Civil Engineering Reference
In-Depth Information
As a result, the strain and stress components are constant within the element.
Various procedures such as the virtual work principle, variational approaches such
as the principle of minimizing the potential energy, and residual methods can be
used to set up the equations to determine the nodal displacements. A detailed de-
scription of the various procedures can be found, for example, in Zienkiewicz (1977)
and Bathe & Wilson (1976).
For the simple example under consideration this will be accomplished by minimizing
the potential energy U of the element given by the difference of the elastic strain energy
E and the work W, resulting from the forces applied to the nodes of the element:
U = E - W = Min!
(10.6)
where
(10.7)
(10.8)
The minimum of U is obtained by setting the derivations of U with respect to the com-
ponents of nodal displacements equal to zero:
(10.9)
This yields a linear system of equations describing the relationship between the
components of the nodal displacements and the corresponding components of
the nodal forces:
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