Professor Fabrice PIERRON LMPF Research Group, ENSAM Chlons en Champagne, France THE VIRTUAL FIELDS...
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Transcript of Professor Fabrice PIERRON LMPF Research Group, ENSAM Chlons en Champagne, France THE VIRTUAL FIELDS...
Professor Fabrice PIERRON
LMPF Research Group, ENSAM Châlons en Champagne, France
THE VIRTUAL FIELDS METHOD
The principle of virtual work
ParisChâlons en Champagne
V
*ii
V
*ii
V
*ijij 0dVufdSuTdV
or
Equilibrium equations (static)
0fij,ij + boundary conditions strong (local)
weak (global)
Valid for any KA virtual fields
Illustration of the PVW
01n
Section SF
e1
e2
l
L0
221
112
2221
1211211 dx.edx.e
01
dx.e.TF
Over element 1
1
F1
211edx
221edx
1
2
3
01
n
Local equilibrium: 0xx 2
12
1
11
21
Forces exerted by 2 over 1
)xL.(FM
F0
F
10e
12
12
3
F
e1
e2 Section S
L0-x1
Resultant of internal forces
2/l
2/l 2211e
12
2/l
2/l 221
2/l
2/l 21112
dxxeM
dxe
dxeF
3
1
F1
211edx
221edx
21 F
e1
e2 Section S
L0-x1
Equilibrium
)xL(Fdxxe
Fdxe
0dx
10
2/l
2/l 2211
2/l
2/l 221
2/l
2/l 211
)xL.(FM
F0
F
10e
12
12
3
2/l
2/l 2211e
12
2/l
2/l 221
2/l
2/l 21112
dxxeM
dxe
dxeF
3
Valid over any section S of the beam: integration over x1
)xL(Fdxxe
Fdxe
0dx
10
2/l
2/l 2211
2/l
2/l 221
2/l
2/l 211
2FLdxdxxe
FLdxdxe
0dxdx
20L
0
2/l
2/l 21211
0
L
0
2/l
2/l 2121
L
0
2/l
2/l 2111
0
0
0
Eq. 1
Eq. 2
Eq. 3
Principle of virtual work (static, no volume forces)
0dSu.TdVfV
*ii
V
*ijij
Let us write a virtual field:
0u
xu*2
1*1
e1
Fe2
L0
l
0
0
1
*12
*22
*11
0dSu.TdVfV
*ii
V
*ijij
0L
0
2/l
2/l 2111V
*1111 dxdxedV 0
0dxdx0L
0
2/l
2/l 2111 Eq. 1
e1
Fe2
L0
l
Let us write another virtual field:
1*2
*1
xu
0u
2/1
0
0
*12
*22
*11
F
e1
e2
L0
l
0dSu.TdVfV
*ii
V
*ijij
0L
0
2/l
2/l 2112V
*1212 dxdxedV2 0L.F
0
L
0
2/l
2/l 2112 FLdxdxe 0 Eq. 2
F
e1
e2
L0
l
F
e1
e2
L0
l
Let us write a 3rd field: virtual bending
2xu
xxu21*
2
21*1
0
0
x
*12
*22
2*11
0dSu.TdVfV
*ii
V
*ijij
0L
0
2/l
2/l 21211V
*1111 dxdxxedV 2
L.F 20
2FLdxdxxe
20L
0
2/l
2/l 212110 Eq. 3
F
e1
e2
L0
l