Reduced Stiffness & P Delta

The element stiffness can be partitioned into structure ($s$) and retained ($r$) degrees of freedom

$$\begin{Bmatrix} \mathbf{f}_{s} \\ \mathbf{f}_{r} \\ \end{Bmatrix} = \begin{bmatrix} \mathbf{K}_{ss} & \mathbf{K}_{sr} \\ \mathbf{K}_{rs} & \mathbf{K}_{rr} \\ \end{bmatrix}\begin{Bmatrix} u_{s} \\ u_{r} \\ \end{Bmatrix}$$

So

$$\mathbf{u}_{r} = {\mathbf{K}_{rr}}^{- 1}\mathbf{f}_{r} - {\mathbf{K}_{rr}}^{- 1}\mathbf{K}_{rs}\mathbf{u}_{s}$$

Giving the reduced equation

$$\left( \mathbf{f}_{s} - \mathbf{K}_{sr}{\mathbf{K}_{rr}}^{- 1}\mathbf{f}_{r} \right) = \left( \mathbf{K}_{ss} - \mathbf{K}_{sr}{\mathbf{K}_{rr}}^{- 1}\mathbf{K}_{rs} \right)\mathbf{u}_{s}$$

or

$${\widetilde{\mathbf{f}}}_{s} = {\widetilde{\mathbf{K}}}_{ss}\mathbf{u}_{s}\left\lbrack {\widetilde{K}}_{ss} \right\rbrack$$

When creating the structure stiffness matrix the element matrix can be assembled and then reduced as above before being included in the structure equations.

$$\mathbf{f}_{S} = \mathbf{K}_{SS}\mathbf{u}_{S}$$

Once the structure displacements are calculated the element displacements can be established from

$$\mathbf{u}_{e} = \begin{Bmatrix} \mathbf{u}_{s} \\ {\mathbf{K}_{rr}}^{- 1}\mathbf{f}_{r} - {\mathbf{K}_{rr}}^{- 1}\mathbf{K}_{rs}\mathbf{u}_{s} \\ \end{Bmatrix}$$

And the element forces as

$$\mathbf{f}_{e} = \mathbf{K}_{ee}\mathbf{u}_{e}$$

For a P-delta analysis the global solution is modified to

$$\mathbf{f}_{S} = \left( \mathbf{K}_{SS} + \mathbf{K}_{gSS} \right)\mathbf{u}_{S} = {\widehat{\mathbf{K}}}_{SS}\mathbf{u}_{S}$$

but the element force calculation is unchanged. This means that once the structure displacements are calculated the element displacements and forces are calculated from

$$\mathbf{u}_{e} = \begin{Bmatrix} \mathbf{u}_{s} \\ {{\widehat{\mathbf{K}}}_{rr}}^{- 1}\mathbf{f}_{r} - {{\widehat{\mathbf{K}}}_{rr}}^{- 1}{\widehat{\mathbf{K}}}_{rs}\mathbf{u}_{s} \\ \end{Bmatrix}$$

$$\mathbf{f}_{e} = \mathbf{K}_{ee}\mathbf{u}_{e}$$