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SAVIJANJE KRUZNE CILINDRICNE LJUSKE PRI ROTACIONO SIMETRICNOM OPTERECENJU -uslovi ravnoteze-
0)2(
0)1(
=++
=-
ZdxdT
aN
TxdMj K
Zwdx
wdDJ =+ 444
4: b
-deformacija-
dxduz
dxwdxz +-= 2
2
e imamo sile:jjjj
jjMNTxN
MNTxMx===
,,,
awz -=ej
dxdMxTx
awEhN
dxwdKMx
=
-=-= j;22
)sin4cos3()sin2cos1(0 xCxCexCxCeww
xx bbbb bb ++++= - -duga cilindricna ljuska-(l>=5)
( )[ ]
[ ]
[ ]
[ ]
EhaxZw
xxToxMoeKdx
wd
xToxxMoeKdx
wd
xxToxMoeKdx
dw
xMoxMoToeK
ww
ha
x
x
x
x
a
2
0
3
3
2
2
2
30
42
2
)(
)sin(cossin21
sin)sin(cos1
)sin(coscos22
1
sincos2
1
)1(3
-=
--=
++-=
++=
-+-=
-=
-
-
-
-
bbbb
bbbbb
bbbbb
bbbb
ub
b
b
b
b
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SAVIJANJE KRUZNIH PLOCA -rotaciono simetricno opterecenje i granicni uslovi: DJ:
KrZ
drdw
rdrwd
rdrwd
rdrwd )(112
32
2
23
3
4
4
=+-+
w=wo+w1 w1=A+Blnr+Cr2+Dr2lnr w1=C1+C2ln+C32+C42ln (=r/a)
( )
( ) 44
000
0
64
int1
rKaZ
wconstZrZ
egralnipartikulardrrMrdr
Kw
drrrZrdrM
===
--=
-=
SILE U PRESEKU:
+--=
+-=
+-=
2
2
23
3
2
2
2
2
11
1
drwd
rdrdw
rdrwdKTr
drwd
drdw
rKM
drdw
rdrwdKMr
uj
u
-proizvoljno opterecenje:
DJ: KrZ
dwd
rdrdw
rdrwd
dd
rdrd
rdrd ),(1111
2
2
22
2
2
2
22
2 jjj
=
++
++
w=wo+w1
w1= wo(r)+ ( ) ( ) ( ) ( )
=
=
+1
___
0coscos
mmmrmWmrWm jj
wo(r)-izraz za rotaciono simetricno opt(m=0), nista ne zavisi od -smena r=et =>DJ=>j-na sa konstantnim koeficijentima m>1: 22 +-+- +++= mmmm rDmrCmrBmrAmWm m=1: rrD
rCrBrAW ln1 11
3111 +++=
m=0: rrDrCrBAW lnln 2002000 +++=
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SFERNA LJUSKA OPTERECENA SILAMA PO KONTURI
( ) agaj sin2 00 EhRR ==D -promena poluprecnika paralelnog kruga
-obrtanje preseka po konturi
( )213 ug -=ha
SFERNA LJUSKA OPTERECENA MOMENTIMA PO KONTURI
( )Eh
R agaj sin22
0 ==D - promena poluprecnika paralelnog kruga
( )g
ajcKa
-== - obrtanje preseka po konturi
( )23
112 u-=
EhK
( )Eh
agajc
sin22
-==
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STANjE: 0P
( )
( ) ( )
pravcuradijalunjezaoptereceE
NZa
Na
NaPN
P
YZ
.__0
0)3(
0sin2
)(.........
......
10
2
0
-=
=++
-=
=
==
d
qqj
jpj
j
jj
( )jqqq
ue
ed
NNEh
REREE
-=
=D=1
0010
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DIFERENCNI POSTUPAK -pravougaone ploce-
DJ: ( )K
yxzdy
wddydxwd
dxwd ,2 4
4
22
4
4
4
=++
42112
4
4
32112
3
3
211
2
2
11
464
222
2
2
Sxwwwww
dxwd
Sxwwww
dxwd
Sxwww
dxwd
Sxww
dxdw
kkkkk
k
kkkk
k
kkk
k
kk
k
--++
--++
-+
-+
+-+-
-+-
+-
-
44
4
33
3
22
2
464
222
2
2
Sywwwww
dywd
Sywwww
dywd
Sywww
dywd
Syww
dydw
hiklm
k
hilm
k
ikl
k
il
k
+-+-
-+-
+-
-
22111111
22
4
2111111
2
3
21111
2
3
11112
)(24
222
222
4
SySxwwwwwwwww
dydxwd
SxSywwwwww
dxdywd
SySxwwwwww
dydxwd
SxSywwww
dxdywd
iillkkilk
k
iklikl
k
iiilll
k
iill
k
-+-+-+
---+++
-+-+
-+-+
+++++++-
-+-+-
++-+-
+--
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granicni uslovi -slobodno oslonjena ivica
kk ww -=+2 -ukljestenje
kk ww =+2 -slobodna ivica
( )( )[ ]( ) ( )( )111122
112
___
2222)2(
22)1()2....(0,
)1....(0,
-+-+
-+
+++-+--++=
++--=
=
=
lliiilhm
kkkikl
WWWWWWWWWWWWWW
kyT
kMy
uaua
ua
-kruzna ploca-
mrsm =l
-u svakoj tacki ploce u kojoj nam je nepoznat ugib pisemo DJ:
( ) ( )
( ) ( ) ( )K
Zmsww
wsKcwWm
mmmmm
m
mmmmm
mm
4
21
2
421
2
2
122
22
2622
22)1(
=-
++--
-
+++
-++-+
-+-
++
lll
l
lll
ll
Zm-vrednost povrsinskog opterecenja u tacki m. - za r=0:
=
++- 0
412 163
643
16 wsKcww
KsZ 40
PRESECNE SILE
( ) ( )[ ]2212123
112
112
224222
,
212
2,
212
21,
--++
-+
-+
-+-+-+---=
-+-
+-=
-+-
+-=
mmmmmmmmmm
mmm
mm
mmm
mm
wwwwwSKmTr
wwwSKmM
wwwSKmMr
lllll
lu
luj
ulul
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KRUZNI PRSTEN -RADIJALNA SILA -imamo samo normalnu silu N
auEbhN
bhNXaN ==== ees1
-RASPODELJENI MOMENTI
auE
bhIyI
MaXM
==
===
es
e
s12
3
2
1212
'1222
1111
=
=
=
=
--==
=
Xa
aaX
Xb
ud
uuu
d
cd
d
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MEMBRANSKA TEORIJA LJUSKI ROTACIONO SIMETRICNO OPTERECENJE:
021
)3(
010cos11)0()2(
010cos1)0(1)1(
=++
=+-+
=+++
ZRN
RN
RYRRNRd
dNRN
dd
RXRRNRNddR
ddN
qj
qjq
j
jqjqq
jqj
jjq
PROIZVOLJNI OPT.
-rotaciono simetricno opterecenje-
( ) 0;,;0 == jqj NfZYX SFERA
( )
21
sin2)(
201sincos)(0
RZRNN
RoPN
dRRYZP
+-=
-=
+=
jq
jpj
j
jpjjjj
KONUS
( )
21
cos2)(
cossincos
2)(0
RZRNN
RozPN
dzzYZtgzPz
+-=
-=
+=
jq
apj
aaa
ap
-ugao u vrhu konusa
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ROTACIONO SIMETRICNA DEFORMACIJA ROTACIONO SIMETRICNE LJUSKE
( )
( ) ( )
[ ] jj
juquj
jc
jjuqeq
jqujej
sinsin
12211
11
211
111
++-+=
+=
-==-=
-=-=
CdRRNRRN
Ehv
ddwv
R
wctgvRRo
dRoNNEh
wddv
RNN
Eh
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MEMBRANSKA TEORIJA CILINDRICNIH LJUSKI -uslovi ravnoteze-
=+
=++
=++=++
0)3(
0)2(
0)1(0)1(
ZR
N
Ydx
xdNRddN
XRd
dNxdx
dNxXRd
dNxdx
dNx
j
jjj
jj
jj
( )
+-
+=
+
+-=
-=
=
+
+-=
+
+-=
-=
)(2)(112
1
)(1
,;0
)(2
)(1
2
jjj
jj
j
jjj
j
j
j
jjj
jjj
j
j
Cxd
dCR
xRddNY
dd
RNx
CxRddNYxN
RZN
fZYzaX
CdxRd
xdNXNx
CdxRddNYxN
RZN
-kruzna cilindricna ljuska- R=const=a -opterecenje razvijamo u red
=
=
=
===1 11
;cos;sin;cosn nn
nZnZnYnYnXnX jjj
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-zadrzavamo samo n-te clanove:
( )
( )[ ] )(2)(11cos1)(1sin
jj
jj
jjj
j
Cd
dCa
dxdxZnnYnnaXnna
Nx
CdxZnnYnnxN
aZnN
+-+-=
++-=
-=
-specijalan slucaj-
( )
( )[ ]
( ) j
jjj
jj
j
nAxAxZnnYnanNx
nAxZnnYnxNaZnN
nACnACfZnYnzaXn
cos212
sin1
cos22sin11,;0
2
+
-+=
++-=-=
==
=
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NAPREZANJE U RAVNI U POLARNIM KOORDINATAMA
jj
j
j
j
drdFd
rddF
rN
drFdN
drdF
rdFd
rNr
r
2
2
2
2
2
2
2
11
;
;11
-=
=
+=
ROTACIONO SIMETRICNO OPTERECENJE:
0;0
);ln23(21;
);ln21(21;1
22
2
2
==
=+++-==
=+++==
jj
jjj
g
e
e
rr
r
NrurCB
rAN
drFdN
drdurCB
rANr
drdF
rNr
F=D+Alnr+Br2+Cr 2lnr D=0(ne utice na naprezanje) C=0(za kruzni prsten uz uslov kompatibilnosti) dr
drr jje
ee += A i C=0 za ploce koje 'imaju centar'
Uticaji od temperature: U=ejr ( ) ttNrN
Eh+-= aujej
1 ulazi u prelazne uslove (UI=UII)npr.
( ) ( )( ) ( )aabbCabCBabrdrrCB
rAM
b
a
lnlnln1ln2321 22222 -+-++-=
+++-=-
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NAVIER-OVO RESENJE )sin()sin(),(
1 1 byn
axmAmnyxw
m n
pp
=
=
=
AmnZmn
Kp4 n2
b2m2
a2+
2
Zmn4
a b0
a
y
0
b
xZ x y,( ) sinmpx
a
sinnpyb
d
d
:=
MORIS-LEVY W=W1+W0
aynshDn
aynCn
aynchBn
aynAnyYn pppp
++
+=)(
( ) dxa
xnxZa
Zn
KnaZnWon
axnWonWo
axnyYnW
a
n
n
pp
p
p
sin2
sin
sin)(
0
44
41
11
=
=
=
=
=
=
( )
axn
aynshDn
aynCn
aynchBn
aynAn
KnaZn
WWyxW
n
pppppp
sin
10,
144
4
++
++
=+=
=
- simetricno opterecenje An i Dn - antimetricno opterecenje Bn i Cn (g.u. Y=+-b/2)
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PLOCA OSLONJENA NA GREDU
( )
+-===
-+
-==
=======
2
2
2
22
2
3
3
3___
4
4
2
44
)2(
2)1(
)2////()1(
dxWpld
dyWpldKMy
dxdMt
dxdyWpldGIt
dxd
dydxWpld
dyWpld
EIgK
EIgTy
dxWpld
GItMt
dxgd
dxdyWpldg
dydWpl
dxdWgr
dxdWplwwY gp
u
u
qj
j
SILE U PRESEKU
( )
( )
( )
-+-=
-+-=
--=
+-=
+-=
dydxWd
dyWdKTy
dxdyWd
dxWdKTx
dxdyWdKMxy
dxWd
dyWdKMy
dyWd
dxWdKMx
2
3
3
3___
2
3
3
3___
2
2
2
2
2
2
2
2
2
2
2
1
u
u
u
u
u
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PLOCE NAPREGNUTE U SVOJOJ RAVNI
- naponska f-ja F:
- 22
2
22
;; dx
FdNydy
FdNxdxdy
FdNxy ==-= ; ako su X i Y=0
DJ:za plocu opterecenu silama po konturi:
02 44
22
4
4
4
=++dy
FddydxFd
dxFd (za plocu opterecenu silama po konturi)
- konturni uslovi:
dsdx
dsdy
-== aa sin;cos
)sin(cos
)()(00
aa -+=
=-+-=+=
-=-=
==
dydF
dxdF
dndF
MdsyypdsxxpFdydydFdx
dxdFdF
QydspdxdF
QxdspdydF
s
snx
s
sny
sny
snx
-Naponska f-ja mora da zadovolji DJ. -opterecenje po konturi na osnovu F(x,y):
aa
aa
cossin
sincos
2
2
2
2
2
2
dxdyFd
dxFdp
dsdyNxy
dsdxNyp
dxdyFd
dyFdp
dsdxNxy
dsdyNxp
nyny
nxnx
--=+=
+=+=
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-SILE U PRESEKU:
( )
+-=
+=
+=
dxdv
dyduDNx
dxdu
dydvDNy
dydv
dxduDNx
u
u
u
121
( )21 u-=EhD
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POLURAVAN -opterecenje(periodicno) razvijamo u sin ili cos red sa periodom L. -parno opterecenje:
=
=
+=
=
2/
0
2/
00
1
0
cos)(4
)(4
cos2
)(
L
n
Ln
n
dxa
xnxpL
a
dxxpL
a
axna
axp
p
p
naponska f-ja:
=
+=1
2 cos)(2 n
o
axnyYnx
AF p
-neparno opterecenje
=
=
=
2/
0
1
sin)(4
sin)(
L
n
nn
dxa
xnxpL
b
axnbxp
p
p
naponska f-ja:
=
=1
sin)(n a
xnyYnF p
ayn
ayn
eDna
ynCneBna
ynAnyYnpp pp
++
+=
-)(
Cn i Dn=0 za poluravan GRANICNE USLOVE POSTAVLJAMO PO SILAMA:
;;;2
2
2
2
2
dxdyFdNxy
dxFdNy
dyFdNx -===
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SAVIJANJE KRUZNE CILINDRICNE.pdfSAVIJANJE KRUZNIH PLOCA.pdfsferna ljuska.pdfDIFERENCNI POSTUPAK 1.pdfDIFERENCNI POSTUPAK 2.pdfKRUZNI PRSTEN.pdfljuske-rot-sim.pdfMEMBRANSKA TEORIJA CILINDRICNIH.pdfNAPREZANJE U RAVNI U POLARNIM KOORDINATAMA.pdfNAVIER-MORIS.pdfPLOCE NAPREGNUTE U SVOJOJ RAVNI.pdfPOLURAVAN.pdf