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General FEA Equations

The main FEA equation is given by:

$$ K_gq_g = F_g + T_g + P_g $$

Where:

  • $K_g$ is the global stiffness matrix [DOFs x DOFs]
  • $q_g$ is the global displacement matrix [DOFs x 1]
  • $F_g$ is the global force matrix [DOFs x 1]
  • $T_g$ is the global traction force matrix [DOFs x 1]
  • $P_g$ is the global body force matrix [DOFs x 1]
  • DOFs = Problem Size = Number of Nodes in Problem * DOFs in Each Node

In this equation displacement is generally the unknown to be solved for, except in some cases where a displacement is fixed and the force (reaction force) is solved for instead.

General Element Equations

The following section describes generalized equations that apply to all mechanical FEA elements. For information on specific elements, refer to the following:

There are two general parameters that determine the size of matricies for an element:

  • Node DOFs = Number of Nodes in Element * DOFs in Each Node
  • Element DOFs = Number of local directions inside element (1, 2, 3)

Shape Function

$$ u = Nq $$

Where:

  • $u$ is the local displacement matrix inside the element [Element DOFs x 1]
  • $N$ is the element shape factor matrix [Element DOFs x Node DOFs]
  • $q$ is the node displacement matrix [Node DOFs x 1]

Strain Calculation

Note: size of the stress and strain matricies varies depending on the dimension of the element:

  • 1 Dimension: [1 x 1]
  • 2 Dimension: [3 x 1]
  • 3 Dimension: [6 x 1]

The B matrix size varies accordingly, depending on the number of nodes and size of the output matrix.

$$ \epsilon = Bq $$

Where:

  • $\epsilon$ is the element strain matrix [1 x 1], [3 x 1], [6 x 1]
  • $B$ is the strain/displacement matrix [1 x Node DOFs], [3 x Node DOFs], [6 x Node DOFs]
  • $q$ is the node displacement matrix [Node DOFs x 1]

Stress Calculation

$$ \sigma = D\epsilon = DBq $$

Where:

  • $\sigma$ is the element stress matrix [1 x 1], [3 x 1], [6 x 1]
  • $D$ is the material constitutive matrix [1 x 1], [3 x 3], [6 x 6]
  • $q$ is the node displacement matrix [Node DOFs x 1]

Stiffness Calculation

Element stiffness is determined integrating the $B$ and $D$ matricies over the element's volume. In many cases these matricies do not depend on position inside the element, and the integral simply becomes the element's volume.

$$ K = \int_{V}^{} B^TDB ,dV $$

Where:

  • $K$ is the element stiffness matrix [Node DOFs x Node DOFs]
  • $V$ is the element's volume