Formulation of stiffness with elastic support

Finite element method

Stiffness of member ‘e’

$k^{(e)} = \frac{AE}{l} \begin{bmatrix}1 & -1 \\-1 & 1 \end{bmatrix}$

Member stiffness in global coordinate

$\underline{k} = \underline{T}^{T} \hat{k} \underline{T }$

$\underline{k} = \frac{AE}{l} \begin{bmatrix} \underline{\lambda} & -\underline{\lambda} \\ -\underline{\lambda} & \underline{\lambda} \end{bmatrix}$

With

$\underline{\lambda} = \begin{bmatrix} Cx^2 & CxCy & CxCz \\ CyCx & Cy^2 & CyCz \\ CzCx & CzCy & Cz^2 \end{bmatrix}$

$Cx = cos(\theta_x), Cy = cos(\theta_y), Cz = cos(\theta_z)$

Based on superposition method, the global stiffness of the structure becomes $K = k^{(1)} + k^{(2)} + ... + k^{(n)}$

In case 3 spring elements are added to ith node

$k_{spring} = \begin{Bmatrix}k_x \\k_y \\k_z \end{Bmatrix}$

These 3 values will inclue directly into the diagonal terms of global stiffness corresponding to elements i without disturbing local stiffness of other elements. To prove this claim, each spring element are parallel to each axis, and therefore, only one term of Cx or Cy or Cz can be equal to 1. The other ends of the spring are considered fixed, leaving therefore rows and colums 4,5,6 unaffecting the stiffness matrix of the structure.

For example, a spring parallel to X-axis: Cx = 1, Cy = 0, Cz = 0

$\underline{k} = \begin{bmatrix}1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0\end{bmatrix} k_x \begin{bmatrix}1 & -1 \\-1 & 1 \end{bmatrix} \begin{bmatrix}1 & 0 & 0 & 0 &0 & 0 \\ 0 & 0 & 0 & 1 &0 & 0 \end{bmatrix}$

Programming

Remove spring element from the system
Calculate member stiffness in local coordinate
Calculate member stiffness in global coordinate
Add each member’s stiffness to structure’s stiffness matrix
Add spring’s stiffness on associated nodes’ rows and columns, following each axis

To calculate displacement, force, and reaction

Remove rows and columns relative to undisplaced direction of each nodes
Calculate the displacement by displacment method $K U = F$
member’s internal force: based on Hook’s law
reaction force on support and by spring elements: rebuild full stiffness matrix