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Negative Pivots

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.

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