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