FEA Simulations. Usually based on energy minimum or virtual work Component of interest is divided into small parts 1D elements for beam or truss structures 2D elements for plate or shell structures 3D elements for solids Boundary conditions are applied
Change in energy for a “virtual” displacement, , in
the structure of volume, V , with surface, S, is
is the energy change
is the virtual displacement
is the internal stress
is the virtual strain due to
are the body forces (e.g., gravity, centrifugal)
are surface tractions(e.g., pressure, friction)
are point loads
The principal of virtual work must hold for all possible
virtual displacements. We must convert at the nodes
to in the elements.
For example if we have a beam we can take
is the strain displacement matrix.
Let where U is the internal energy and Vis
the potential energy due to the loads and are given by
Recall then and
and the variation becomes
Since , are the moduli, and
is the same as virtual work.
an element is a linear (or quadratic)
function of the displacements of
the nodes of each element.
each element is zero.
each element to get the total energy of all elements.
For element 1:
For element 2:
same function for displacements
as the coordinates
coordinates at nodes
the shaded area is
- i.e. a determinant