The element stiffness can be partitioned into structure (s) and retained
(r) degrees of freedom
{fsfr}=[KssKrsKsrKrr]{usur}
So
ur=Krr−1fr−Krr−1Krsus
Giving the reduced equation
(fs−KsrKrr−1fr)=(Kss−KsrKrr−1Krs)us
or
fs=Kssus[Kss]
When creating the structure stiffness matrix the element matrix can be
assembled and then reduced as above before being included in the
structure equations.
fS=KSSuS
Once the structure displacements are calculated the element
displacements can be established from
ue={usKrr−1fr−Krr−1Krsus}
And the element forces as
fe=Keeue
For a P-delta analysis the global solution is modified to
fS=(KSS+KgSS)uS=KSSuS
but the element force calculation is unchanged. This means that once the
structure displacements are calculated the element displacements and
forces are calculated from