1. 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 – 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  1 21 311 4 1

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

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

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

12. Gaussian Quadrature 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 nkAkAk kk 11201 21 1 3 21 31 5/9 5 28/90 35/9

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.