!"#$%&' ( ) *+ ,-.)/ 0,1)2). ,+3)/ 4&5%&' 67 678"&!9!:;7<$'8 !" #$!%&%'( #)(!$* +"," -./01 233-4. ! "#$" %&'($)*+ #" #,#$"-&$,.&! *&# "%)&%,+!"# /)" 0+1,"2!&! "* %+-'+2$&-,"!$+ 3" 4,0&#5 ! 0"!"2&* #" 2"%)22" &* 3"!+-,!&3+ -6$+3+ 3" "/),*,12,+ + -6$+3+ 3" *+# 3"#'*&.&-,"!$+#7 /)" %+!#,#$" "! "8'2"#&2 *&# "%)&%,+!"# 3,9"2"!%,&*"# 3" "/),*,12,+ "! 9)!%,:! 3" *+# 3"#'*&.&-,"!$+#5 ;!,%,&*-"!$" #" "#$)3,& "* %+-'+2$&-,"!$+ 92"!$" & %&20&# &8,&*"#7 *)"0+ #" "#$)3,& "* '2+1*"-& 3" <"8,:! = >!&*-"!$" "* 3" $+2#,:!5 ?&# "%)&%,+!"# 3,9"2"!%,&*"# +23,!&2,&# 2"#)*$&!$"#7 %+! #)# %+22"#'+!3,"!$"# %+!3,%,+!"# 3" %+!$+2!+7 ')"3"! ,!$"02&2#" = +1$"!"2 *+# 3"#'*&.&-,"!$+# = 0,2+# 3" )! "*"-"!$+ 3" 4,0& &,#*&3+5 @" -)"#$2&! &*0)!+# "A"-'*+# #,-'*"# 3" %+-+ 2"&*,.&2 3,%B& ,!$"02&%,:!5 +"5" 6-789 91:;/- 89 8 ;9#2;0<19 8=-8>;9 !"!#! $%&'%()* +(,-.-*%('/ C"%+23"-+# *& B,':$"#,# 3" D"2!+)**(E F !"#$%& '# &()"*#+,-$ & !$# .,&/# "&+%# 0)*&%, # # &0(!&"/) #1,' '#0 0&++,)$&0 %"#$02&"0#'&0 .&"*#$&+&$ .'#$#0 3 .#"#'&'#0 # 0, *,0*# G7 *+ %)&* %+!3)%" & /)" $+3+# *+# ')!$+# 3" *& #"%%,:! #+-"$,3& & )! "#9)"2.+ &8,&* "! #) 1&2,%6!$2+ -"%H!,%+ #" 3"9+2-&! )!& -,#-& -&0!,$)3 εx 5 #$& 3"9+2-&%,:! εx ')"3" "#%2,1,2#" "! 9)!%,:! 3" *+# 3"#'*&.&-,"!$+ &8,&*"# u %+-+ du εx = IJ5KL dx M!& "8'2"#,:! 3,9"2"!%,&* /)" 2"*&%,+!& )!& -"3,3& 3" 3"9+2-&%,:! I εx L %+! %+-'+!"!$"# 3" 3"#N '*&.&-,"!$+ IuL #" 3"!+-,!& )!& !"#$%&' () !$*#$%&'+ $%'!,-.%$# 5 ?& "8'2"#,:! 3" *& 3"9+2-&%,:! "#'"%(>%& εx (x) IJ5KL 2"#)*$& 3" %+-'&2&2 I4"2 O,0)2& J5P5KL *& *+!0,$)3 3"* "*"-"!$+ 3,9"2"!%,&* du &!$"# (ds = dx) = 3"#')6# /)" #" 3"#'*&%" ds∗ = dx + dx dx du (x) ds − ds = ε (x) = ds dx ∗ PK IJ5PL x x+dx u+du * ds ds u du x+dx+u+dx dx x+u !"#$% &'() *+,-$.%/!01 2+ #1 +3+.+14- 2!,+$+1/!%3 2+ 5%$$% 6% 4+17!01 +1 /%2% 8#14- 2+ 3% 7+//!01 7+ -54!+1+ % 8%$4!$ 2+ 3% 3+9 2+ :--;+ <&'=> ?1% +@8$+7!01 A#+ $+3%/!-1% #1% .+2!2% 2+ 4+17!01 < σx > /-1 #1% .+2!2% 2+ 2+,-$.%/!01 < εx > 7+ 2+1-.!1% #1% !"#$%&' $(')*%*+*%,# <2+B1+ +3 /-.8-$4%.!+14- .+/C1!/- 2+3 .%4+$!%3 $(')*%*+*%,( >' D! 3% 7+//!01 +7 E-.-"F1+% 7+$C 3% .!7.% 4+17!01 8%$% 4-2-7 3-7 8#14-7 2+ 3% 7+//!01' G+/-$2+.-7 A#+ +3 +7,#+$H- %@!%3 N 7+ 2+B1+ /-.- 3% !14+"$%3 2+ 3%7 4+17!-1+7 %@!%3+7 7-5$+ 3% 7+//!01) σ x = E εx N= = ˆ ˆA σx dA <&'I> Eεx dA <&'J> A 7! 3% 7+//!01 +7 E-.-"F1+% N = σx A = EA εx 7! 3% 7+//!01 1- +7 E-.-"F1+% 7+ 2+B1+ +3 L%3-$ EA = ˆ E dA A <&'K> <&'M> <&'&> N-17!2+$+.-7 #1% 5%$$% 2+ 7+//!01 4$%17L+$7%3 A </-174%14+ - 2+ L%$!%/!01 7#%L+O L+$ !"#$% &'P'(> 7-.+4!2% % #1% /%$"% 2!74$!5#!2% p (x) +1 3% 2!$+//!01 2+3 +Q+ 2+ 3% 5%$$%' D+ E% 7#8#+74A#+ 3% L%$!%/!01 2+ 3% 7+//!01 +7 7#B/!+14+.+14+ 7#%L+ 2+ 4%3 ,-$.% A#+ +7 %/+84%53+ 3% E!804+7!7 2+ R+$1-#33! 2+ A#+ 3% 2+,-$.%/!01 εx +7 #1!,-$.+ +1 /%2% 7+//!01' S3 +3+.+14- 2!,+$+1/!%3 2+ 5%$$% <#1% $+5%1%2%> 7+ 2+B1+ /-.- +3 3!.!4%2- 8-$ 2-7 7+//!-1+7 7+8%$%2%7 #1 2!,+$+1/!%3 dx' S3 +A#!3!5$!- 2+ +74+ +3+.+14- 2!,+$+1/!%3 $+7#34% 2+ 7#.%$ +7,#+$H-7 !14+$1-7 9 ,#+$H%7 +@4+$1%7 %/4#%12- 7-5$+ +3 .!7.dN (x) + p (x) = 0 <&'T> dx G++.83%H%12- <&'K> 9 <&'P> +1 <&'T> $+7#34% N = EA εx du d EA (x) dx dx + p (x) = 0 <&'(U> D! +3 C$+% 2+ 3% 7+//!01 +7 /-174%14+ 3% +/#%/!01 %14+$!-$ 7+ 7!.83!B/% %) EA V#+ +7 #1% +/#%/!01 2!,+$+1/!%3) d2 u + p (x) = 0 dx2 PP <&'((> !"#$%!#% "& '&()$" *#$&%* !" #$%&'(% )! $%* +%'&* &,,-)!%*)* !"&$ x. !/ 012'0, ,-)!% )! )!-'3*&'(% 4$! *5*-!&! !" 6. %, 7*8 5-,)$&9," !%9-! /*" 3*-'*:/!" , !%9-! /*" 3*-'*:/!" 8 "$" )!-'3*)*" % + &,+#&$-&' + $'-%$-&' /," &,!;&'!%9!" 4$! 0$/9'5/'&*% * /* '%&(<%'9* 8 "$" )!-'3*)*" %, )!5!%)!% )! /* &,,-)!%*)* x= >*-* -!",/3!- !"9* !&$*&'(% )!:! &,%,&!-"!. *)!01" )! /* &*-<* !29!-%* + $"#+# $&' "& + $- !$ . !"& = p (x). &$*/!" ",% /*" ?* &*%9')*) )! &,%)'&',%!" )! &,%9,-%, 4$! 5$!)!% 8 )!:!% ;@*-"! !" 6 A!/ ,-)!% )! /* !&$*&'(%B 8 !% <!%!-*/ $%* !% &*)* !29-!0, )! /* :*--*= C"9*" 5$!)!% "!- )! )!"5/*D*0'!%9, A;@*- !/ 3*/,- )! uB , )! #$!-D* A;@*- !/ 3*/,- )! N , !4$'3*/!%9!0!%9! !/ )! εB= !"!"! #$%&'()*+ ,+%+-.-,/%+ E$*%), !/ 5-,:/!0* !" '","919'&,. !"9, !" &$*%), !" "$;&'!%9! &,% /*" &,%)'&',%!" )! !4$'/':-', 5*-* )!9!-0'%*- /," !"#$!-D,". 5$!)! -!"$/9*- 01" "!%&'//, 5-'0!-, ,:9!%!- /," !"#$!-D," !"9," /*" )!#,-0*&',%!" ε (x) $"*%), /* /!8 )! F,,G! 8 /$!<, /," )!"5/*D*0'!%9," u N (x). &,% '%9!<-*%), /* !&$*&'(% &'%!019'&*= C" )!&'- !" )! /*" )," &,%)'&',%!" )! &,%9,-%, )!:! H= * 5*-9'- )! /* !&$*&'(% )! !4$'/':-', I=J ),%)! "!- )! #$!-D* A"$5,%<*0," !% !/ !29-!0, K'%*/ x = LB. N (x) = N (L) + ˆ "! ,:9'!%! N (x) &,0, L p (x) dx x 6= E,% /," !"#$!-D," "! ,:9'!%!% /*" )!#,-0*&',%!" $"*%), /* /!8 )! F,,G! I=L ε (x) = N (x) EA M= N%9!<-*0," /* !&$*&'(% &'%!019'&* I=6 $9'/'D*%), /* "!<$%)* &,%)'&'(% )! &,%9,-%, A!% !/ !29-!0, ,5$!"9, * /* )! #$!-D*"B 4$! )!:! "!- )! )!"5/*D*0'!%9," A!% !"9! &*", "$5$!"9, !% !/ !29-!0, !% x = 0B u (x) = u(0) + ˆ x ε (x) dx 0 N+ N p(x) dN dx dx X dx K'<$-* I=6 C4$'/':-', )! $% !/!0!%9, )'#!-!%&'*/ )! :*--* 6M !"!#! $%&'()*+(,) -. /%01+(%)./ ! "#$!% &%'($ )$ *+, &'+"*$-%, )$ .!/$!.$'0% '$,#*(% %1$&(%"*$ *% 2.&3($,., )$ *.!$%*.)%) #(.4 *.5%)% $! $,(% &%'($ )$* 1#',+6 ! (%* 1%,+ $, &+,."*$ ,#-%' *%, ,+*#1.+!$, )$ #!% -.,-% $,('#1(#'% 1+! ).,(.!(%, 1%'/%, 78+ ).9$'$!($, 1+!).1.+!$, )$ 1+!(+'!+ &%'% +"($!$' #!% !#$:% ,+*#1.3!6 , )$1.' ;#$ ,. )%)% #!% "%''% )$<!.)% &+' ,# /$+-$('0% =*+!/.(#) 7 ,$11.3!> 7 -%($'.%*? ,$ 1+!+1$! )+, ,+*#1.+!$, u1 (x) 7 u2 (x) &%'% $,(%)+, )$ 1%'/% p1 (x) 7 p2 (x) 7 1+!).1.+!$, )$ 1+!(+'!+ cc1 7 cc2 '$,&$1(.:%-$!($ d2 u + p1 (x) = 0 dx2 d2 u EA 2 + p2 (x) = 0 dx EA + cc1 =⇒ u1 (x) + cc2 =⇒ u2 (x) $!(+!1$, u (x) = α u1 (x) + β u2 (x) $, ,+*#1.3! )$ EA d2 u + [α p1 (x) + β p2 (x)] = 0 dx2 + [α cc1 + β cc2 ] )+!)$ α 7 β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x = L *% ,+*#1.3! @ (.$!$ #!% 1+!).1.3! )$ 9#$'5% N1 (L) = P 7 *% ,+*#1.3! B (.$!$ #!% 1+!).1.3! )$ )$,&*%5%-.$!(+ u (L) = ū? *% 1+!).1.3! '$,#*(%!($ )$"$ .!($'&'$(%',$ 1+-+ u (L) = α u1 (L) + β u2 (L) + N (L) = α N1 (L) + β N2 (L)6 !"!2! 34.&50%/ +,-,.,/, 0#11# 23# %* #"4' %561%"' 7 '"%6)$# # 8% ' 81'8)' J$%-+, #! &'.-$' $I$-&*+ )$ *% ,+*#1.3! )$ *% $1#%1.3! =K6@@>6 L%)% #!% 1+*#-!% 1.*0!)'.1% .-&$).)% )$ )$,&*%5%',$ $! %-"+, $G('$-+, 7 "%I+ *% %11.3! )$* &$,+ &'+&.+ =:$' M./#'% K6B6N6@>? .!($'$,% )$($'-.!%' *% ).,('."#1.3! )$ ($!,.+!$, $! *% %*(#'%6 * $I$ x 2% ,.)+ +'.$!(%)+ )$ %"%I+ % %''."% 7 ,# +'./$! $,(O $! $* $G('$-+ .!9$'.+'? *% 1%'/% &+' #!.)%) )$ *+!/.(#) $, p (x) = −γA )+!)$ γ = ρg $, $* &$,+ $,&$10<1+ )$* -%($'.%* 1+!,(.(#(.:+6 P+(%' ;#$ $! $,($ &'+"*$-% A $, 1+!,(%!($ *#$/+ *% $1#%1.3! ).9$'$!1.%* '$,#*(% EA d2 u = Aγ dx2 BN + γΑ x − u ε σ N !"#$% &'() *+,#-.% /%0+ ,% %11!2. 34 546+ 5$+5!+ 7 ,% !.84"$%1!2. 34 ,% -!6-% $46#,8% 64.1!,,%-4.84 γ d2 u = dx2 ˆE x γ x γ du (x) γ = dx = x + C = x + C = ε (x) dx E 0 E 0 E γ 2 u (x) = x + Cx + D 2E 9&':;< =% 3484$-!.%1!2. 34 ,%6 1+.68%.846 34 !.84"$%1!2. 9 C 7 D < 64 ,+"$% !-5+.!4.3+ ,%6 1+.3!1!+.46 34 1+.8+$.+> 4. .#468$+ 1%6+ 6! ,+6 4?8$4-+6 34 ,% 1+,#-.% .+ 5#434. 3465,%@%$64 $46#,8% u(x=0) = D = 0 γ 2 u(x=L) = L + CL + D = 0 2E 34 ,% 5$!-4$% D = 0 > ,,4A%.3+ % ,% 64"#.3% C = − γ L 2E 7 468+6 A%,+$46 % 9&':;< 64 8!4.4 γ x (x − L) 2E du L γ εx (x) = x− = dx E 2 L N (x) = EAεx (x) = γA x − 2 u (x) = ! "#$%&'!( )% *(!#$%! %#$'% %( *'"+%' %,$'%+- . /# 0!(-' &%#1'"2- 3% x ˆ 0 x d2 u dx = dx2 4/% )% '%5%)2'"6'% 3% (! +")+! 7-'+! ˆ 0 x d dx du dx x ˆ x qx (x) du du du = (x) − (0) = − dx dx = dx 0 dx dx EA 0 du (x) = − dx ˆ 0 x ˆ 0 x du qx (x) dx + (0) EA dx ˆ x du du qx (x) (x) dx = u (x) − u (0) = − dx + (0) x dx EA dx 0 ˆ xˆ x du qx (x) dxdx + (0) x + −u (0) u (x) = − EA dx 0 0 . 8!6"$/!(+%#$% )% '%%+*(!9! %( 0!(-' 3% du dx (0) . u (0) *-' 2-#)$!#$%) C . D ;B !"#$ %&"!&'%() *+% %, -%(.,#/#01%&"! u (x) 2#$3# %& 4!$0# '+#-$5"1'#) 2#,% 6 %& ,!( %7"$%0!( 8 %( 05710! # ,# 01"#- -% ,# '!,+0&# 9(1%0.$% &%:#"12!;< =# -%4!$0#'1>& ε (x) 2#$3# ,1&%#,0%&"% 98 .!$ ,! "#&"! ,# "%&(1>& σ 8 %, %(4+%$/! 1&"%$&! N ;) %( &+,! # ,# 01"#- -% ,# '!,+0&#) 05710! .!(1"12! 9"$#''1>&; %& %, %7"$%0! (+.%$1!$ 8 03&10! &%:#"12! 9'!0.$%(1>&; %& ,# ?#(%< =#( $%#''1!&%( %& ,!( %7"$%0!( (% !?"1%&%& -1$%'"#0%&"% '!0! %, 2#,!$ -% N %& "#,%( .+&"!() &!"#&-! *+% %& %, .$10%$ %7"$%0! x = 0 @#8 *+% '#0?1#$,% %, (1:&! .!$*+% ,# $%#''1>& %( %, %(4+%$/! (!?$% ,# '#$# &%:#"12# -% ,# (%''1>& 9%( -%'1$ *+% ,# &!$0#, (#,1%&"% # ,# (%''1>& 2# %& ,# -1$%''1>& &%:#"12# -%, %A% x; L 2 L = γA 2 RL = N(x=L) = γA R0 = −N(x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εx = 0 %& %("% '#(!;< =# (!,+'1>& :%&%$#, -% ,# %'+#'1>& -14%$%&'1#, &! (% 0!-1D'# 9%'< E<FG;) ,! *+% @#8 *+% $%'#,'+,#$ %( %, 2#,!$ -% ,#( '!&("#&"%( -% 1&"%:$#'1>& C 8 D -% #'+%$-! # ,#( &+%2#( '!&-1'1!&%( -% ?!$-%< H@!$# "%&%0!( u(x=0) = D = 0 du γ = L+C =0 dx (x=L) E -% -!&-% $%(+,"# D = 0) 8 γ C = − L) E '!& ,! '+#,I γ x x −L E 2 du γ = (x − L) εx (x) = dx E N (x) = EAε (x) = γA (x − L) u (x) = !"#$ *+% %, -%(.,#/#01%&"! u (x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x 9-!&-% ro %( %, $#-1! %& ,# ?#(%; x r (x) = ro 1 − L GK γΑ - x − u ε σ N !"#$% &'() *+,#-.% /%0+ ,% %11!2. 34 546+ 5$+5!+ 7, 8$4% 34 ,% 6411!2. 46 2 A (x) = π [r (x)] = 9% 41#%1!2. % $46+,:4$ 46 πro2 x 2 1− L d x 2 du x 2 2 =0 Eπro 1 − − γπro2 1 − dx L dx L d x 2 du x 2 2 − γπro2 1 − =0 Eπro 1 − dx L dx L } | {z {z } | p N $4+$34.%.3+ !.;4"$%.3+ #.% :4< d x 2 du x 2 2 Eπro 1 − = γπro2 1 − dx L dx L N= Eπro2 πro2 γL x 3 x 2 du =− 1− 1− +C L dx 3 L 7. 4, 4=;$4-+ ,!/$4 34/4 1#-5,!$64 >#4 4, 5$!-4$ -!4-/$+ 64 %.#,4 ? N B46540%.3+ ,% 34$!:%3% 4 !.;4"$%.3+ = 0@A 34 3+.34 C = 0' x γL du 1− =− dx 3E L γL2 u (x) = − 3E x 1 x 2 − L 2 L +D :%,#%.3+ 4. 4, /+$34 !.C4$!+$ ? ux=0 = 0@ $46#,;% D = 0A D.%,-4.;4 γL2 x 1 x 2 u=− − 3E L 2 L E!. 4-/%$"+ 4. 46;4 1%6+ 4, 5$+/,4-% 46 !"!#$# %" F ,% 6+,#1!2. 5#434 +/;4.4$64 4. C+$-% 64.1!,,% 4:%,#%.3+ %@ 5$!-4$+ ,+6 46C#4$<+6A /@ ,#4"+ ,%6 ;4.6!+.46A 1@ 1+. 4,,%6 ,%6 34C+$-%1!+.46 #6%.3+ ,% 41#%1!2. 1+.6;!;#;!:% F 3@ D.%,-4.;4 !.;4"$%.3+ ,+6 3465,%<%-!4.;+6 % 5%$;!$ 34 ,%6 34C+$-%1!+.46' G4%-+6 % 1+.;!.#%1!2. ,+6 34;%,,46' HI ! "#$%&"&'$ %( ()*&+&,-&# .+#,!+ (/&.( )*( ($ "!%! 0*$1# x (+ (23*(-4# N (x) ()*&+&,-( (+ 0(2# %( +! 0!-1( 2*0(-&#-5 6#7# +! "#+*7$! (2 "'$&"! %&"8# 0(2# 9!+( :(21# (2 ()*&9!+($1( ! &$1(.-!p (x) ($1-( x ; (+ (/1-(7# +&,-( x = L< N (x) = −γ ˆ x L 1 A (x) dx = −γV (x) = −γ A (x) h (x) 3 %#$%( h (x) = L − x (2 +! !+1*-! %(+ "#$# 0#- ($"&7! %( +! 2(""&'$5 ! 1($2&'$ ; +! %(3#-7!"&'$ 9!+($ -(20("1&9!7($1( N (x) 1 = −γ (L − x) A (x) 3 σ (x) γ ε (x) = =− (L − x) E 3E σ (x) = +*(.# +! ("*!"&'$ "&$(7=1&"! >5? 0(-7&1( (2"-&,&-@ du γ =− (L − x) dx 3E x2 γ 1x γ Lx − + C = − Lx 1 − +C u=− 3E 2 3E 2L +! "#$21!$1( %( &$1(.-!"&'$ C 2( #,1&($( *2!$%# +! "#$%&"&'$ u(x=0) = 0A )*( "#$%*"( ! C = 0 "#$ +# "*!+ 2( "#70+(1! +! 2#+*"&'$ %(+ 0-#,+(7!5 B+ %(20+!4!7&($1# %( +! 0*$1! u(x=L) -(2*+1!@ 1 γL2 γ 2 =− u (L) = − L 1 − 3E 2 6E !"!#!#! $%&'()* +,%)-%.-/)0-* 1*2% 345% 3,%30% B+ 0-(2($1( (C(70+# 7*(21-! "#7# "#7,&$!- %#2 2#+*"&#$(25 D*0#$.!7#2 *$! "#+*7$! 2&7&+!! +! !$1(-&#- 0(-# 1-*$"!%! ! *$! !+1*-! H 5 F γΑ γΑ = H + x E&.*-! >5F@ 6#+*7$! 1-#$"#G"'$&"! ,!C# 0(2# 0-#0&# H!%# )*( +! ("*!"&'$ %&3(-($"&!+ (2 +&$(!+A 0#%(7#2 #,1($(- +! 2#+*"&'$ %(+ 0-#,+(7! "#7# +! 2*7! %( +! 2#+*"&'$ %(+ (C(70+# !$1(-&#- 7!2 +! 2#+*"&'$ %(+ 1-#$"# %( "#$# 2#7(1&%# ! +! 3*(-4! F &.*!+ !+ 0(2# %(+ "#$# 0#- ($"&7! %( +! !+1*-! H 2 γ H 2 F = γA (H) (L − H) = (L − H) πro 1 − 3 L ?> !"#"$%& ū " '()" ('*+$%" (&!+,-.$/ '()" (+0*' %' 0'(&!1'0 x 2 dū dū = Eπro2 1 − dx L dx N (x) = F = EA %'(2'3"$%& F dū 1 = 2 dx Eπro 1 − x 2 L -$)'*0"$%& ū (x) = !" ,&$()"$)' C (' &4)-'$' ,&$ !" ,&$%-,-.$ 1 FL +C 2 Eπro 1 − Lx ū(x=0) = 0 ,&$ !& ,+"! FL Eπro2 " # 1 FL −1 ū (x) = 2 Eπro 1 − Lx C=− 0''#2!"5"$%& F 2&0 (+ 1"!&0/ (+#"$%& !" (&!+,-.$ %'! '3'#2!& "$)'0-&0 6 0'&0%'$"$%& γL2 u (x) = − 3E ( #) 3 " 1 x 1 x 2 H 1− − + 1− L 2 L L 1 − Lx 7&)"0 8+' '()" (&!+,-.$ 1"!' 2"0" '! )0&$,& %' ,&$& 9 x ∈ [0 : H]: ;&0 &)0& !"%& (- (' 8+-(-'0" &4)'$'0 !" (&!+,-.$ %' !" ,&!+#$" )0&$,&<,.$-," 4"3& 2'(& 20&2-& 2'0& 0'()0-$*-%" '$ "#4&( '=)0'#&(/ 2+'%' &4)'$'0(' %' !" (-*+-'$)' >&0#"/ "2'!"$%& $+'1"#'$)' " 8+' !" ',+",-.$ %->'0'$,-"! '( !-$'"! 6 8+' 2+'%'$ ,&#4-$"0(' !-$'"!#'$)' (&!+,-&$'(? @A %' !" (&!+,-.$ 4"3& 2'(& 20&2-& ,&$ 4&0%' !-40' %')'0#-$"#&( '! %'(2!"5"#-'$)& %'! '=)0'#& (+2'0-&0 u(x=H) γL2 =− 3E (" H 1 − L 2 H L 2 # H + 1− L 3 " 1 1− 1− H L #) BA %' !" (&!+,-.$ ,&$ 4&0%' 4"3& !" ",,-.$ %' +$" ,"0*" F = 1 &4)'$'#&( '! %'(2!"5"#-'$)& %'! 4&0%' (+2'0-&0 " # ū(x=H) = L 1 1− 2 Eπro 1− H L CA !" 0'()0-,,-.$ %' 8+' '! 4&0%' (+2'0-&0 $& (' %'(2!",' -#2!-," +$" 0'",,-.$ R 9+$" >+'05" 2+$)+"! "2!-,"%" '$ x = H : )"! 8+'? u(x=H) + Rū(x=H) = 0 %' %&$%' 2+'%' &4)'$'0(' !" 0'",,-.$ ,&00'(2&$%-'$)' R=− u(x=H) ū(x=H) 6 !" (&!+,-.$ ,&#2!')" '( !" (+#" %' !" (&!+,-.$ ,&$ '! 4&0%' !-40' #D( !" (&!+,-.$ %'4-%" " !" 0'",,-.$ R 2 ( 1 x 2 3 " #) " 1 1 RL 1− + x 2 Eπro 1− L 1 − Lx # " 3 # " 2 2 γL x 1 x L R γL H 1 =− + − − 1− 1− 2 3E L 2 L E πro 3 L 1 − Lx u (x) = − γL 3E x − L 2 L + 1− H L !" 1− # !"!#!$! %&'()*+ ,-+ .* +)/&0 .123.)&0 4&* (*+ 4+35+ 6(*2(+'! !"#$% "&$'" ($#$ ($)%*+!'"' !, ("%$ +! -) ("'." /-)0-", "/,*("+" " -)" ",0-'" a1 2" ($,-#)" !%03 '!%0'*).*+" +! +!%/,"4"'%! !) "#5$% !60'!#$% 7 %- %!((*8) !% ($)%0")0!1 Pa/L − P + ε σ N a x + u P(1-a/L) 9*.-'" :1;< =$,-#)" ($) ("'." /-)0-", 2" !(-"(*8) +*>!'!)(*", )$ 0*!)! 0?'#*)$ *)+!/!)+*!)0! !) 0$+$ !, +$#*)*$ EA d2 u =0 dx2 @!5*+$ " A-! ," ("'." /-)0-", P *#/,*(" -)" +*%($)0*)-*+"+ !) N !) x = a, /"'" *)0!.'"' ," !(-"(*8) +*>!'!)(*", '!%-,0" )!(!%"'*$ +*B*+*' !, +$#*)*$ !) +$% /"'0!% [0 : L] = [0 : a] + [a : L]C *)0!.'")+$ !) [0 : a] N = EA du = C1 dx 0≤x<a !) a !%03 ," ("'." /-)0-", A-! #$+*D(" !, B",$' +! N 1 2" ("'." /-)0-", /-!+! *)0!'/'!0"'%! ($#$ %* !) -) !)0$')$ δ +! x = a &"7 -)" ("'." +*%0'*5-*+" p = Pδ +! 0", >$'#" A-! ," *)0!.'", !) +*(&$ !)0$')$ δ !%< ˆ a+ du N(x=a+ δ ) = EA = C1 + 2 dx +! 0", >$'#" A-! !) !, %!.-)+$ 0'"#$ N = EA δ 2 − a− 2δ du = C1 − P dx P δ dx = C1 − P a<x≤L *)0!.'")+$ )-!B"#!)0! !) ("+" 0'"#$ /$' %!/"'"+$ EAu = C1 x + C2 EAu = C1 x − P (x − a) + C2 0≤x<a a<x≤L 2" +!0!'#*)"(*8) +! ,"% ($)%0")0!% !% #-7 %!)(*,,"C B",-")+$ ," /'*#!'" ! x = 0 EAu(x=0) = C2 = 0 ,,!B")+$ " ," %!.-)+" 7 B",-")+$ !) x = L EAu(x=L) = C1 L − P (L − a) a C1 = P 1 − L EF ! "#$%&'() *" *)+#)&*" P 1 − La x 0≤x<a u (x) = EA P 1 − La P u (x) = x− (x − a) EA EA xi Pa h 1− a<x≤L = EA L ! ,!-'!&'() .* $#" .*"/$!0!1'*)+#" *" 2'3$')*!$4 #" *"5%*-0#" *) $#" *6+-*1#" ,!$*) *) x = 0 *) x = L a N0 = C 1 = P 1 − L a NL = C1 − P = −P L 7#+!- 8%* $! &#),*)&'() /#"'+',! /!-! P *" $! .'-*&&'() /#"'+',! .*$ *9* x: !"; /!-! %)! &!-<! /#"'+',! *$ *"5%*-0# *) *$ /-'1*- +-!1# *" .* +-!&&'() = *$ +-!1# "%/*-'#- .* &#1/-*"'()4 !" -*!&&'#)*" "#) '),*-"!1*)+* /-#/#-&'#)!$*" ! "% .'"+!)&'! !$ /%)+# .* !/$'&!&'() .* $! &!-<! >,*?'<%-!@4 P !"!#!$! %&'()*+ ,&* )&-.)./*0&1 2/ /304/)& ?')!$1*)+* &#)"'.*-*1#" *$ &!"# .* %)! &#$%1)! .* $#)<'+%. L: 8%* )# +'*)* &!-<! !/$'&!.! >p (x) = 0@ /*-# .* $! &%!$ "* &#)#&*) $#" .*"/$!0!1'*)+#" u0 = uL .* "%" *6+-*1#"4 ! *&%!&'() .'5*-*)&'!$ *" >"%/#)'*).# 8%* $! &#$%1)! *" .* "*&&'() &#)"+!)+*@ EA du =0 dx &%=! ')+*<-!$ *" "*)&'$$!1*)+* du =ε=C dx u (x) = Cx + D # /-'1*-# 8%* .*2* )#+!-"*: $# &%!$ *" "*)&'$$# * ')+%'+',#: *" 8%* !$ )# A!2*- 5%*-0!" .'"+-'3 2%'.!" *$ *"5%*-0# )#-1!$ N *" &#)"+!)+*4 %*<# !$ A!2*- "%/%*"+# AE &#)"+!)+*: $! .*5#-1!&'() *" +!12'B) &#)"+!)+* *) +#.! $! /'*0!4 !" &#)"+!)+*" .* ')+*<-!&'() "* &!$&%$!) ! /!-+'- .* $!" &#).'&'#)*" u(x=0) = D = u0 u(x=L) = CL + D = uL .* .#).* u L − u0 C= =ε L uL − u0 x x u (x) = + uL x + u0 = u0 1 − L L L D = u0 = .#).* /%*.* ,*-"* 8%* *$ .*"/$!0!1'*)+# ,!-;! $')*!$1*)+* &#) ?')!$1*)+* *$ *"5%*-0# )#-1!$ *" N = EAε = EA (uL − u0 ) L ED x L *)+-* u0 = uL 4 >C4DE@ ! "#$%&%'(#! %')&% *+, "%,-*!.!/#%')+, "% *+, %0)&%/+, (uL − u0 ) %, *! %*+'1!(#2' e "% *! 3!&&! 4 !* (+(#%')% EA ,% *+ "%'+/#'! *! &#1#"%. !0#!* K "% *! 3!&&!5 (+' "#(6! '+)!(#2' L N = Ke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x − yF 4 78% %* %=% x (+#'(#"% (+' %* %=% "% *! <#1!9 ;%'+/#'!&%/+, (+' v ! *+, "%,-*!.!/#%')+, %' *! "#&%((#2' y9 J9 !, $8%&.!, %0)%&'!, !()>!' %' *! "#&%((#2' y E'+ 6!4 $8%&.!, %0)%&'!, %' *! "#&%((#2' !0#!* x5 ,# *!, 683#%&! *! ,+*8(#2' "% )!* -&+3*%/! %, *+ )&!)!"+ %' *! ,%((#2' !')%&#+&F9 K9 !, )%',#+'%, '+&/!*%, %' *! "#&%((#2' )&!',<%&,!* ! *! <#1! E σy F ,+' "%,-&%(#!3*%,5 %,)+ #'(*84% *!, )%',#+'%, "% (+')!()+ "%3#"!, ! *!, (!&1!, !-*#(!"!,5 *8%1+ %, #'"#,)#')+ 78% *!, (!&1!, ,% !-*#78%' ,+3&% *! -!&)%, ,8-%&#+&5 #'$%&#+& + ,+3&% %* %=% "% *! <#1!9 L9 !, ,%((#+'%, ,% /!')#%'%' -*!'!, !* "%$+&/!&,% *! <#1! @9 !, "%$+&/!(#+'%, "%3#"!, !* (+&)% )&!',<%&,!* ,+' "%,-&%(#!3*%, γ = 0 9 H, "%(#& 78% *!, ,%((#+'%, ,% /!')#%'%' '+&/!*%, !* %=% "%$+&/!"+9 !, >*)#/!, "+, EL 4 @F (+'$+&/!' *! 6#-2)%,#, "% M%&'+8**#AN!<#%&5 *! ELF %0-&%,! 78% *+, "%,-*!.!/#%')+, u %' *! "#&%((#2' x E"%3#"+, ! *! D%0#2'F "%-%'"%&C' "%* 1#&+ "% *! ,%((#2' φ 4 <!&#!&C' *#'%!*/%')% %' *! !*)8&! "% *! <#1! (+' <!*+& '8*+ %' %* %=% u (x, y) = −φ (x) y = − dv (x) y dx EO9J@F "+'"% *! ,%18'"! #18!*"!" &%,8*)! "% E@F9 H' 3!,% ! *+ !')%&#+& *!, >'#(!, "%$+&/!(#+'%, &%*%<!')%, ,+' *!, "%$+&/!(#+'%, "% D%0#2' %' *! "#&%((#2' x5 78% "%'+/#'!&%/+, ,#/-*%/%')% (+' ε9 H,)!, "%$+&/!(#+'%, <!&?!' *#'%!*/%')% %' %* %,-%,+& %' $8'(#2' "% *! "#,)!'(#! !* 3!&#(%')&+ "% *! ,%((#2' 4 ,+' -&+-+&(#+'!*%, ! *! (8&<!)8&! "%* %=%9 du dv (x) d ε (x, y) = − = y = −χy EO9JPF dx dx dx LK φ Y u=- φy 1 dv dx v 1 y X !"#$% &'() *+,-.%/%0!+123, +1 4!"%,' 5.%13 678 9319+ .% :#$4%2#$% 9+. +;+ 3$!"!1%.0+12+ $+:23 <#+9% +1231:+, 9+=1!9% -3$ χ (x) = d2 v dφ (x) = 2 dx dx >&'?@A B#+"3 .%, 2+1,!31+, +1 .% 9!$+::!C1 %6!%. 4%.+1 >&'?(A σ(x, y) = Eε (x, y) = −E χ (x) y D. +,E#+$/3 13$0%. -3$ F!-C2+,!, 4%.+ 0G .3 <#+ ,+ 4+$!=:% 8% <#+ N (x) = ˆ σ (x, y) dA = A ˆ −Eχ (x) ydA = −Eχ (x) A ˆ y dA A >&'?&A 9319+ .% H.2!0% !12+"$%. !19!:%9% +, 0 -3$<#+ +. +;+ -%,% -3$ +. I%$!:+12$3 9+ .% ,+::!C1' D. 030+123 J+:23$ $+,#.2% 9+ !12+"$%$ +. 030+123 9+ +,2%, 2+1,!31+, +1 +. K$+% 9+ .% ,+::!C1 M (x) = − ˆ σ (x, y) y dA = Eχ (x) A ˆ y 2 dA = Eχ (x) I A >&'?LA D,2% H.2!0% +:#%:!C1 13, -$34++ .% $+.%:!C1 :31,2!2#2!4% +12$+ +. +,E#+$/3 "+1+$%.!/%93 > M A 8 .% 9+E3$0%:!C1 "+1+$%.!/%9% > χA' B% +:#%:!C1 9+ +<#!.!I$!3 % .% 2$%,.%:!C1 >4+$2!:%.A $+,#.2% dT (x) + q (x) = 0 dx >&'MNA D1 2%123 <#+ .% +:#%:!C1 9+ +<#!.!I$!3 9+ 030+123, %.$+9+93$ 9+. +;+ 13$0%. > z A %. -.%13 9+ 034!0!+123 >x − yA +, dM (x) =0 >&'M?A T (x) + dx dM (x) T (x) = − dx >&'MMA d2 M (x) + q (x) = 0 dx2 >&'MOA B.+4%193 +,2% H.2!0% % .% +6-$+,!C1 >&'MNA 9+ +<#!.!I$!3 % .% 2$%,.%:!C1 − OO y q(x) T+ T dT dx dx x dM M+ dx dx M dx !"#$% &'&( )*#!+!,$!- ./ 0!"%1 % 1# 0.2 $..34+%2%/5- +% .64$.1!7/ 5.+ 3-3./8- ./ 9#/:!7/ 5. +% :#$0%8#$% ;&'<=> − d2 EIχ (x) + q (x) = 0 dx2 ;&'?@> A ./ ,%1. % +% B!478.1!1 5. *#. +% 1.::!7/ .1 :-/18%/8. ./ 8-5% +% 4!.2% −EI d2 χ (x) + q (x) = 0 dx2 ;&'?C> D/%+3./8. $..34+%2%/5- %*#E +% :#$0%8#$% ./ 9#/:!7/ 5. +-1 5.14+%2%3!./8-1 ;&'<F> −EI d4 v (x) + q (x) = 0 dx4 ;&'?F> 8./.3-1 +% .:#%:!7/ 5!9.$./:!%+ 5. .*#!+!,$!- 5. +% 0!"% % G.6!7/ ./ 9#/:!7/ 5. +-1 5.14+%2%3!./8-1' )18% .:#%:!7/ 5!9.$./:!%+ -$5!/%$!%H +!/.%+H 5. @ -$5./ $.*#!.$. 5. @ :-/5!:!-/.1 5. ,-$5.H ./ "./.$%+ ? 4-$ .68$.3-' )18%1 :-/5!:!-/.1 4#.5./ 1.$ 5. 5-1 8!4-1( ! "#$#%&'!() !*!+%#,'!*) %#+!-.(#%,* / 0!/-1($#%,* ' E1!:%3./8. 4-5.3-1 !34-/.$ +-1 5.14+%2%3!./8-1 ./ #/ .68$.3-' )18-1 5.14+%2%3!./8-1 4#.5./ 1.$ ./ +% 5!$.::!7/ yH ;.1 5.:!$ 4-5.3-1 !34-/.$ v>H - ./ +% 5!$.::!7/ xH ./ .18. I+8!3- :%1- :-3- u 5.4./5. 5.+ "!$dv φ ;&'<@> +- *#. 4-5.3-1 !34-/.$ .1 dx ' ! 2!3-,++) +,(3$,'!* / ! 43!$5,* ' E1!:%3./8. 4-5.3-1 !34-/.$ +%1 9#.$2%1 ./ #/ .68$.3-' )18%1 9#.$2%1 4#.5./ 1.$ .+ .19#.$2- 5. :-$8. 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() -& +,&- &+$N&)5 (3 4(+'% q(x) = qy (x) T (x) = Ty (x) 9(+#%4&% &4(!O3 6,( -& 4'3$%'0,+'.) 4( $()3'#)(3 4( +#%$( $%&)3:(%3&-(3 &- (A( 4( -& :'7& 3( +&-+,-& & "&%$'% 4( -& (/"%(3'.) 4( P#,%&3Q> <$(#%2& 4( R#--'7)#)=8 *6,2 3( ')+-,>( -& B'".$(3'3 4( 6,( )# B&> D,(%;&3 $&)7()+'&-(3 &"-'+&4&3 3#0%( -&3 +&%&3 4( -& :'7&5 -,(7# "#% %(+'"%#+'4&4 4( $()3'#)(3 $&)7()+'&-(3 (- :&-#% 4( -&3 $()3'#)(3 4( +#%$( (3 +(%# () -&3 +&%&3 3,"(%'#% ( ')D(%'#%8 *4(!O3 )#$&% 6,( &- B&0(% 4(3"%(+'&4# -&3 4(D#%!&+'#)(3 $%&)3:(%3&-(3 4( +#%$( < γ = 0= )# B&> ,)& %(-&+'.) +#)3$'$,$':& 6,( ",(4& -'7&% T +#) γ 5 "#% -# +,&- (- +#%$( T 3( #0$'()( 4( -& +#)4'+'.) 4( (6,'-'0%'# 4( !#!()$#3 <(+,&+'.) F8KK= ST !"!#! $%&'()*+, -,&,./.-0&, !"! #$% $& '"()*%+! ,%! -,(,./.-0( %, &%0%,!"-( #$% 1(, 1% *!, 0(&1-0-(&%, 1% 0(&.("&( ,%!& 1% 2$%"3! 4&!.$"!*%,5 6 1(, 1% 1%,'*!3!+-%&.(, 4%,%&0-!*%,57 8&.(&0%, %, '(,-)*%9 :7 ;%.%"+-&!" "%!00-(&%, 6 1-!<"!+!, 1% %,2$%"3(, %& '!".-0$*!" %* +(+%&.( =%0.(" 0(+( 2$&> 0-?& 1% x 4M (x)5 $,!&1( *!, 1(, 0(&1-0-(&%, 1% 0(&.("&( 1% 2$%"3!,9 @7 A.-*-3!" *! "%*!0-?& 0(&,.-.$.-B! <%&%"!*-3!1! '!"! ().%&%" *! 0$"B!.$"! χ (x) = M (x) EI C7 D&.%<"!" *! %0$!0-?& 0-&%+/.-0! '!"! ().%&%" <-"(, 6 1%,'*!3!+-%&.(, d2 v M (x) = χ (x) = 2 dx ˆ EI x dv M (x) = φ (x) = dx + C1 dx EI 0 ˆ xˆ x M (x) 2 dx + C1 x + C2 v (x) = EI 0 0 E7 A,!" *!, 0(&1-0-(&%, 1% 0(&.("&( 1% 1%,'*!3!+-%&.(, '!"! 1%.%"+-&!" *!, 0(&,.!&.%, C2 7 C1 6 !"!"! 12)*3(&,! !"!"!#! $%&' (%)*+,-./'0' F(+( $& '"-+%" %G%+'*( ,%&0-**( (),%"B%+(, 0(+( ().%&%" *! ,(*$0-?& 1% $&! B-<! )->%+'(."!1! 0(& 0!"<! $&-2("+%7 q d4 v (x) = 4 dx EI q x L H-<$"! I7J9 K-<! %+'(."!1! )!G( 0!"<! $&-2("+% 0$6!, 0(&1-0-(&%, 1% 0(&.("&( ,(& v(x=0) = 0 dv =0 dx (x=0) v(x=L) = 0 dv =0 dx (x=L) CL !"#$%&!'( #)"& #*+&*,-! ',.#%#!*,&/ )# (0",#!# +!& 1#2 d3 v = dx3 1 EI ´x '() 1#*#) d2 v = dx2 1 EI ´ ´x "%#) 1#*#) dv = dx *+&"%( 1#*#) v= 1 EI 1 EI (x) = − TEI q (x) dx + A 0 q (x) dxdx + Ax + B 0 = M (x) EI 345678 ´ ´ ´x 0 ´ ´ ´ ´x 0 2 q (x) dxdxdx + Ax + Bx + C 2 q (x) dxdxdxdx + =φ Ax3 Bx2 + + Cx + D 6 2 9",/,2&!'( /& :%& ; /& <"&= >&%& ,?>(!#% &//@ /&) *(!',*,(!#) '# 0,#?>("%&?,#!"( "#!'%#?() 0 0 L3 0 0 L2 2 6 L2 2 1 A B 0 1 C D 0 0 1 L 1 L A(!'# B#?() '#!(?,!&'( *(! 0 = − 0q v φq L qx4 qL4 dx = = 24EI 0 24EI x=0 ˆ L 3 L q qx qL3 φq = dx3 = = EI x=0 6EI 0 6EI q v = EI ˆ q L 4 C! #/ ),)"#?& '# < #*+&*,(!#) *(! < ,!*-$!,"&) %#)+/"&!"#= /&) '() >%,?#%&) #*+&*,(!#) )(! '# %#)(/+*,-! ,!?#',&"&= C D = 0 0 vq φq /( D+# >+#'# //#1&%)# & /&) '() %#)"&!"#)= %#)+/"&!'( #!"(!*#) L3 6 L2 2 L2 2 L A B '# '(!'# A=− =− qL 2EI qL3 =− 6EI B= L/4 1 qL2 12EI C! *(!)#*+#!*,& #/ ?(?#!"( E#*"(% 1&/# x x2 qL qL2 dxdx + EI Ax + EI B = q − x+ 2 2 12 0 2 x x 2 qL −6 +1 = 6 12 L L M (x) = q ˆ ˆ D+# 1&/+&'( #! /() #F"%#?() ; #/ *#!"%( 1&/# M(x=0) = qL2 12 M(x= L ) = − 2 qL2 24 M(x=L) = + qL2 12 G&%& 1#% #/ 1&/(% '# /() ?(?#!"() )(0%# /() #?>("%&?,#!"() '#0# %#*(%'&%)# D+# B&; D+# *&?0,&% #/ ),$!( &/ D+# &*"H& #! #/ #F"%#?( ,2D+,#%'( 3*&%& !#$&",1&85 C/ *(%"# >+#'# (0"#!#%)# '#%,1&!'( #/ ?(?#!"( ( T (x) = − ˆ x q (x) dx − EI A = −qx + 0 :I qL 2 !"#$%" &"' ($) *"$++,- ") #"*&,+$(") ." $/-0- *")!(&$ ." "#$(!$* "( +-*&" " (-) "1&*"%-) qL 2 qL =− 2 Ry(x=0) = −T(x=0) = − Ry(x=L) = T(x=L) 2 &$ &- 3!" (-) .")/($4$%," &-) )qL4 x 4 qL4 x 3 qL4 x 2 − + 24EI L 12EI L 24EI L x 3 x 2 qL4 x 4 = −2 + 24EI L L L v (x) = 2( %51,%- .")/($4$%," &- ") " "( +" &*- 0 #$(" v(x= L ) = vmáx = 2 !"!"!#! qL4 384EI $%&' (%)*+,),-., '*/0'1' 2'3/ 4'5&' 6-%7/5), 6$ )-(!+,7 8" "*$( ") ,.9 &,+$ $( +$)- $ &"*,-* :;<=>?' +$%@,$ ($) +- .,+,- ") ." @-*."< AB-*$ " #"4 ." $ !($* "( 8,*- " (-) "1&*"%-) B$0 3!" $ !($* ($) +!*#$&!*$) ˆ xˆ x q dxdx + Ax + B χ (x) = EI 0 0 χ(x=0) = B qL2 + AL + B χ(x=L) = 2EI 6$) +!$&*- "+!$+,- ") /$*$ -@&" "* ($) +- )&$ &") *")!(&$ $B-*$ ." ($) .-) /*,%"*$) 0 0 3 L 6 L 0 1 L2 2 1 0 0 L 0 1 A B 0 1 C D 0 0 2 = − qL 02 24EI L 12 B=D=0 ." ($ +!$*&$ /!"." .")/"C$*)" A A=− 0 ." ($ &"*+"*$ C 1 C= L qL 2EI L3 qL qL4 qL3 + − = 24EI 6 2EI 24EI D $(%" &" "( .")/($4$%," &- 0 "( %-%" &- *")!(&$ x 3 x qx4 qL3 qL 3 qL4 x 4 v (x) = −2 + − x + x= 24EI 12EI 24EI 24EI L L L 2 2 2 x qx x qL qL − M (x) = − x= 2 2 2 L L E; ! "#$" %&#'( )*" "# +'#,-." "/&.*&0 .'# "#1*"02'# & +&0$,0 "3%.*#,/&4"!$" 5" .&# %'!5,%,'!"# 5" ")*,.,-0,' 6"#$0*%$*0& ,#'#$7$,%&8( #" +*"5" 9&%"0: R0 = − qL 2 x qL q dx = T (x) = −R0 − − qx 2 0 ˆ x ˆ x Lx x2 L T (x) dx = −q M (x) = − − x dx = −q − 2 2 2 0 0 ˆ !"#$%$& '!( )$(*'+,+-.$%#!( .%#$/&+%)! '+ )$0!&-+1.2% /$%$&+'.,+)+3 M (x) d2 v qL2 x 2 x = − = dx2 EI 2EI L L 2 ˆ x 2 qL qL3 1 x 3 1 x 2 C1 dv x x = dx = − − + dx 2EI 0 L L 2EI 3 L 2 L L x 3 ˆ x 4 qL qL 1 x 4 1 x 3 1 x 3 1 x 2 v= + C2 − + C1 dx = − + C1 2EI 0 3 L 2 L 2EI 12 L 6 L L 4' 15'16'! )$ '+( 1!%(#+%#$( C1 C2 ($ 7+1$ 8+'6+%)! '+( 1!%).1.!%$( )$ 1!%#!&%! )$ )$(*'+,+9 -.$%#!( $% +-"!( $:#&$-!( ; v(x=0) = 0< v(x=L) = 0=< '! 16+' 1!%)61$ + C2 = 0 '6$/! C1 = 1 12 x 3 x qL4 x 4 −2 + v (x) = 24EI L L L 4' -5:.-! )$(*'+,+-.$%#! $( $% $' 1$%#&! 8+'$ v(x= L ) = vmáx = 2 5qL4 384EI >6$ $( ? 8$1$( $' )$(*'+,+-.$%#! -5:.-! )$' 1+(! ".$-*!#&+)!@ Simp. Apoy. Empotr. A./6&+ B@CD3 4'5(#.1+ )$ '+ 8./+ "+E! 1+&/+ 6%.0!&-$ !"!"!"! #$%& '$()*+(+,-+ &)./&0& 1&2. 3&4%& )5,-5&* F+ 8./+ $( )$ '!%/.#6) L '+ 1+&/+ *6%#6+' $(#5 +*'.1+)+ + 6%+ ).(#+%1.+ aL )$' +*! ! .,>6.$&)! G+)! >6$ '+ $(#&61#6&+ $( .(!(#5#.1+< *!)$-!( ).&$1#+-$%#$ $(1&.".& d3 v P T (x) )!%)$H $( '+ 06%1.2% $(1+'2% >6$ 8+'$ = [(1 − a) − H (x − La)] =− 0 *+&+ x < La 1 *+&+ x ≥ La dx3 EI EI HI aL P T(x) P(1-a) Pa L x M(x) !"#$% &'(() *!"% +!,-./,/01/ %-23%4% 5%62 7%$"% -#01#%. Ei 4204/ .% 8#07!90 h i/+ 0#.% +! /. %$"#,/012 M (x) x Dx PL h χ (x) = (1 − a) − = −a /+ 0/"%1!:2 3 :%./ /. %$"#,/012 +! /+ > 0 EI EI L L ;H (x − La) +/ -#/4/ /+7$!5!$ 1%,5!<0 Lx − a 0 = -%$% /:%.#%$ /. "!$2 !01/"$%,2+ #0% :/> x 2 D x E2 P L2 − (1 − a) + C1 −a φ (x) = 2EI L L !01/"$%042 #0% +/"#04% :/> /0 82$,% +!,!.%$ E3 x 3 D x P L3 x −a (1 − a) + C1 L + C2 v (x) = − 6EI L L L ?%+ 720+1%01/+ C1 3 C2 $/+#.1%0 4/ %-.!7%$ .%+ 7204!7!20/+ 4/ 72012$02' ?% -$!,/$% 7204!7!90 4/ 72012$02 27#$$/ /0 x = 0@ A#/ 72$$/+-204/ %. /B1$/,2 !>A#!/$42@ :%.#%042 /01207/+ /0 x = 0 $/+#.1% C2 = 0' ?% +/"#04% 7204!7!90 4/ 72012$02 27#$$/ /0 x = L@ A#/ 72$$/+-204/ %. /B1$/,2@ 4/$/7C2@ :%.#%042 /01207/+ /0 x = L P L3 1 − a − (1 − a)3 + C1 L 6EI P L3 a (1 − a) (2 − a) + C1 L = 6EI P L2 a (1 − a) (2 − a) C1 = − 6EI v(x=L) = 4/ 4204/ E3 P L 3 x 3 D x x P L3 −a a (1 − a) (2 − a) − (1 − a) v (x) = − 6EI L L 6EI L D E 3 3 PL x 2 x x = −a − a (2 − a) (1 − a) − 6EI L L L D0 /+1/ 7%+2 /0 A#/ .% 7%$"% 02 /+1E 7/01$%4%@ /. ,EB!,2 4/+-.%>%,!/012 02 +/ -$24#7/ 5%62 .% 7%$"%@ -%$% 4/1/$,!0%$.2 C%3 A#/ /07201$%$ /. -#012 4204/ +/ %0#.% /. 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SA( +"# *& !;#. & & +"#-.#/&+.9#H F/%"#0&4", /#& 1.0& ,.# +&'0& !# !* .#-!'."' q = 0 d4 v (x) =0 dx4 +*42; 92 98:2; 72927*. +/7&, +"# .+."#!, ! ="' ! ,"# !,%*&3&4.!#-", #/*", !# *", !6-'!4", 7 .0/&*!, 7 "%/!,-", & *", B/! ,! "=-.!#!# +"# *& ,"*/+.9# ,.4%*!4!#-! &%"7& &H 52; v(x=0) = 0 v(x=L) = 0 dv 1 P L2 3 a − 3a2 + 2a = φ1 = −φSA(x=0) = dx (x=0) 6 EI dv 1 P L2 = φ2 = −φSA(x=L) = a a2 − 1 dx (x=L) 6 EI I"4" 7& 1.4", &#-!,2 .#-!0'&# " !,-& !+/&+.9# .:!'!#+.&* ,! "=-.!#! $. A#-.+& & $J?KL( %!'" ,.# q ( /#& 1!3 ", 1!+!, -'!, 1!+!, d3 v =A dx3 (x) = − TEI d2 v = Ax + B dx2 = dv Ax2 = + Bx + C dx 2 =φ +/&-'" 1!+!, v = Ax3 Bx2 + + Cx + D 6 2 MN M (x) EI !"#"$%&'( #% )*% + #% ,!%- .%*% "/.(&0* %##1 #%2 3(&'"3"(&02 '0 3(&!(*&( !0&'*0/(2 0 0 L3 6 L2 2 0 0 L2 2 L 1 A B 0 1 C D 0 0 1 L 1 3(& #( 34%# '0 #%2 '(2 .*"/0*%2 20 !"0&0 0 φ1 = 0 φ2 C = φ1 D=0 ##05%&'( % #%2 '(2 6#!"/%2 *02(#5"0&'( L3 6 L2 2 L2 2 L A B = −φ1 L φ2 − φ1 6 (φ1 + φ2 ) L2 2 B = − (2φ1 + φ2 ) L A= + 3(& 0##( x 3 x 2 x − L (2φ1 + φ2 ) + Lφ1 v (x) = L (φ1 + φ2 ) L L L x 2 x − 2 (2φ1 + φ2 ) + φ1 φ (x) = 3 (φ1 + φ2 ) L L i io h x o EI n h x EI n x 6 (φ1 + φ2 ) − 2 (2φ1 + φ2 ) = φ1 6 − 4 + φ2 6 − 2 M (x) = L L L L L EI T (x) = −6 2 {φ1 + φ2 } L 7" %8(*% 24/%/(2 %/9%2 2(#43"(&02 *00/.#%$%&'( φ1 + φ2 .(* #(2 5%#(*02 '0:&"'(2 %&!02- 20 (9!"0&0 #% 2(#43";& '0 4&% 5"<% 3(& 4&% 3%*<% .4&!4%# 34+(2 0=!*0/(2 &( 20 '02.#%$%& + &( <"*%&> ?# .0*:# '0 '02.#%$%/"0&!( !*%&250*2%# .%*% 0# 3%2( .%*!"34#%* a = 0,4 20 /402!*% 0& #% @"<4*%> A40'0 &(!%*20 #% B40*!0 '"2/"&43";& '0 #(2 '02.#%$%/"0&!(2 %# 3%/9"%* #%2 3(&'"3"(&02 '0 3(&!(*&(> φ1−φ2 a=0.4L P Empotr. 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"# ! D = φ1 C=− qz e L (φ2 − φ1 ) + GJ 2 L $"# %" $&'% ()!") !#'# "* qz e x x (x − L) + φ1 + (φ2 − φ1 ) GJ 2 L L (φ2 − φ1 ) qz e x− + θ (x) = GJ 2 L (φ2 − φ1 ) L Mt (x) = qz e x − + GJ 2 L φx (x) = +!,- " ' %' %-#!'%- ' ! %' !$&'$-.# -/!)!#$-'%0 '% 1-21" )!2&%3' " 2! %%!4' 2- 2! '#'%-5'# 6") 2!6')' " %' -#7&!#$-' ! %' $')4' 8 ! %'2 $"# -$-"#!2 ! $"#3")#" ! 4-)"2 -16&!23"29 :' 2"%&$-.# !#$"#3)' ' 6&! ! ;!)2! $"1" %' $"1,-#'$-.# (2&1'* ! %'2 2-4&-!#3!2 2"%&$-"#!2 !,- " '% 1"1!#3" 3")2") -23)-,&- " $"# $"# -$-"#!2 ! $"#3")#" ! 4-)"2 #&%"2 !# %"2 !<3)!1"29 =23" 2! $"#"$! $"1" 2"%&$-.# ! %' !$&'$-.# -/!)!#$-'% #">?"1"4@#!' 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