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7/24/2019 s8 PPD1 Teorie Polin Navier
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PLCI PLANE
DREPTUNGHIULARE
1
7/24/2019 s8 PPD1 Teorie Polin Navier
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ECUAIA FUNDAMENTALa plcilor plane dreptunghiulare
Ipoteze
x
y
z
h
x
yl
l
p x,y( )= ct
yx lla
a
h
,min
20
1
80
1
5
1max h
w
Plci plane subiri cu deformaii mici
7/24/2019 s8 PPD1 Teorie Polin Navier
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Ipoteze
Ipoteza lui Kirchhoff:O dreapt normal la planul
median nainte de deformare rmne dreapt inormal la planul median dup deformare
Planul median este inextensibil n planul su
zeste neglijabil n raport cu x i y
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Aspectul static
xz
x
x
yz
dx
yd
h/2
h/2
zdz
1
1
y
xy
yz
yx
Txz
MxxyM
yzT
yM
Myx
x
y
z
7/24/2019 s8 PPD1 Teorie Polin Navier
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Aspectul static
Txz
Mx
xyMyzT
yM
Myx
x
y
z
7/24/2019 s8 PPD1 Teorie Polin Navier
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Aspectul geometric
y
w
x
w
y
x
x, y
z
h
w
u, v
x
y
,
z
zv
zu
y
x
7/24/2019 s8 PPD1 Teorie Polin Navier
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Ecuaia Sophie Germain - Lagrange
D
yxpyxw
,,22
2
2
2
22
yx
D
yxp
y
w
yx
w
x
w ,2
4
4
22
4
4
4
ecuaie diferenial liniar de ordinul IV,neomogen, cu coeficieni constani
23
112
EhD
rigiditatea cilindricla ncovoiere a plcii
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Eforturi secionale
2
2
2
2
y
w
x
wDMx
2
2
2
2
xw
ywDMy
wx
Dy
M
x
MT
yxxxz
2
wy
D
y
M
x
MT
yxyyz
2
momente ncovoietoare
momente de torsiune yx
wDMM yxxy
2
1
fore tietoare
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CONDIII DE CONTUR
x
y
z
h
b
aO A
B C
latur ncastrat
latur simplurezemat
latur ncastratelastic
latur liber
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Condiii de conturx
y
z
h
b
aO A
B C
0w 0
y
wy 02
2
x
wx
002
yx
w
x y
2
2
y
wDMx
2
2
y
wDMy
0xyM
yx MM
3
3
y
wD
y
MT
yyz
Latura ncastrat OA (y = 0)
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Condiii de conturx
y
z
h
b
aO A
B C
0w 0xM 022
2
2
y
w
x
w
02
2
ywy
02
2
x
w
Latura simplu rezemat OB (x= 0)
Fora tietoare generalizat
010022
yx
wDM
yx
w
y xyxyx
2
3
3
3* 2
yx
w
x
wD
y
MTT
xy
xzxz
laturiilunguln.ctx
y
z
xyM dy
dyxyM
dy
(M + ...)xy
(M + )xy
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Condiii de conturx
y
z
h
b
aO A
B C
Latura cu rezemare elastic AC (x = a)
y
xzR
Mx
Rxz
xM
-solicitarea la ncovoiere a grinzii n planul yOz :
xzRxq
EI
xq
y
w
;
4
4
gr
xz
EI
R
y
w
4
4
2
3
3
3
4
4
2yx
w
x
w
EI
D
y
w
gr
-solicitarea la torsiune a grinzii:
y
xtt
t dyyMMGI
M0
; xt
t
t
MGIdy
dM
GIy
11
2
2
2
2
2
3
y
w
x
w
GI
D
yx
w
t
yx
w
y
x
2
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Condiii de conturx
y
z
h
b
aO A
B C
Latura liber BC (y = b)
yx
w
y
wD
x
MTR
yx
yzyz 2
3
3
3
2
Fora tietoare generalizat
0,0 yzy RM sau 00 , QRMM yzy
DM
xw
yw 0
2
2
2
2
/0
D
Q
yx
w
y
w 02
3
3
3
/02
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- plci libere de legturi, ncrcate pe contur cu momente ncovoietoarei/sau de torsiune distribuite uniform
0,22 yxw feydxcybxyaxyxw 22,
REZOLVAREA PLCILOR PLANEDREPTUNGHIULARE
Soluii exacte cu polinoame algebrice:
caDy
w
x
wDMx
2
2
2
2
2
acDx
w
y
wDMy
2
2
2
2
2
bDyx
wDMxy
11
2
1M
2M12M
21M
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0M
0M
Exemple:O
b /2
y
b /2
x
z
a /2 /2a
placa supus la ncovoiere
0; 12021 MMMM
012
0
bD
Mca
feydxyx
D
Mw
220
12
.ctyx
paraboloid de revoluie
d = e = f = 0 dac se consider c n punctul O (lax=0 iy=0)
0w
0x
w
0
y
w
220
12yx
D
Mw
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placa supus la torsiune
01221 ;0 MMMM
1
0
0
D
Mb
ca
xy
D
Mw
1
0 .ctxy
0M
0M
M02 2M0
M2 02M0
paraboloid hiperbolic
d = e = f = 0
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serii duble trigonometriceNavier:
plciplane dreptunghiulare simplu rezemate pe contur; origineasistemului de axe trebuie sfie ntr-un colal plcii
depinde dencrcare
ncrcaredistribuitsinusoidal ncrcaredistribuituniform, localsau pe toat suprafaa plcii ncrcare distribuit liniar forconcentrat
ybn
xa
m
wyxwn
mnm
11 sinsin,
mnw ybn
xa
mpyxp
nmn
m
11
sinsin,
ydxdybn
xa
m
yxpabp
ba
mn
sinsin,
4
00
REZOLVAREA PLCILOR PLANEDREPTUNGHIULARE
Soluii cu serii Fourier = exacte pentru un numr infinit de termeni
7/24/2019 s8 PPD1 Teorie Polin Navier
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serii simple trigonometrice: Lvy plci plane dreptunghiulare simplu rezemate pe dou laturi paralele,
orice tip de rezemare pe celelalte laturi; axa pe laturile simplu
rezemate trebuie s fie ax de simetrie
yxwyxwyxw p ,,, 0 0,0
22 yxw
D
yxpyxwp
,,22
02sin,4
44
2
22*
10
mm
IVm
mm Y
amY
amYx
amyYyxw
a
myyDyyCyByAyY mmmmmmmmmm
,shchshch)(
(*)
(**)
analog grinda-perete
dac
D
xp
dx
wdxpp
p 4
4**dezvoltare n serie simpl trigonometric
Obs. Seriile simple trigonometrice sunt mult mai rapid convergente dectseriile duble trigonometrice.
7/24/2019 s8 PPD1 Teorie Polin Navier
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SOLUII APROXIMATIVE
metode variaionale: Ritz - Galerkinteorema minimului energiei poteniale totale Galerkinprincipiul lucrului mecanic virtual
metode numerice: metoda diferenelor finite metoda elementului finit metoda fiilor finite
7/24/2019 s8 PPD1 Teorie Polin Navier
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ax
y
b
z
A B
C D
p( )x,yAPLICAREASOLUIEINAVIER
,3,2,1,;;sinsin,11
nminecunoscuicoeficienwyb
nx
a
mwyxw mn
nmn
m
Verificarea condiiilor de margine:AB, CD(y= 0,y= b)
0
0sinsin
00
12
22
12
2
0
2
2
2
2
y
nmn
m
y
y
M
yb
nx
a
m
b
nw
y
w
x
w
y
wDM
Mw
AC,BD(x= 0,x= a)
0
0sinsin
00
12
22
12
2
0
2
2
2
2
x
nmn
m
x
x
M
yb
nx
a
m
a
mw
x
w
yw
xwDM
Mw
D
yxpyxw
,,22
2
3
112;
Eh
D
2224
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yb
nx
a
m
a
mw
x
w
nmn
m
1
2
2
22
14
4
sinsin
yb
nx
a
m
b
n
a
mw
yx
w
nmn
m
12
22
2
22
122
4
sinsin
yb
nx
a
m
b
nw
y
w
nmn
m
1
2
2
22
14
4
sinsin
y
b
nx
a
m
b
n
a
mww
nmn
m
1
2
2
22
2
22
1
22 sinsin
D
yxp ,
Dezvoltarea ncrcrii n serie dubl trigonometric:
yb
nx
a
mpyxp
nmn
m
11
sinsin,
D
p
b
n
a
mw mnmn
2
2
2
2
24 dxdy
b
nx
a
myxp
abp
ba
mn
sinsin,
4
00
I
bamn
mn ydxdyb
nx
a
myxp
bn
amDab
bn
am
p
Dw
sinsin,
41
002
2
2
2
2
4
2
2
2
2
24
a
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Exemplu
ax
y
b
x0
0y
c c
dd
z
pctyxp ,
dydyy
cxcxx
00
00
,
,
21sinsin0
0
0
0
IpIdxdyb
nx
a
mpI
dy
dy
cx
cx
ca
mx
a
m
m
a
a
m
xa
m
xdxa
mI
cx
cx
cx
cx
sinsin2cos
sin 01
0
0
0
0
dbny
bn
nb
b
n
y
b
n
ydybnI
dy
dy
dy
dy
sinsin2
cos
sin 02
0
0
0
0
db
ny
b
nc
a
mx
a
m
b
n
a
mDmn
pwmn
sinsinsinsin16
002
2
2
2
26
b
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Caz particular: ncrcare distribuit uniform pe toat placa2
;2
00
bdy
acx
,5,3,1,;16
2
2
2
2
26
nm
b
n
a
mDmn
p
,5,3,1,;sinsin
16,
12
2
2
2
21
6
nm
b
n
a
mmn
yb
nx
a
m
D
pyxw
nm
,5,3,1,;sinsin16
12
2
2
2
2
2
2
2
2
142
2
2
2
nmyb
nx
a
m
b
n
a
mmn
bn
amp
y
w
x
wDM
nm
x
,5,3,1,;sinsin161
2
2
2
2
2
2
2
2
2
142
2
2
2
nmy
bnx
am
b
n
a
mmn
a
m
b
n
pxw
ywDM
nm
y
,5,3,1,;coscos
1611
1
2
2
2
2
2
1
4
2
nm
bn
am
yb
nx
a
m
ab
p
yx
wDM
nm
xy
2sin
2sin
16 222
2
2
2
26
nm
b
n
a
mDmn
pwmn
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Plac ptrat din oel (a = b; = 0,3):
,5,3,1,;
sinsin16
,1
2221
6
4
nmnmmn
yb
nx
a
m
D
payxw
nm
,5,3,1,;sinsin
16
1222
22
14
2
nmyb
nx
a
m
nmmn
nmpaMM
nm
yx
,5,3,1,;
coscos16
1 1 22214
2
nmnm
yb
nx
a
m
pa
M nmxy
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n centrul plcii, la x=y= a/2:
,5,3,1,;16
116
4
max
nmtD
paw
n
mn
m
2221
2
222
12sin
2sin
nmmnnmmn
nm
t
nm
mn
D
pa
ttttttttttttttttD
pat
D
pa
D
pattttttttt
D
pat
D
pa
D
patttt
D
pat
D
pa
Dpat
D
patD
pa
w
nmn
m
n
mn
m
nmn
m
nmn
m
4
777557733771175553355115333113116
47
1
7
16
4
4
5553355115333113116
45
1
5
1
6
4
4
333113116
43
1
3
16
4
4
116
41
1
1
16
4
max
004045931,0
1616
0040647,01616
004055,01616
004161,01616
,5,3,1,;16
114
2
max,max,
nmt
paMM
n
mn
m
yx
222221
2
222
22 1
2sin
2sin
nmmn
nmnm
nmmn
nmt
nm
mn
2
2
2
max
0482,0
04692,0
05338,0
pa
pa
pa
M
0,5,3,1,;2cos
2
cos
1611
2221
4
2
xy
nm
xy Mnmnm
nm
paM
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La colurile plcii:
A(0,0):
,5,3,1,;16
1
11
4
2
nmtpa
M
n
mn
m
xy
22
2
222
032,0
0314,0
02877,01
pa
pa
pa
Mnm
t Axymn
B(a,0):
A
xy
B
xymn MMnm
t
222
1
C(0,a):
AxyCxymn MM
nmt
222
1
D(a,a):
A
xy
D
xymn MM
nm
t
222
1
n axele de simetrie:
x=a/2sauy = b/2
Mxy
= 0
ax
y
a
Mx yM
Mmax pa2= 0,047
y
a
ax
xMy
0,032pa
2
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S se determine wmax,Mmax iMxy,max pentru o placptrat din beton armat
simplu rezemat pe contur cu
a= 6 m;E= 3e7 kN/m2; = 0,2; h= 14 cm;p= 20 kN/m2
lund m= n= 7, respectiv 50 i 100.
Sse compare soluiile cu cele furnizate de MEF.
Tem: termen de predareS12 (17.12.2014)