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How do I eliminate negative eigenvalues from an eigenvalue
buckling analysis?
- It is essential to perform a linear static
analysis prior to the eigenvalue analysis. This eliminates the
added complexities of the eigen-analysis and enables a check
on the basic stiffness matrix of the structure. Any warning or
error messages in the LUSAS output file should be
investigated. See the additional section on LUSAS Output
Errors in this User area for more information
- Check the output file for any warnings or
errors from the eigen-buckling analysis
- If using the Subspace eigensolver, increase
the number of starting iteration vectors. If any of the modes
are close together the default magnitude for this parameter
may not be sufficient to allow accurate resolution in their
extraction. Increasing this parameter is also essential if
only requesting a small number of eigenvalues (1-10)
- Ensure that the solution converged correctly.
If not, and the solution history was converging, then increase
the number of iterations permitted
- Tighten the convergence tolerance, since some
modes may be close together and require greater numerical
resolution. This may also require an increase in the number of
iterations permitted
- Negative eigenvalues computed during an
eigenvalue buckling analyses can imply genuine numerical
difficulties in the solution procedure which can be rectified
by using the alternative eigenvalue buckling
solution in which the original buckling problem is recast to
an alternative form in which, if certain rules are adhered to,
all the computed eigenvalues will be positive (see the theory
manual for more information). The alternative buckling
procedure is available for the subspace iteration
eigensolver and can be invoked from the advanced button
on the eigenvalue control form as solve for 1/(1-buckling
load).
Because the load factor for alternative
buckling is calculated from 1/(1-eigenvalue), negative
eigenvalues can still be computed. From the equation, this can
be seen to indicate that the applied load is higher than the
buckling load, hence producing buckling load factors of less
than unity - typically in the range 0.95-0.99. As a result,
the load level must be specified to ensure that the load
factors calculated are close to, but greater
than unity (i.e. the load applied should be slightly less than
the buckling load).
The reason for the recommendation of
ensuring that the load factors are close to unity is
that if the load factor is too large, the eigenvalue will be
approaching unity which implies that a small error in the
eigenvalue may produce a large error in the computed load
factor.
- Reduce the load applied to ensure that it is
below the lowest expected buckling mode of the structure
- Negative eigenvalues can indicate bifurcation
in tension or bifurcation that would occur if the loading is
reversed in sign, i.e. the applied loading is in the opposite
direction to that which would cause buckling of the structure
- Mechanisms can be excited in Semiloof shell
(QSL8, TSL6) elements than give rise to negative eigenvalues
– particularly for thin, cylindrical structures. The use of
the fine integration rule for these elements will overcome
such element mechanisms
- In certain situations the eigenvalues of the
recast solution may be very closely spaced and cause
convergence problems in the iterative solution. The number of
permitted iterations should be increased from the default in
this situation.
- Some shear dominated buckling analyses cannot
be solved using the subspace method (even with alternative
buckling) and the recommendation is then to use either the
fast block Lanczos solver (if your licence allows) or the inverse
iteration range method in which a range of
eigenvalues or frequencies is specified in which the
eigensolution will be computed. Note that the number of
eigenvalues field should be specified as zero to ensure that
all eigenvalues within the specified range are computed.
Specifying a non-zero value that is less than the actual
number in the range will not necessarily provide the lowest
eigenvalues, i.e. there may be gaps
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