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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

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

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

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.

Energy Minimization

Let where U is the internal energy and Vis

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.

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 all nodes.

Interpolation Function

- Isoparametric elements have
same function displacements

as the coordinates

- Transform to an element with
coordinates at nodes

Interpolation Function (continued)

- 2D isoparametric element
- 1D isoparametric element
- 3D isoparametric element in a similar manner

Volume Integration

- For the parallelepiped the
the shaded area is

- The volume is
- i.e. a determinant

- For a rectangular
parallelepiped

- The volume is

Volume Integration (continued)

- Using the isoparametric coordinates
- Use similar expressions for
- The volume becomes

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

Gaussian Quadrature (continued)

- Then the integral becomes
- We can write the integral as

Gaussian Quadrature (continued)

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

Governing Equations (continued)

- Then
- Let
- And
- Hence

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

Boundary Conditions (continued)

- The energy minimum
- Leads to

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.

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