GJt|drs/downrev.xmlPKypB   PJ?Line 6"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!yU#drs/shapexml.xmlUn1}Z%,e%(4B~ .l/|}H*evg:M3xh[Yu*T{w `ķ0>a7ܜG9Xl·BpL&ST7q]e3R]KK6 s/cQslHrz_CӃIRHRUyϩ J܅㰥*;?;= |RH.;Îs/tEK)IIɫW\uoاda:HS"tFa ffGFyl|vbh\Ů Ng4;,I_uƴh&3b0dPܑ'BMI-<[M@ :ǣhA/ 0+亜L֍oX֜l6["53>j .+Sy(Ys"!,RZSE)?UŽ zQ3ՋmӴr78$&b I$dO!UnPK!!:rdrs/downrev.xmlDJ1!b3+2mZDSEqw:9sdHBp1c qtǍק;!"k4Iw-ڍ\pH#rTR%az[IFjcFVZ8=CK*cӸS1|<P(u97 "M?>hl}~>|K D/PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!yU#*drs/shapexml.xmlPK-!!:r{drs/downrev.xmlPK{pB   Po?Line 7"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!S͠$drs/shapexml.xmlUn0 }:tq.ڠnA}-ˉ] IΥ_?Jr,mQ!yxtsSluvfpL)2+J]4tfFuNWףNDZWྚk\U.;k)eٰ@hzGͼ`Q洗]^RAaԩМ|#9"v- -vh3^^ ju"4#\IiJ7vU 瘪"bO9t F| ހK ljwn@A9Xl·CpL&Scs5d3R]KK6 s#07%yscA*}RH.aQ ffx|7RFy"]j6c>AW]#]jv\_բAvFDͲ~8r$8] Mf9*VCUZy[lhJjjJ lXHyBo^9 `lWu9 %-R߰9l7)"Ejf|69$\.V6hQ6\ExC 4˥uYܩå" _C_@k3kLmXr1MZf0'/0*~w/~Xfdiȴ/J_pHJܻZzD:JABPK!pdrs/downrev.xmlDN1&CsMi%BF F:LG ۅ˓o]' q@޴h8>,@ĄlLa,4{yclJ})e-9SgÔeh x㮓3abOϖi->݋>]jsn D1[aZǘ(hD/\PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!S͠$*drs/shapexml.xmlPK-!p|drs/downrev.xmlPKypB   P81?Line 8"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!7#drs/shapexml.xmlUn0 }:tq.ڠnA}-ˉ] IΥ_?Jr,mQyxtsSluvfpL)2+J]4tfFuNWףNDZWྚk\U.;k)eٰ@hzf^lff(s.hPu*4'Wz$g@S֮ųA@@/|5_k :FdCؤ4%<vU qLU]N'e#ao@ å6 u;7gc !PN%V? 6SC )B:m&Bs&afYDe@p:,Ғ Ȝ>{o˸rꦄHrz_CӃRHRUyϹ ÕL G0S%A> ).;dG9rB ]tDuJRRjcp)>yscA*}RH.c0F3 ߍz^H͘Olly}tx{Hts5;W/~j ;)Q[]E33*VCUZy[lhJjjJ lXHyBo^9 `lWu9 %-R߰9l7)"Ejf|69՘$\.V6hQ6\ExC 4˥uYܩå" _C_@k3kLmXrPI Ҙ& 3v7/0*~w7^0FT/v Jڑig/J_pHJܻZzD:JABPK!sPdrs/downrev.xmlDN1&CsLIG)$^ ܰ;L\tl0?-t>Vb(E\xpests*&dg2psO~*%#s4PZǢ&q;bJ&6I]Y6;z@˷բ߯˪]Ï1WT!گVzTr>n0& H PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!7#*drs/shapexml.xmlPK-!sP{drs/downrev.xmlPKzpB   PJ?Line 9"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!3$drs/shapexml.xmlUn1}Z%,4B~ .l/|}H*ev=99s|}Uluv3Jfz_dz+J]4tfuNWNDZWM5U*jrǵlQ 4Pz3g6Xi3D9ehPu*4'#i=3 ak€QZxC@@/|5_j :FdCؤxiJ7xo+N☪"ۜbOv9t/(' {J.%)op߹9 r*\:JpHi(FakWݎepͥhoG4-G<3b0TPܑ'BMI-<[=2H;[u<7ңhP[iaNpr]~&F e~7,kNky6- pHMN5& k u(OkUvs ~>|K D/PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!3$*drs/shapexml.xmlPK-!96|drs/downrev.xmlPK|>p?B   R$?Line 10"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!yU"drs/shapexml.xmlUn0 }:tq.ڠnA}-ˉ] IΥ_?Jr,mQyxtsSluvfpL)2+J]4tfFuNWףNDZWྚk\U.;k)eٰ@hzf^lff(s.1YBshuIހ]Gia{h3^^ jטu"6#^IiJ7x*☪"bS9 F| ހK ljwn@BJ,<l·StL8MsÐmN,vF2 0]cidNd\9uSsKP9/}š|eDAdc%=*<\a<݃)!Aȵ4rz*s;!Å)R`hn)qJ$n1pzYFH=YKf'\_<>:0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK!%RWT%drs/shapexml.xmlUn0 }:tq.͚nA}-ˉ] IΥ_?Jr,mQ!yxt}WluvfpL)*Ӌ+J]4fuNNDZ5Wྚk\Uծ:k)eٰ@hzG`Q洗](Ѡ0LhN]i]7 aQZaro m&k+XC͗C`M#4DOx&)q4`>0WV *csM9?B`0> (al $Syv|()DTHyn15=TWa;#Eήd 2[((fM -B* T5A!M*<0]i ޔ$[s=U?H_;!)Hv4!(d%jG4AP$%%P,s:z>%'o3}o7a,He[ pe<8L= FHYK'.[~y5%pfջů]-dgt{M,Ö/.Ex~A}EӢbģUPBqG<Zx2H <[u<7ңhQ[iaJpr]~&F e~7,kNky1- pHMN5 "Zl w^ vikwpH PSE)?$iLV;ُ7 ?J/~Xfdiȴ/J_pHJܻZzD:JQBPK! 4drs/downrev.xmlD]k0p†vFݘu۳ 6IIl._/nS;E\q Ddt;HDv w{ɍH#.e )cՒŸ=>X$C#u!NY"-ZRu+cLQ\w\kI+ȳ5cF_02(%6_PK-!Zf[Content_Types].xmlPK-!1_a /_rels/.relsPK-!%RWT%*drs/shapexml.xmlPK-! 4}drs/downrev.xmlPKz7p8B   RJ?Line 12"PK!Zf[Content_Types].xmlMO &2WR=cJ`F0iK`#̼vLw 9uSq:w`G ^i ½KI)c/ $oVjTMRc|}042ҥCƔM̏P~*ka/8^DkHbL8e i"K\XN\6rco4y@_;oPK!1_a _rels/.relsj0 ѽqCNo^K [ILcX&m߾0XFo;>0xM e`|X}đ I`߽N4aG2$RKIZ)4(M9`ctB{m:f@`3n|O,ܗr޾jxR0T ,0@}WBLǬ5vPK! 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I4 zB (T <D?"U I zB )T <D?"  zB *T <D?"X|  zB +T <D?" 2 zB ,T <D?"L ag zB -T <D?"  zB .T <D?"ED D O 0T <5N  ?" N I ,$  0 In a further analysis we shall adopt the assumption of replacing kinematics conditions by statics ones (reaction torque); the bar is considered as being in the equilibrium but free to be twisted (free torsion). z>0 d2JCCCCTCCC0JsB l G A TF  2TA G TF ,$@  02 T BTN C"?RR   P*> dG3 z T S BoCfDEXF*oEgmcd)RZ<Qf&[oS46*xy<aCE@       `"<` ?G YF `B T 0Do S  t2 T 6"`% ^ t2 T 6"`% ^ z2 T <1"`AK z `B T 0Dg I TS `B T 0Dg  ^  `B T 0Dg U  `B T 0Dg }  `B T 0Dg |  `B T 0Dg 1 <b `B T 0Dg 9* D `B T 0Dg  % `B T 0Dg F Q `B T 0Dg A  `B T 0Dg B  B !T s *D  X ,$D  0 3T S BoCfDEXF*jJEgmcd)RZ<Qf&[oS46*xy<aCE@       `"<` ? t ,$D  02  T S  N BԙCENGgnḢItEQ3> ԙw$3ԙw$3tE`Tw$3tE`T"`L +} ,$D  0 P*> dG 2 4T@ 3 N BԙCENGgnḢItEQ3Ԕ ԙw$3ԙw$3tE`Tw$3tE`T  ,$D  0 L&> dC  5T 0N  p ,$D 0 YM(>0 d2D3 e 6T <\N  ?"W&,$D   0 ]We will make also use of de Sain-Venant principle replacing distributed loading with a torque^ 2^C 7 7T <N  ?" 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G C rX B?#" `Kn [ ,$D$ 0 sX B?#" `\ o l ,$D* 0 vX HlGH@ ?"hf\,$DA 0 cDistortion function  G  B wX BDo?" l,$D  0 xX BAn ? ?pqfH $D 0 H X 0޽h ?XX#X$XIXJXMX mX vX 33~v___PPT10V.2p:X+FOD' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-g6B fade*<3<*XDs' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*NX%(D' =-s6Bwipe(down)*<3<*NXD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9X%(Du' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*OX%(D' =-u6Bwipe(right)*<3<*OXD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*PX%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-s6Bwipe(down)*<3<*XD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*dX%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-u6Bwipe(right)*<3<*XD' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*X%(D'  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' 8 $D 0 \ HA  ?o ?S"`?S, TH8  $D  0 \ HA  ?o ?S"`?P 9 E8  $D  0 \ 6o; 8 p,$D 04 \ 0u,$  0 0 j - distortion functionT>0 d2ACH \ 0޽h ? 3300___PPT100.2p:X+bىDd0' = @B D0' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-s6Bwipe(left)*<3<*\D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-o6Bdissolve*<3<*\D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =-s6Bwipe(left)*<3<* \D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =-s6Bwipe(left)*<3<* \D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-s6Bwipe(left)*<3<*\D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-s6Bwipe(left)*<3<*\D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-o6Bdissolve*<3<*\+8+0+\0 +B  0  #`(  ` ` HA ?Ԕ ?S"`?>H $D 0  ` HA  ?o ?S"`?EX  $D  0z 0 ` HA#  ?Ԕ ?S"`?5 zY H  $D 0 z H pp  ` 000,$D 0` ` 0H pp   ` 0L3 h  S0">0 d2@ z H pp   ` H0 H,$D 0`  ` 0H pp   ` 0 h  S0">0 d2@ z H pp   ` 8`8,$D 0` ` 0H pp  ` 0_ h  S0">0 d2@ z H pp  ` X0 X,$D  0` ` 0H pp  ` 0L h  S0">0 d2@ z H pp  ` HXH,$D  0` ` 0H pp  ` 0,{ h  S0">0 d2@ z H pp  ` @0@,$D  0` ` 0H pp  ` 0` h  S0">0 d2@  ` HA$ !?3o ?S"`?g  t 8 !$D  0z H pp  ` P0P ,$D  0` ` 0H pp  ` 0 h  S0">0 d2@  ` HA% "?o ?S"`?9 J 1 8 "$D   0 ` HA  ?o ?S"`?V8  $D  0 ` 6 ot z,$D  0 |8The governing equation of torsion boundary value problem(9>8 d29C  ` HA' $?3o ?S"`? t 8 $$D  0 !` 0 8 ,$ 0 Eor:&>0 d2C #` H GH] ?" 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V ZB | s *D3o8ZB | s *D3oN? |T  ^   |# 2C  !| # BTCDE(F3o TSPhKF?3'>  @   S"  C ^  "| # BC|DE(F3o ^}obM8`| @   S"  | |T  ^  #|#   $| # BTCDE(F3o TSPhKF?3'>  @   S"  C ^  %| # BC|DE(F3o ^}obM8`| @   S"  |  z oq &| hlC,$D 0ZB '| s *D0ZB (| s *DoqZB )| s *DZB *| s *DJ(Rlz    +|  ,$D  0ZB ,| s *D i ZB -| s *DZ/ 8 ZB .| s *D   ZB /| s *DH  ZB 0| s *D b  "z lN  1| N7,$@  0`B 2| 03D/. `B 3| 03DTpN `B 4| 03Dl`B 5| 03D Q/  6| 03P / ,$ 0 FD(>0 d2 7| 0P  -,$ 0 fd1F>0 d2 8| 0P ,$ 0 fd2F>0 d2B 9| 0Da `y ,$@ 0 :| 0P X 8>x,$ 0 Td6>0 d2 ;| 0 ,P ,$ 0 ft1F>0 d2, <| 0P F ,$ 0 FAssume: constant distribution of shear stress across of tube thickness6G>0 d2CACz r   =| r ,$D  0`B >| 0'Dr P2 `B ?| 0'D s `B @| 0'D L `B A| 0'Dq O `B B| 0'D+  6 z  C|# c ,$D 0ZB D| s *DԔ"J J ZB E| s *DԔ) z2 F| <=Ԕ"`&2 z2 G| <BԔ"`0 `B H| 0DjJ  fN %  I|  z2 J| B=jJ"`% 2 K| BBjJ"`C L| # BCDEFXԔ::QGW$ <c*!;CNeat )H=`bx ?` DPrvxdP)No),@                   S" 3 Krz    M| z I ,$D 0ZB N| s *Dd fB O| 6uD  P| 6<%  6   HD*>0 d2ZB Q| s *D . 6 z  R|# c ,$D  0ZB S| s *DԔ"J J ZB T| s *DԔ) z2 U| <=Ԕ"`&2 z2 V| <BԔ"`0 `B W| 0DjJ  fN %  X|  z2 Y| B=jJ"`% 2 Z| BBjJ"`C [| # BCDEFXԔ::QGW$ <c*!;CNeat )H=`bx ?` DPrvxdP)No),@                   S" 3 K \| 0%  ,$  0 ft2F>0 d2 ]| 0%%  ,$ 0 0Assume: prismatic tube of varying wall thickness61>0 d2C+C ^| 0#%   ,$  0 _From equilibrium condition:(>8 d2C _| <A[  ??  8  $D  0 `| 0C"? C,$D 0 a| 63op 4,$D 0 b| <A\ a?? [  8 a$D 0 c| BA] b?3Ԕ?\ 0 8 b$D  0 d| <x0%  ?#" ` "Closed thin-walled cross-sections"#( 2#GlB e| 6D?"KK l A a  h|A a ,$D   0ZB | s *D7 a ZB | s *D ZB | s *D^AU | ZB | s *D tB f| 6D?"K E tB g| 6D?" y 0H | 0޽h ? 33HH___PPT10H.dpnL+ehbDF'  = @B DAF' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*]|%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C|%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9|%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:|%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =-s6Bwipe(down)*<3<*|D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*M|%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h|%(D' =-s6Bwipe(down)*<3<*h|DL' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* |%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*R|%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8|%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*<|%(D' =%(D(' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1|%(D' =-s6Bwipe(down)*<3<*1|D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;|%(D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&|%(D' =-u6Bwipe(right)*<3<*&|D3' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\|%(D' =-s6Bwipe(down)*<3<*\|D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=|%(D' =-s6Bwipe(down)*<3<*=|D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+|%(D' =-s6Bwipe(down)*<3<*+|D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*^|%(D' =-s6Bwipe(left)*<3<*^|D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b|%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*_|%(D' =-s6Bwipe(left)*<3<*_|D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`|%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*a|%(D' =-o6Bdissolve*<3<*a|D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c|%(D' =-s6Bwipe(left)*<3<*c|++0+6|0 ++0+7|0 ++0+8|0 ++0+:|0 ++0+;|0 ++0+<|0 ++0+\|0 ++0+]|0 ++0+^|0 +U 0 %1{(  Fz g [   ,$D 0ZB  s *Do o FZB  s *Dg N[2  BZjJ"`-< ,$D 0  0p% C,$ 0 Vtd6>0 d26 z  # jm ,$D 0ZB  s *DԔ"J J ZB   s *DԔ) z2   <=Ԕ"`&2 z2   <BԔ"`0 `B   0DjJ  fN %     z2  B=jJ"`% 2  BBjJ"`C  # BCDEFXԔ::QGW$ <c*!;CNeat )H=`bx ?` DPrvxdP)No),@                   S" 3 KR  6j& ,$D 0B  0D3Ԕ ,$D   0  0ܛ% Z,$ 0 \ds$>0 d2@  0% cD ,$ 0 BdA$>0 d2@  0%  i6 ,$  0 ^r(s)$>0 d2@B   s *D  ,$@  0 ! 0$%  & ,$ 0 Es(>0 d2DB " s *D @- ,$D  0 # <A^ Z ??Y2p8 Z $D 0 $ <As d ??\ P 8 d $D 0 % BA` \ ?3 ?C3E \ %  & <Aa ] ??X { T] 8 ] $D 0 ' 0 _% X ,$  0 lS  area of the figure embedded within central curve sB7>0 d2@4CD ( BAb ^ ?Ԕ?H JM 8 ^ $D 0 ) <Ac _ ??  8 _ $D 0 * <Ad ` ??m 6 8 ` $D 0B ,@ BDԔ?"< ,$D 0 - 0%  ,$  0 Td6>0 d2B . s *D ,u ,$D   0 0 BA_  ?o? 8  $D 0 1 <p%  ?#" ` "Closed thin-walled cross-sections"#( 2#GH  0޽h ? 33I;A;___PPT10!;.dpnL+Dm9'  % = @B D(9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D)' =4@BB#BB%()?)?D}' =.E7 BBBBBWM 1.94444E-6 2.22222E-6 L -0.31719 0.2537 *3>*B ppt_xB ppt_y=@0BBAApBB(}"Bt><*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(DX' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-s6Bwipe(down)*<3<*"D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =%(DX' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bwipe(down)*<3<*.D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(down)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-o6Bwipe(up)*<3<*'D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(++0+0 ++0+0 ++0+0 ++0+0 ++0+!0 ++0+'0 ++0+-0 +5  0 1!)!+- (  z   X ,$D 0`  0`  0 `  0  6[22 ,$D  0  6P2 ,$D  0   63,$D  0   0f- JI,$ 0 bb1D>0 d2CKC   0l-  ,$ 0 bh1D>0 d2CKC   0q- p10 W,$  0 bb2D>0 d2CKC   0Pw- n9 8 U,$  0 bh2D>0 d2CKC  0|- /f e ,$ 0 bb3D>0 d2CKC  0l-  ,$ 0 bh3D>0 d2CKC  <At s ??plp8 s $D 0  <Af f ??OlN_8 f $D 0  <Ah  ??e8  $D 0  07 o ,$D"  0  0- Zp,$ 0 q/Solutions for torsion of rectangular bars obey:&0>0 d20C  <Au u ??"`  8 u $D 0  <Aw { ??"` ! ? 8 { $D 0  HAe  ?3o ?S"`?[  8  $D  0  HAi  ?1 ?S"`?[ E 8  $D   0  0T-  r ,$! 0 XA2&>0 d2C   <Aj l ??b  j 8 l $D# 0  07 0 ,$D$  0  BAk m ? ?"` J H m $D% 0 -   0-  s~,$& 0 XA3&>0 d2C   BAl n ? ?9 T N8 n $D( 0  0 V-,$D'  0   0 W >,$D)  0 ! BAm o ? ?A V8 o $D* 02 " 63o ,$@+  02 # 63oq ,$D,  0 % <ܛ-  ?" ,$  0 H Assumptions: ( 2 C & <-  ?"j,$  0 WCross-section partitioning: ( 2C ' 0- ,$D 0 XA1&>0 d2C  ( 0- dK,$ 0 DA2&>0 d2C ) 0- ,$D 0 DA3&>0 d2C * 0W!,$D  0 + 0- Zp,$ 0 DA1&>0 d2C<2 ,  l$2BYENG%xI2Q ? ҰY)Y`TҰY)Y`T2`T)Y`T2`T"> H,$D- 0 - <X-  ?#" `  ]Open thin-walled cross-sections" ( 2 GH  0޽h ?, 33kk___PPT10k.#6n+<%D0g'  = @B Df' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-o6Bwipe(up)*<3<*&D ' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D> ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bwipe(left)*<3<*%D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-s6Bwipe(left)*<3<**DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-s6Bwipe(down)*<3<*"D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bwipe(down)*<3<*#D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-u6Bwipe(right)*<3<*,+(+0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+%0 ++0+&0 ++0+(0 ++0++0 +?5  0 <m(    <Aw  ??"` ! ?     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