Professor Fabrice PIERRON LMPF Research Group, ENSAM Châlons en Champagne, France THE VIRTUAL FIELDS METHOD The principle of virtual work Paris Châlons en Champagne

or Equilibrium equations (static) + boundary conditions strong (local) weak (global) Valid for any KA virtual fields

Illustration of the PVW Section S F e1e1 e2e2 l L0L0

Over element 1 1F1F Local equilibrium:

21 Forces exerted by 2 over 1 F e1e1 e2e2 Section S L 0 -x 1

Resultant of internal forces 1 F1F1 21 F e1e1 e2e2 Section S L 0 -x 1

Equilibrium

Valid over any section S of the beam: integration over x 1 Eq. 1 Eq. 2 Eq. 3

Principle of virtual work (static, no volume forces) Let us write a virtual field: e1e1 F e2e2 L0L0 l

Eq. 1 e1e1 F e2e2 L0L0 l

Let us write another virtual field: F e1e1 e2e2 L0L0 l

Eq. 2 F e1e1 e2e2 L0L0 l

F e1e1 e2e2 L0L0 l Let us write a 3rd field: virtual bending

Eq. 3 F e1e1 e2e2 L0L0 l