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O  =5 INFORMACJA!Udostpniane materiaBy pomocnicze do nauki przedmiotu WytrzymaBo[ MateriaBw s przeznaczone w pierwszym rzdzie dla wykBadowcw. Dla wBa[ciwego ich wykorzystania konieczny jest komentarz osoby rozumiejcej tre[ci zawarte w prezentacjach. Dla studentw jest to tylko materiaB uzupeBniajcy do studiw w bezpo[rednim kontakcie z prowadzcymi, a tak|e uBatwiajcy zrozumienie tre[ci podrcznikw. Przedstawiana wersja jest pierwsz edycj wykBadw przeprowadzonych w roku ak. 2009/10 i wymaga mo|e poprawek i uzupeBnieD. Pobierajcy te materiaBy proszeni s o przesyBanie swoich uwag na adres e-mailowy autora: mc@limba.wil.pk.edu.pl. ~Z~>y  ,UGICIA ZGINANEJ BELKIN R{NICZKOWE RWNANIE UGITEJ OSI BELKI6Geometria ugicia osi belki 9bNapr|enia normalne i krzywizna osi przy zginaniu22LRwnanie r|niczkowe ugitej osi belki''MNZnakowanie w rwnaniu ugitej osi belki((NNZnakowanie w rwnaniu ugitej osi belki((PNZnakowanie w rwnaniu ugitej osi belki((RJCaBkowanie rwnania ugitej osi belki&&X JCaBkowanie rwnania ugitej osi belki&&Y JCaBkowanie rwnania ugitej osi belki&&Z JCaBkowanie rwnania ugitej osi belki&& ,UGICIA ZGINANEJ BELKIF METODA OBCI{EC FIKCYJNYCH (MOHRA)\Statyka a ugicia belek; podobieDstwa formalne//\Statyka a ugicia belek; podobieDstwa formalne//<Statyka belki fikcyjnej a przemieszczenia belki rzeczywistej==VDobr obci|enia i schematu belki fikcyjnej,, Dobr schematw belek fikcyjnych!! Dobr schematw belek fikcyjnych!!   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OdksztaBcenie liniowe 2l RV JpRV,$D 0r 3p S A +??"?c V + Bp 0R? & v$2. Z prawa Hooke a 2 l R  KpR ,$D 0r 8p S A .??"?c   . Cp 0RO 6  xH3. Napr|enie normalne przy zginaniu% 2% R C  Lp#  RC ,$D 0r 6p S A -??"?c C  - Fp 0XR   F4. Z porwnania 2 i 3: 2l Rz  Npz R,$D 0x :p c &A H??"?c z  H Gp 6R 2 f(5. Krzywizna prta k" 2H p 0޽h ? 33 ~ ___PPT10^ .^\@n+xwD2 ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ip%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Jp%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Kp%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Lp%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Np%(+    +l, (  l~ l s * * `}   ^ l 6^ l 6^ l 6##^ l 6((^ l 6^  l 6((^ l 6 #l0  bA C??"? a H C$@ 0  l hy *lhy,$D 0 l2  bA J??"?y J  'l 0/hJY 1 p@1. Zwizek krzywizny i momentu:! 2!l h +lh,$D 0 !l2 TA B?  ?3"`?a  B  (l 05hBY ) j:2. Wzr na krzywizn krzywej: 2l )l 0p9 hY Q ,$ 0 x3. Rwnanie r|niczkowe dla wyznaczenia ugi osi belki w(x)"= 288H l 0޽h ? 33  ___PPT10 .[R+DS ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**l%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+l%(D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#l%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)l%(+8+0+)l +)  @0HV(  ~  s * 6l H E H HE,$D 0@  Bg  /H8  ZB  s *Dj j_ `B  0ZDS BS  % 0 Ym g  Tw(x) 2 * 0<]   7x 2N  : ) a  ;  BCDE@F Ԕ!2CW^|y*T~ @     TB < c $DJ @ 0a% E rw>0, w >0D 2l G[  E[G ,$D 0}@ t  0G[mZB  s *DPP_ `B  0ZDG $ 0$h Tw(x) 2 ( 0Tlt  7x 2@  3G   BCDE@F Ԕ!2CW^|y*T~ @     TB 2 c $DJ A 0pb  rw>0, w <0D 2l  y  F y ,$D 0@ X !  - yIZB  s *D `B  0ZDX    & 00, w >0D 2l  yD G yD,$D 0@ y !  . J y ZB   s *D  q `B ! 0ZDy f f  ' 0T !  Tw(x) 2 ) 0 w  7x 2N  7 D& .^  8  BCDE@F Ԕ!2CW^|y*T~ @     TB 9 c $DJ C 0 $ jD rw>0, w <0D 2H  0޽h ? 33  ___PPT10b .[R+LD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(DP' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#blinds(vertical)*<3<*D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G%(+8+0+ +P%  ~vP-dP(  ~  s *8)   ^  6^  6##^  6((^  6  0&1,$  0 &Niezale|nie, znakowanie momentu M(x) wynika z przyjtej umowy (M jest dodatni gdy rozciga  spody )"d 2c A{l M~ dM~,$D 0`B 1 0DԔfB 2 6DjJT _   7# ;MAZB 3 s *DjJ_  ZB 5 s *DjJ_  ZB 6 s *DjJ  2 8 CENH)QQo `T `T `T `T-\A Z 6p~ 3M>0 2l  7  c 7 ,$D 0@    _  `B R 0DԔ \ \ fB S 6DjJ ; ; T _   T# f ZB U s *DjJ_  ZB V s *DjJ_  ZB W s *DjJ  2 X CENH)QQo `T `T `T `TX  [ 0 P 7  3M>0 2ul  =~ a= ~,$D 0`B B 0DԔ fB C 6DjJ T _   D# f=1ZB E s *DjJ_  ZB F s *DjJ_  ZB G s *DjJ  2 H CENH)QQo `T `T `T `TXL1 \ 0l  ~ 3M<0 2l ` 7  b 7 ,$D 0@   ^`  `B J 0DԔc c fB K 6DjJB B T _   L# ? ZB M s *DjJ_  ZB N s *DjJ_  ZB O s *DjJ  2 P CENH)QQo `T `T `T `T1  ] 0®P 7  3M<0 2H  0޽h ?O@8HPX 33z r ___PPT10R .[R+, ɡD ' = @B D ' = @BA?%,( < +O%,( < +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<*d%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(+8+0+ +,  `Qp(  ~  s *\Ѯ)   ^  6^  6##^  6((^  6  6Ԯf1,$  0 4Przyjmiemy umow, |e zachowujc przyjte znakowanie momentw, rwnanie r|niczkowe osi belki bdziemy zapisywali w postaci:"| 2{ /0  bAR ? ?"?Aa PH $D 0  2 6ۮkS,$  0 zJEZLI dodatni zwrot osi w bdzie zgodny ze zwrotem dodatniego momentu t.j. bdzie skierowany w stron spodw:bn 2 )F%!l    KW[ ,$D 0`B 5 0DԔ&  fB 6 6DjJ#  T _   7#    ZB 8 s *DjJ_  ZB 9 s *DjJ_  ZB : s *DjJ  `B ? 0D* * fB @ 6ZD   A 6   Tw(x) 2 B 6 z  7x 2 L < ?"6@`NNN?N .m,$   0 rPrzyjcie dodatniego zwrotu osi x nie ma wpBywu na znak w pow. rwnaniu. Nale|y jednak pamita, |e zmiana skrtno[ci ukBadu w,x powoduje zmian znaku pierwszej pochodnej funkcji w(x). V 2 [3,}36l G ic  O ,$D 0B M BDjJ?"0@NNN?NG G6  N <L ?"6@ NNN?Ni ic  VMy8 2  P B o?"6@ NNN?NS ,$D 0~ Q <Ѽ ?"6@ NNN?N 70v ,$  0 pW przeciwnym przypadku w rwnaniu nale|y przyj znak +.9 29H  0޽h ? 33___PPT10.[R+7_D' = @B D<' = @BA?%,( < +O%,( < +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<*/%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-o6Bwipe(up)*<3<*2D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-g6B fade*<3<*PD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =-s6Bwipe(left)*<3<*QD' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D'  =-o6Bwipe(up)*<3<*L++0+ ++0+2 ++0+L ++0+Q +L1  %F( 8(  ~  s *A   ^  6^  6##^  6((^  6D  06,$  0 rDla belki z jednym przedziaBem charakterystycznym wyznaczenie ugicia osi belki i obrotu przekroju poprzecznego prta (por. rys.), wymaga dwukrotnego scaBkowania rwnania r|niczkowego:" 2^  6  64 6i\,$  0 Jednokrotne scaBkowanie okre[la nachylenie stycznej do osi ugitej belki (a wic obrt przekroju poprzecznego)o 2o  c &A U?? "? 8 U$D  0  c &A X?? "?5  j8 X$D  0*  6 ^ 6E ,$  0 `Ponowne scaBkowanie daje w wyniku ugicia belki:1 21  6pͼ  > 2  80  bA _? ?"?<H _$D 0  l  Q @ W ,$D 0 52  bA \??"?N F  \ T  Bg  # s S ZB  s *Dj j_ `B  0ZDS BS    0@m g  Tw(x) 2 ! 6{  7x 2 # # BCDE@F Ԕ!2CW^|y*T~ @     d hZB $ s *D `B ) 0ZD 4 " QT _   *#  4 ? ( ZB + s *Do_  ZB , s *Do_  ZB - s *Do  `B . 0DԔ ZB / s *DA 8 A QZB 0 s *D(v ( 1  BCDEFAA!7My@  ! 0<  2  BCDEF!7My@  7 l x 3 c &A Y??"?;  Y`B >B 0D 0 ? BP   5w F < ?"6@ NNN?NV6,$   0 Dla wyznaczenia dwu staBych caBkowania konieczne jest okre[lenie dwu warunkw brzegowych.Z 2ZH  0޽h ? 33___PPT10.[R+D' = @B D' = @BA?%,( < +O%,( < +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<*8%(D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*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' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D'  =-s6Bwipe(left)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =-o6Bwipe(up)*<3<*F++0+ ++0+ ++0+ ++0+F +-    :b r (  ~  s *`)i   ^  6^  6^  6  ^  6  0@,1m,$ 0 |Dla wyznaczenia dwu staBych caBkowania: C i D trzeba ustali dwa warunki brzegowe wynikajce z podparcia belki w sposb zapewniajcy jej geometryczn niezmienno[ w pBaszczyznie rysunku, np:F 2*~^  6^  6'' l   `,$D 0z@ z/o -z/oZB ) s *Do //`R * 0</o`R + 0z7N z/o . 7ZB / s *Do //`R 0 0</o`R 1 0z7`B 3 0Do77fR 4 67;fR 5 6F5yTB : c $D5 TB ; c $DFTB < c $DK E 0le Aw=0" 2ZB G s *D ZB H s *D ZB L s *D ZB M s *D  N 0k4  Aw=0" 2 O 0o Aw=0" 2 P 0DtP & Aw=0" 2 Q 0wd  Aw=0" 2 T 00N Aw=0" 2ZB U s *D;; ZB V s *Dy y  \ 0 S 1A 2M l    b ,$D 0`B 6 0Do< l l `B 7 0Do { { `B 8 0Dow w lR 9 < {  = tA,Szeroki uko[ny w gr"   T H  @#  (  > nA,Szeroki uko[ny w gr" L  ? nA,Szeroki uko[ny w gr" H F T H  A# E   B nA,Szeroki uko[ny w gr" L  C nA,Szeroki uko[ny w gr" H F ZB DB s *D   F 6|!   hw=0 w =08 2`B I 0Dx x j `B J 0D ~ `B K 0DL L~  Z 6'Ft 4  hw=0 w =08 2 [ 6\,g  hw=0 w =08 2 ] 6$1   1B 2f _ 04 1V,$ 0 UWAGA: poniewa| nie uwzgldniamy wpBywu sil podBu|nych na ugicia, 3 przypadki w wierszu A i 3 przypadki w wierszu B s sobie rwnowa|ne (mimo, |e niektre belki s chwiejne lub statyczne niewyznaczalne!) 2H  0޽h ? 33Z R ___PPT102 .[R+%)D ' = @B DQ ' = @BA?%,( < +O%,( < +D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<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<*b%(D' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =-6B'blinds(horizontal)*<3<*_+p+0+ ++0+_ +~  < 4  * (  ~  s *V   ^  6^  6##^  6((^  6^  00&1,$  0 W przypadku gdy rwnanie momentu nie da si zapisa w sposb analityczny jednym rwnaniem dla caBej belki i musi by zapisywane w przedziaBach charakterystycznych, dla ka|dego z tych przedziaBw trzeba zapisa rwnanie r|niczkowe ugi (oznaczajc ugicia odpowiednim indeksem) i dokona dwukrotnego caBkowania w ka|dym przedziale. N 2N^  6//  ) < ?"6@ NNN?NN1< ,$  0 .W rezultacie otrzymujemy 2n staBych caBkowania (gdzie n oznacza liczb przedziaBw charakterystycznych) i trzeba uBo|y 2n-2 (2 warunki mamy z warunkw podparcia belki) dodatkowych warunkw  zszycia na brzegach ssiednich przedziaBw charakterystycznych. 2 * < ?"6@ NNN?N 1>,$  0 W wyniku takiego postpowania otrzymujemy ukBad 2n algebraicznych rwnaD liniowych dla wyznaczenia 2n staBych caBkowania. Procedura ta jest uci|liwa i warta zastosowania tylko wtedy, gdy chcemy mie rwnanie linii ugicia dla caBej belki (pozwala to na analityczne wyznaczenie maksymalnych ugi i miejsca ich wystpowania.E 2EH  0޽h ? 33  ___PPT10 .[R+% JD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-o6Bwipe(up)*<3<*)D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-o6Bwipe(up)*<3<**++0+ ++0+) ++0+* +o  U@R(  |l ^   ^,$D  0B q HD1?"0@NNN?N B nB HD1?"0@NNN?N B #B HD1?"0@NNN?N  B $B HD1?"0@NNN?N B k HD1?"0@NNN?N B m HD1?"0@NNN?N B sB HD1?"0@NNN?N^ ^B ' BD1?"0@NNN?NN5,$D  0~  s *O `   ^  6^  6^  6  0ku Ilustracja wykorzystania warunkw brzegowych i warunkw  zszycia B 2B^  6B  <D?"0@NNN?NRR l iV Yi,$D 0B   BD>?"0@NNN?Ni4p 4B  BD>?"0@NNN?N 44  H 1?"6@ NNN?Nq  _  N o?"6@ NNN?NI:  A*?Ciemny uko[ny w gr"6@`NNN?N~B  HD>?"0@NNN?NV0B  BDjJ?"0@NNN?NwX2  S  B[CENH_IfQ >? [fm[fmf`Tfmf`T"6@ NNN?ND1T  mS  # mS  B o?"6@ NNN?N mS B  BDo?"0@NNN?N L B  BDo?"0@NNN?N8 8K B  BDo?"0@NNN?N P B  BDo?"0@NNN?N O B  BDo?"0@NNN?N N B  BDo?"0@NNN?N M B  BDo?"0@NNN?Nm mR d  < l l2! }2l!,$D  0 * B˜ ?"6@ NNN?Nl2 1n 2 - BϜ ?"6@ NNN?N2 11 2 . BҜ ?"6@ NNN?N66 12 2 / BH֜ ?"6@ NNN?N-:! 13 2 1 BB ?"6@ NNN?N6  14 2 2 B`̜ ?"6@ NNN?N : ! 15 2 3 BhC ?"6@ NNN?N 2 16 2K 4 <N?"6@ NNN?N+Q%,$D 0 mw1=w2L 2d 5 <?"6@ NNN?N u ,$D 0 w 1=w 2\ 2K 6 <7?"6@ NNN?NYS,$D 0 mw2=w3L 2H 7 <h0?"6@ NNN?N C ,$D 0 j w 6=0D 2 K 9 <䛜?"6@ NNN?NY S,$D  0 mw4=w5L 2K : <?"6@ NNN?NY 3S,$D  0 mw5=w6L 26 ; <?"6@ NNN?NYS,$D  0 Xw6=08 2 d = <H ?"6@ NNN?N k ,$D  0 w 2=w 3\ 2d > <&?"6@ NNN?N  ,$D 0 w 5=w 6\ 2d ? <xJ?"6@ NNN?N ` ,$D 0 w 3=w 4\ 2K 8 <?"6@ NNN?NYS,$D 0 mw3=w4L 26 @ <d?"6@ NNN?N6 0 ,$D  0 Xw1=08 2 B B <D?"0@NNN?Nlysl,$D)  0 C  r0e0e    BCDE4F A@  o 8c8c     ?A)BCD|E|| Rs ,@ sZ @    # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abo ,$D1  0 D  0e0e    B}CDE@F  A@  o 8c8c     ?A)BCD|E||}^@   J  )3L=IVc#| @     # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab ,$D0  0 F  r0e0e    BCDE4F A@  o 8c8c     ?A)BCD|E|| 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6D?"0@NNN?N %Q ,$@  0B { 6D?"0@NNN?N  ,$D  0   x0e0e    BC/DE4F A@  o 8c8c     ?A)BCD|E|| C ]w#),/.. @    # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abd,$D,  02  B?"6@ NNN?Np ,$D/ 0R  B?"6@ NNN?Nk,$D+ 0  xA*?Ciemny uko[ny w gr"6@ NNN?NB,$D2 0H  0޽h ? 33___PPT10w.[R+Gg?D' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*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<*}DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(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<*8%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(DA' =%(D' =%(D' =A@BBBB0B%(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<*@%(DA' =%(D' =%(D' =A@BBBB0B%(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<*>%(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<*5%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D4' =%(D' =%(D' =4@BBBB%(D' 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=1:Bvisible*o3>+B#style.visibility<*C%(D' =-s6Bwipe(down)*<3<*CD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(Da' =%(D ' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*CD' =1:Bhidden*o3>+B#style.visibility<*C%(D)' =4@BB BB%(D' =-g6B fade*<3<*DD' =1:Bhidden*o3>+B#style.visibility<*D%(D)' =4@BB BB%(D' =-g6B fade*<3<*FD' =1:Bhidden*o3>+B#style.visibility<*F%(D)' =4@BB BB%(D' =-g6B fade*<3<*JD' =1:Bhidden*o3>+B#style.visibility<*J%(D)' =4@BB BB%(D' =-g6B fade*<3<*LD' =1:Bhidden*o3>+B#style.visibility<*L%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-u6Bwipe(right)*<3<*CD' =%(D9' =4@BBBB%(D' 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Wykresy momentw i siB poprzecznych od takiego obci|enia nie mog wic zawiera niecigBo[ci (nie ma obci|enia w postaci skupionych momentw czy siB). JP 2[U  <g" ?"6@ NNN?N;S `",$ 0 5[1/m] 2  0`k" ql ,$ 0 V&SpeBnienie warunkw 2  0i" *q ,$  0 polega na dobraniu podpr belki fikcyjnej tak, aby w charakterystycznych punktach byBy speBnione zwizkii 2it  0 u",$  0 "I tak, je[li w jakim[ punkcie belki rzeczywistej wR=0 , to na belce fikcyjnej musi by w tym punkcie MF=0. Podobnie je[li w R =0 to i QF =0 itd.  21 zv  <" ?"6@ NNN?NZ,$ 0 RPodstawowym warunkiem wykorzystania analogii Mohra jest aby zmienna x w belce fikcyjnej miaBa taki sam zakres wa|no[ci jak w belce rzeczywistej (0 d" xR d" l, 0 d" xF d" l). 2$LAaAI a  ! 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' H0 B r M@G3q R;r H / a ;":fp   5! {% ) 4. 2 6 : < A E J N 2S U Y P] +` d 9f k m r zt "v x e} 2 & ˅ n x ~  % ̐ X A m _ h |  w  <    G +  |q MiF9 [ p ?  ;Oh+'0^ `hx   Slajd 1Marcin ChrzanowskiMarcin Chrzanowski59cMicrosoft Office PowerPoint@ b@|9[@WG|]g  .  -- @ !--'@"Arial-. 2 NWM2_06/  ."System4-@"Arial-.  2 18 .---^$-----V8----\8----($----*$----8----8+----8$----8$----8----B8    ----*$,*)('&&% "'&&'()*+,----X836789:998642/-,+,,-/133234555443210001122----0$FFDCCCEDB@;=;<>>??ABDF----$IHCDI----&$jfbee_[ZZXSW]^^cj----$pojkp----Z$+~}|{zyxxxyz{|}~~~{ywvussstvxz}----8$----*$----$ ----p8'----<$----V8----&$----d$0---- $     ----8----($----B8 -5789::::987520.)--12345554111----V8BEGHIHHGECA><;:;;<>@BBABCDDDCCBA@???@@AA----$RNJMR----8ZYUUZZYVRTYY----8$dca```abcb`_]\[[[[\[[]]cbd----`8" sqpnljigfeeeffhijmpqrssssjjjklmnssooonmlkjjoo----N$%}|zzzzz{|}}~~~{ywuuuuvwxz{}----8$----<$----8---- $----t8)---- $----*$----p8'---- $     ----V8!!! ----&$92.--)%#''',01149----`$.FBA@??>??ACDEEECA><:98==?@AAA@@?<;:::;<=?ACDEF---- $XRVPOLLGKOMNRX----t8)YY[]_acddeffeedee``^\ZXXWXXY[\^`aaaa_^]YY`_^]]\\\]^_```--www--^%---www--0%(%--www--6%(%--www--(%--www--*%--www--%%--www--Z%+(%--www--8%--www--8%--www--%%--www--*%   % --www--*%,*)('&&% "'&&'()*+,--www--2%36789:998642/-,+,,-/133(%234555443210001122--www--0%FFDCCCEDB@;=;<>>??ABDF--www--%IHCDI--www--&%jfbee_[ZZXSW]^^cj--www--%pojkp--www--Z%+~}|{zyxxxyz{|}~~~{ywvussstvxz}--www--8%--www--*%--www--% --www--R%' %--www--<%--www--0%(%--www--&%--www--d%0--www-- %     --www--%%--www--(%--www--*%-5789::::987520.)--% 12345554111--www--0%BEGHIHHGECA><;:;;<>@BB(%ABCDDDCCBA@???@@AA--www--%RNJMR--www--%ZYUUZZ%YVRTYY--www--8%dca```abcb`_]\[[[[\[[]]cbd--www--H%"sqpnljigfeeeffhijmpqrssssjjjklmnss% ooonmlkjjoo--www--N%%}|zzzzz{|}}~~~{ywuuuuvwxz{}--www--8%--www--<%--www--%%--www-- %--www--V%) %--www-- %--www--*%--www--R%' %--www-- %     --www--0%!!! 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Lb@"Arial--.I2 Lb,rozumiejcej treci zawarte w prezentacjach.         .--.@"Arial-.  2 x<8 .@"Arial-. xb@"Arial--.X2 xb6Dla studentw jest to tylko materia uzupeniajcy do             .--.@"Arial-. b@"Arial--.T2 b3studiw w bezporednim kontakcie z prowadzcymi, a d           .--.@"Arial-. b@"Arial--.R2 b2take uatwiajcy zrozumienie treci podrcznikw.         .--.@"Arial-.  2 <8 .@"Arial-. b@"Arial--.T2 b3Przedstawiana wersja jest pierwsz edycj wykadw d          .--.@"Arial-. +2 bprzeprowadzonych w roku      .@"Arial-.  2 ak.@"Arial-. @"Arial--..2 . 2009/10 i wymaga moe      .--.@"Arial-. 4b@"Arial--.]2 4b9poprawek i uzupenie. Pobierajcy te materiay proszeni             .--.@"Arial-. Xb@"Arial--.@2 Xb&s o przesyanie swoich uwag na adres          .--.@"Arial-.  2 Xe8.@"Arial-.  2 X-8 .@"Arial-. 2 Xmailowy .@"Arial-. 2 XOautora:    .@"Arial-. *2 |bmc@limba.wil.pk.edu.pl."     .՜.+,0    Pokaz na ekranieiokak 8A  ArialSymbolTimes New RomanProjekt domylnyMicrosoft Equation 3.0 INFORMACJA!UGICIA ZGINANEJ BELKIGeometria ugicia osi belki2Naprenia normalne i krzywizna osi przy zginaniu'Rwnanie rniczkowe ugitej osi belki(Znakowanie w rwnaniu ugitej osi belki(Znakowanie w rwnaniu ugitej osi belki(Znakowanie w rwnaniu ugitej osi belki&Cakowanie rwnania ugitej osi belki&Cakowanie rwnania ugitej osi belki&Cakowanie rwnania ugitej osi belki&Cakowanie rwnania ugitej osi belkiUGICIA ZGINANEJ BELKI/Statyka a ugicia belek; podobiestwa formalne/Statyka a ugicia belek; podobiestwa formalne=Statyka belki fikcyjnej a przemieszczenia belki rzeczywistej,Dobr obcienia i schematu belki fikcyjnej!Dobr schematw belek fikcyjnych!Dobr schematw belek fikcyjnych Slajd 20 Slajd 21 Slajd 22_Zastosowanie metody Mohra dla wyznaczania ugi i ktw obrotu przekroju w wybranych punktach^Zastosowanie metody Mohra dla wyznaczania ugi i ktw obrotu przekroju w wybranych punktachHWzory dla wyznaczania powierzchni i rodkow cikoci obcie cigych Slajd 263Przykad analizy ugi belki o zmiennej sztywnoci Uywane czcionkiSzablon projektuOsadzone serwery OLETytuy slajdw*_ YMarcin ChrzanowskiMarcin Chrzanowski  !"#$%&'()*+,-./0123456789:;<>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./012345689:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefhijklmnpqrstuvRoot EntrydO)PictureswyCurrent UseroSummaryInformation(7^PowerPoint Document(= DocumentSummaryInformation8g