Capítulo 8 ECUACIONES DIFERENCIALES Cálculo de

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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
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du
&!$"# (ds = dx) = 3"#')6# /)" #" 3"#'*&%" ds∗ = dx + dx
dx
du (x)
ds − ds
=
ε (x) =
ds
dx
∗
PK
IJ5PL
x
x+dx
u+du
*
ds
ds
u
du
x+dx+u+dx dx
x+u
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2+1-.!1% #1% !"#$%&' $(')*%*+*%,# <2+B1+ +3 /-.8-$4%.!+14- .+/C1!/- 2+3 .%4+$!%3 $(')*%*+*%,( >'
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σ x = E εx
N=
=
ˆ
ˆA
σx dA
<&'I>
Eεx dA
<&'J>
A
7! 3% 7+//!01 +7 E-.-"F1+%
N = σx A = EA εx
7! 3% 7+//!01 1- +7 E-.-"F1+% 7+ 2+B1+ +3 L%3-$
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
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+ $"#+# $&' "& + $- !$
. !"& =
p (x).
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εB=
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ε (x)
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u
N (x). &,%
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!&$*&'(% &'%!019'&*= C" )!&'-
!" )! /*" )," &,%)'&',%!" )! &,%9,-%, )!:!
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"!- )! #$!-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
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!/ !29-!0, !%
x = 0B
u (x) = u(0) +
ˆ
x
ε (x) dx
0
N+
N
p(x)
dN
dx
dx
X
dx
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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 ,+! 1+$<1.$!($, %'".('%'.+,6
, .-&+'(%!($ !+(%' ;#$
@6 A% $,('#1(#'% %.,*%)% )$"$ ,$' *% -.,-% =-.,-% /$+-$('0% 7 -%($'.%*>
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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
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+,-,.,/,
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.-&$).)% )$ )$,&*%5%',$ $! %-"+, $G('$-+, 7 "%I+ *% %11.3! )$* &$,+ &'+&.+ =:$' M./#'% K6B6N6@>?
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#!.)%) )$ *+!/.(#) $, 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
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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
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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
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du
dx
(0)
. u (0) *-' 2-#)$!#$%) C . D
;B
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%( 05710! # ,# 01"#- -% ,# '!,+0&# 9(1%0.$% &%:#"12!;< =# -%4!$0#'1>& ε (x) 2#$3# ,1&%#,0%&"% 98
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%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=0)
B, .%(! -% ,# '!,+0&# %( %&"!&'%( (!.!$"#-! .!$ 01"#-%( %& '#-# %7"$%0!<
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,1?$%) 8 -%?% DA#$(% %, %(4+%$/! 9 N = 0 %& %("% '#(!; ! %& 4!$0# %*+12#,%&"% ,# -%4!$0#'1>& 9 εx = 0
%& %("% '#(!;<
=# (!,+'1>& :%&%$#, -% ,# %'+#'1>& -14%$%&'1#, &! (% 0!-1D'# 9%'< E<FG;) ,! *+% @#8 *+% $%'#,'+,#$
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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) 2#,% 6 %& ,# ?#(% 8 '$%'% %& 4!$0# '+#-$5"1'# @#("# %, %7"$%0!
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#"+$#,0%&"% "!-! %, .%(! -% ,# '!,+0&# %("5 #@!$# (!.!$"#-! .!$ %, #.!8!<
!"!#!8!
9/1+.*% :;*2:% 3%(/ 7)5/ 7&/72/
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%& %("!( '#(!( .+%-% -%0#&-#$ @%$$#01%&"#( 0#"%05"1'#( 05( '!0.,%A#(< =# %'+#'1>& -14%$%&'1#,
# +"1,1/#$ #@!$# %( ,# 2%$(1>& 9E<F6;< B, $#-1! %( ,1&%#, %& 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
78% !,#/#*! %* (+/-+&)!/#%')+ "% 8'! 3!&&! !* "% 8' &%,+&)% "% &#1#"%. K 9
:%(+&"!& 78% "%3#"+ ! *! 6#-2)%,#, "% *#'%!*#"!" %, -+,#3*% ,8-%&-+'%& *!, !((#+'%, 4 &%,-8%,)!,9
;% %,)! $+&/! %, -+,#3*% %<!*8!& *! &%,-8%,)! "% 8'! (+*8/'! 3!=+ -%,+ -&+-#+ "% *! (8!* ,% (+'+(%'
*+, "%,-*!.!/#%')+, "% ,8, %0)&%/+, (+/+ *! ,8/! "% *!, ,+*8(#+'%, "%* -&#/%& %=%/-*+ 4 "% %,)%
>*)#/+9
!"! #$%&' () *+(,$-)
!"!#! $%&'() *+,-.*) /% 0.1)- %2 3%4.52 67')
:%,8/#%'"+ *+ <#,)+ %' *+, (!-?)8*+, @AB5 *!, 6#-2)%,#, /C, #/-+&)!')%, "%* (+/-+&)!/#%')+
"% <#1!, %' D%0#2' ,+' E!"%/C, "% *!, "% *#'%!*#"!"FG
H* %=% "% *! <#1! %, &%()+
! ,%((#2' '+ (!/3#! %' )+"+ %* )&!/+9
! "#&%((#2' '+&/!* !* -*!'+ "% *! <#1! %, 8'! "% *!, "#&%((#+'%, -&#'(#-!*%, "% #'%&(#! "% *!
,%((#2'
I8-+'"&%/+, E,#' '#'18'! -%&"#"! "% 1%'%&!*#"!"F 78% %* -*!'+ "% /+<#/#%')+ E+ -*!'+ "% (!&1!F
"% *! <#1! %, %* -*!'+ Ex − 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
'
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dv
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v=0
M
d2 v
=χ= 2 =0
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dx
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1 dM
d3 v
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EI dx
dx
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1 dM
d3 v
T
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EI
EI dx
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dv
=0
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<(+,&+'.) F8GJ= > () -& (/"%(3'.) 4( -&3 $()3'#)(3 () D,)+'.) 4(- !#!()$# @(+$#%
σx (x, y) = −
M (x)
y
I
<F8KL=
M')&-!()$( -& (-(++'.) 4( -& +#):()+'.) "#3'$':& "&%& -& +,%:&$,%& χ +#')+'4( +#) -& 4(- !#!()$#8
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6,( (A( (3$&!#3 $#!&)4# !#!()$#
ˆ
Iz =
y 2 dA
<F8KF=
A
M')&-!()$( & -&3 D,(%;&3 (/$(%)&3 > & -#3 (3D,(%;#3 ')$(%)#3 3( -(3 &7%(7&%O ,) 3,02)4'+( ')4'+&)4#
-& 4'%(++'.) () -& +,&- &+$N&)5 (3 4(+'%
q(x) = qy (x)
T (x) = Ty (x)
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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
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0
0
L2
2
6
L2
2

1
A
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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! #/ ),)"#?& '# < #*+&*,(!#) *(! < ,!*-$!,"&) %#)+/"&!"#= /&) '() >%,?#%&) #*+&*,(!#) )(! '#
%#)(/+*,-! ,!?#',&"&=
C
D
=
0
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/( D+# >+#'# //#1&%)# & /&) '() %#)"&!"#)= %#)+/"&!'( #!"(!*#)
L3
6
L2
2
L2
2
L
A
B
'# '(!'#
A=−
=−
qL
2EI
qL3
=−
6EI
B=
L/4
1
qL2
12EI
C! *(!)#*+#!*,& #/ ?(?#!"( E#*"(% 1&/#
x
x2 qL
qL2
dxdx + EI Ax + EI B = q −
x+
2
2
12
0 2
x
x 2
qL
−6 +1
=
6
12
L
L
M (x) = q
ˆ ˆ
D+# 1&/+&'( #! /() #F"%#?() ; #/ *#!"%( 1&/#
M(x=0) =
qL2
12
M(x= L ) = −
2
qL2
24
M(x=L) = +
qL2
12
G&%& 1#% #/ 1&/(% '# /() ?(?#!"() )(0%# /() #?>("%&?,#!"() '#0# %#*(%'&%)# D+# B&; D+#
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'#%,1&!'( #/ ?(?#!"( (
T (x) = −
ˆ
x
q (x) dx − EI A = −qx +
0
:I
qL
2
!"#$%" &"' ($) *"$++,- ") #"*&,+$(") ." $/-0- *")!(&$ ." "#$(!$* "( +-*&" " (-) "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$) +!$&*- "+!$+,- ") /$*$ -@&" "* ($) +- )&$ &") *")!(&$ $B-*$
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." ($) .-) /*,%"*$)
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0 
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1  C
D
0
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0
2 

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 = − qL  02 

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24EI L 
12

B=D=0
." ($ +!$*&$ /!"." .")/"C$*)" A
A=−
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1
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+
−
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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#.% /. "!$2@ -2$ /6/,-.2 +! a = 0,4
;+/ %-.!7% /0 .% -$!,/$% ,!1%4= /. 4/+-.%>%,!/012 ,EB!,2 27#$$/ /0 /. +/"#042 1$%,2 4204/ /.
%$"#,/012 4/ hi /+ -2+!1!:2
2 a
x 2 x
P L2
φ (x) =
−
− a − (1 − a) (2 − a)
(1 − a)
2EI
L
L
3
.2 7#%. !,-.!7% %0#.%$ /. 72$7C/1/@ /. A#/ 2-/$%042 $/+#.1%
0=
x 2
L
−2
x
L
FG
+
1 2
a +2
3
! "# !
x
L
máx
p
=1−
1 − (a2 + 2) /3 = 0,471
vmáx = −(0,01975)
)&'& !* +&," %&'-.+/*&' ! a =
!, 456.4" 7 1&*!
1
2
P L3
EI
$%&'& a = 0,4(
$+&'0& !# !* +!#-'" ! *& 1.0&(2 !* !,%*&3&4.!#-" !* +!#-'"
vmáx = −
!"#$
1 P L3
48 EI
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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
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6
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x 3
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v (x) = L (φ1 + φ2 )
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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
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L
L
L
L
L
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T (x) = −6 2 {φ1 + φ2 }
L
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du
d
ε (x, z) =
=
dx
dx
χy (x) =
dw (x)
z
dx
dw (x)
−
z
dx
= χy z
dφy
d2 w
=− 2
dx
dx
σ(x, z) = Eε (x, z) = Eχy z
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ˆ
σ (x, z) z dA = Eχy (x)
A
ˆ
z 2 dA = Eχy (x) Iy
A
dTz (x)
+ qz (x) = 0
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dMy (x)
dx
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+ qz (x) = 0
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ˆ
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A
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θ=
dφx
dx
GG
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2
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dφx (x)
qz e
=
x + C = θ (x)
dx
GJ
qz e x 2
+ Cx + D
φx (x) =
GJ 2
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φx (x = 0) = D = φ1
qz e L 2
φx (x = L) =
+ CL + D = φ2
GJ 2
6M
! "# !
D = φ1
C=−
qz e L (φ2 − φ1 )
+
GJ 2
L
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x
(x − L) + φ1 + (φ2 − φ1 )
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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) =
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qz e x
(x − L)
GJ 2
L
P
Mt (x) = qz e x −
2
φPx (x) =
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;'%!
x
(φ2 − φ1 )
L
(φ2 − φ1 )
MtH (x) = GJ
L
φH
x (x) = φ1 +
0 $"1" %' )!%'$-.#
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u (x.y) =
#$%&'()(*+$,-.%
− dv(x)
y
dx
34
v (x)
#$' $/$ #$ '( 0+1( 2 #$%&'()(*+$,-.%
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z
u (x.z) = − dw(x)
dx
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•
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φx
•
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y
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v
w
=
u (y, z)
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'$"/( 8)0)
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


 dv(x) 
 dw(x) 

u (x)
u
u (y, z)
− dx y
− dx z
 +  −φx z 
u (x) =  v  (x, y, z) =  0  +  v (x)  + 
0
0
w
φx y
0
w (x)
N
Mx
Mz
My

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