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Illustration of the PVW Section S F e1e1 e2e2 l L0L0
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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