# Vartiational Formulation

• Marcus Aßmus
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

## Abstract

In the previous chapters, the local behavior of an elastic, threelayered composite structure was derived, which resulted in the description of the initial boundary value problem. The principle of virtual work is a formulation equivalent to the balances of forces and moments, and represents a weak form of the balance (Bathe, Finite-elemente-methoden. Springer, Berlin, 2002, [1]). It is obtained by weighting the equations of motion with test functions equivalent to the vectors of degrees of freedom and subsequent partial integration over the area considered (Oñate, Structural analysis with the finite element method linear statics: vol 2. Beams, plates and shells. Springer, Dordrecht, 2013, [5]). The test function can be interpreted as infinitesimal deformation field (virtual displacements, virtual deflections, and virtual rotations). This field is arbitrary, but must satisfy the geometric boundary conditions and have to be continuously differentiable (Bathe, Finite-elemente-methoden. Springer, Berlin, 2002, [1]). There is no further assumption in this principle (Oden and Reddy, Variational methods in theoretical mechanics. Springer, Berlin, 1983, [4]). The following process serves as a basis for the numerical implementation.

## References

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Bathe KJ (2002) Finite-elemente-methoden, 2nd edn. Springer, Berlin
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Oden JT, Reddy JN (1983) Variational methods in theoretical mechanics, 2nd edn. Springer, Berlin.
5. 5.
Oñate E (2013) Structural analysis with the finite element method linear statics: vol 2. Beams, plates and shells. Springer, Dordrecht.