Even a well-conditioned stiffness matrix can still produce a negative
pivot if the system is unstable, that is, it is passing through a bifurcation or limit
point, e.g.,

Such points in an analysis can permit other solution paths to be followed. In the case
of the limit point, the solution could progress by increasing or decreasing the
displacement for a change in load. For points of bifurcation a completely different
solution path could be followed. In both cases the ensuing solution path may be
non-physical, meaning that some of the results will not be consistent with the known laws
of physics. In general, these points of instability can be seen as the stiffness matrix
expressing a preference for an alternative solution path on the basis that it represents
an easier option.

For example, an axially loaded straight strut will generate one negative pivot if
loaded in a geometrically nonlinear analysis to just beyond the first buckling load.
Without additional perturbation to force the lateral buckling deformation, the strut will
remain straight and resist the buckling path offered at this point. Further increase of
the load to just beyond the second buckling load will generate two negative pivots and so
on.

A count of the number of negative pivots is given in the LUSAS log and output files
(parameter NSCH). When NSCH is greater than zero an unstable solution point (limit or
bifurcation) has been reached. The variable PIVMN will give the corresponding value of the
most negative pivot. A negative CSTIF value, together with a negative PIVMIN value
corresponds to a limit point but a positive CSTIF and a negative PIVMIN correspond to a
bifurcation point (although this is only the first one located in each case since limit
points are detected by a change in sign).

Some general points…

- Negative pivots can occur during the iterative solution but disappear when the solution
has converged, indicating that a unstable point was reached and was resolved during
subsequent iterations. This will not affect the integrity of the solution but if this
occurs during many load increments the rate of convergence may be detrimentally affected
and the causes should be investigated. Typically this indicates that the load step is too
large (causing massive nonlinearity from which it is not possible to recover numerically)
and should be reduced
- If negative pivots occur during each iteration of an increment and the solution will not
converge this may, again, indicate that the load step is too large and should be reduced
- If the solution does not converge, even with a reduced load step,
the solution procedure may need to be changed. Running
the problem under arc length control gives the best chance
of negotiating a limit or bifurcation point. A limit point
can also be overcome by using prescribed displacement
loading. Before modifying the solution procedure to arc
length, the list of the more frequent causes and remedies
of pivots should
be used

Note that the use of the MODELLER option to ignore negative pivots
(File> Model Properties> Solution> Nonlinear options…)
is not recommended until all other checks
have been carried out to ensure model integrity.

Because negative pivots can also be generated through poor conditioning
as a result of modelling errors, it is recommended that
use is made of the checks
provided that give a list of the more frequent causes and
their remedies.