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Releases

Releases are applied at the nodes (or pseudo nodes in the case of offset elements) so that the elements do not have any moment connection. This can be applied to the elements by recognising that any moment applied to node is not resisted by the element. This condition is used to partition the element stiffness matrix.

{fsfe}=[kssksekeskee]{usue}\begin{Bmatrix} \mathbf{f}_{s} \\ \mathbf{f}_{e} \\ \end{Bmatrix} = \begin{bmatrix} \mathbf{k}_{ss} & \mathbf{k}_{se} \\ \mathbf{k}_{es} & \mathbf{k}_{ee} \\ \end{bmatrix}\begin{Bmatrix} \mathbf{u}_{s} \\ \mathbf{u}_{e} \\ \end{Bmatrix}

Where the subscripts ss refer to the structure and ee refer to the element. Once partitioned the degrees of freedom related to the structure are combined into the structure stiffness matrix while the element degrees of freedom are included in the structure stiffness matrix (but do not interact with the stiffness matrix for any other element).

In the case of stiffnesses at the releases we add the stiffness terms for the release into the partitioned stiffness matrix. So in the case of a beam with releases at end 2 and stiffness associated with the released degrees of freedom the matrix is partitioned as below.

[ks11ks12ks21ks22]→[ks11k~s12k~e12k~s21k~s22k~e21k~e22]+[kr−kr−krkr]\begin{bmatrix} \mathbf{k}_{s11} & \mathbf{k}_{s12} & & \\ \mathbf{k}_{s21} & \mathbf{k}_{s22} & & \\ & & & \\ & & & \\ \end{bmatrix} \rightarrow \begin{bmatrix} \mathbf{k}_{s11} & {\widetilde{\mathbf{k}}}_{s12} & & {\widetilde{\mathbf{k}}}_{e12} \\ {\widetilde{\mathbf{k}}}_{s21} & {\widetilde{\mathbf{k}}}_{s22} & & \\ & & & \\ {\widetilde{\mathbf{k}}}_{e21} & & & {\widetilde{\mathbf{k}}}_{e22} \\ \end{bmatrix} + \begin{bmatrix} & & & \\ & \mathbf{k}_{r} & & - \mathbf{k}_{r} \\ & & & \\ & - \mathbf{k}_{r} & & \mathbf{k}_{r} \\ \end{bmatrix}

The stiffness terms at nodes where releases are applied are split, so

ksij→k~sij+k~eij\mathbf{k}_{sij} \rightarrow {\widetilde{\mathbf{k}}}_{sij} + {\widetilde{\mathbf{k}}}_{eij}

The release stiffness terms are similar to a spring stiffness matrix:

kr=[kxkykzkxxkyykzz]\mathbf{k}_{r} = \begin{bmatrix} k_{x} & & & & & \\ & k_{y} & & & & \\ & & k_{z} & & & \\ & & & k_{xx} & & \\ & & & & k_{yy} & \\ & & & & & k_{zz} \\ \end{bmatrix}