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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

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fea simulations
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
FEA Simulations (Contintued)
  • Solution of governing equations
    • Static: solution of simultaneous 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
Principal of Virtual Work

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

principal of virtual work continued
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



is the strain displacement matrix.

energy minimization
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 , are the moduli, and


is the same as virtual work.

finite elements
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 (i.e., integrate) the energy (or virtual work) of

each element to get the total energy of all elements.

finite elements continued
Finite Elements (continued)

For example:

For element 1:

For element 2:


interpolation function
Interpolation Function
  • Isoparametric elements have

same function for displacements

as the coordinates

  • Transform to an element with

coordinates at nodes

interpolation function continued
Interpolation Function (continued)
  • 2D isoparametric element
  • 1D isoparametric element
  • 3D isoparametric element in a similar manner
volume integration
Volume Integration
  • For the parallelepiped the

the shaded area is

  • The volume is

- i.e. a determinant

  • For a rectangular


  • The volume is
volume integration continued
Volume Integration (continued)
  • Using the isoparametric coordinates
  • Use similar expressions for
  • The volume becomes
  • The determinant is

the Jacobian

gaussian quadrature
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


gaussian quadrature continued
Gaussian Quadrature (continued)
  • Then the integral becomes
  • We can write the integral as
governing equations
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
boundary conditions
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
Boundary Conditions (continued)
  • The energy minimum
  • Leads to
  • The governing equations are based on virtual work or the minimization of energy.
  • Each displacement functions are zero outside its element.
  • 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.