1. FEA Simulations Boundary conditions are applied
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
Force or stress (i.e., pressure or shear)
Displacement
Multi-point constraints

2. FEA Simulations (Contintued)
Solution of governing equations
Static: solution of simulataneous equations
Vibrations: eigenvalue analysis
Transient: Numerically step through time
Nonlinear: includes buckling uses an iterative solution
Evaluation of stress and strain
Post-processing: e.g., contour or history plots

3. Principal of Virtual Work
Change in energy for a “virtual” displacement, , in the structure of volume, V , with surface, S, is where 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

4. Principal of Virtual Work (continued)
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 and or is the strain displacement matrix.

5. Energy Minimization
Let where U is the internal energy and V is the potential energy due to the loads and are given by and . Recall then and and the variation becomes . Since , and ,
is the same as virtual work.

6. Finite Elements Assume displacements inside
an element is a linear (or quadratic)
function of the displacements of
the nodes of each element.
Assume the function outside
each element is zero.
Add the displacements of each
element to get the displacements
of the nodes.

7. Finite Elements (continued)
For example: For element 1: For element 2: Hence

8. Interpolation Function
Isoparametric elements have
same function for displacements
and coordinates
Transform to an element with
coordinates at nodes
Node x h 1 -1 2 3 4

9. Interpolation Function (continued)
2D isoparametric element
1D isoparametric element
3D isoparametric element in a similar manner

10. For the parallelepiped the the shaded area is The volume is
Volume Integration
For the parallelepiped the
the shaded area is
The volume is
- i.e. a determinant
For a rectangular
parallelepiped

11. Volume Integration (continued)
Using the isoparametric coordinates
Use similar expressions for
The volume becomes
*Determinant is Jacobian
of coordinates

12. Gaussian Quadrature and
Numerical integration is more efficient if both the multiplying factor and the location of the integration points are specified by the integration rule.
The rule for integration along x can be expanded to include y and z.
For example, in one dimension, let
Transform to isoparametric coordinates
and

13. Gaussian Quadrature (continued)
Then the integral becomes
We can write the integral as

14. Gaussian Quadrature (continued)
Exact for polynomial of degree n k Ak xk 1 2 3
5/9 5
8/9

15. Governing Equations Use virtual work or energy minimization
Sum over each element since each element has no influence outside its boundary
Let where is the vector of nodal displacements for element k
Then the strains are

16. Governing Equations (continued)
Then
Let
And
Hence

17. Boundary Conditions Surface forces have been included
So far the FE model is not restrained from rigid body motion
Hence displacement boundary conditions are needed
Recall
Let the displacement constraints be
Develop an augmented energy

18. Boundary Conditions (continued)
The energy minimum
Leads to

19. Summary
The governing equations are based on virtual work or the minimization of energy.
Displacement functions are zero outside elements.
Isoparametric elements use the same function for the coordinates and displacements inside elements
A Jacobian is required to complete integrations over the internal coordinates.
Gaussian quadrature is typically used for integrations. Hence stresses and strains are calculated at integration points.
Enforced displacements can be included through the use of Lagrange multipliers.