Civil Engineering Reference
In-Depth Information
is a linear perturbation procedure. The analysis can be the first step in a global
analysis of an unloaded structure, or it can be performed after the structure
has been preloaded. It can be used to model measured initial overall and local
geometric imperfections or in the investigation of the imperfection sensitiv-
ity of a structure in case of lack of measurements. Eigenvalue buckling is
generally used to estimate the critical buckling loads of
stiff
structures (clas-
sical eigenvalue buckling). Stiff structures carry their design loads primarily
by axial or membrane action, rather than by bending action. Their response
usually involves very little deformation prior to buckling. However, even
when the response of a structure is nonlinear before collapse, a general
eigenvalue buckling analysis can provide useful estimates of collapse mode
shapes.
The buckling loads are calculated relative to the original state of the
structure. If the eigenvalue buckling procedure is the first step in an analysis,
the buckled (deformed) state of the model at the end of the eigenvalue buck-
ling analysis step will be the updated original state of the structure. The
eigenvalue buckling can include preloads such as dead load and other loads.
The preloads are often zero in classical eigenvalue buckling analyses. An
incremental loading pattern is defined in the eigenvalue buckling prediction
step. The magnitude of this loading is not important; it will be scaled by the
load multipliers that are predicted by the eigenvalue buckling analysis. The
buckling mode shapes (eigenvectors) are also predicted by the eigenvalue
buckling analysis. The critical buckling loads are then equal to the preloads
plus the scaled incremental load. Normally, the lowest load multiplier and
buckling mode is of interest. The buckling mode shapes are normalized vec-
tors and do not represent actual magnitudes of deformation at critical load.
They are normalized so that the maximum displacement component has a
magnitude of 1.0. If all displacement components are zero, the maximum
rotation component is normalized to 1.0. These buckling mode shapes
are often the most useful outcome of the eigenvalue buckling analysis, since
they predict the likely failure modes of the structure.
Some structures have many buckling modes with closely spaced eigen-
values, which can cause numerical problems. In these cases, it is recom-
mended to apply enough preload to load the structure to just below the
buckling load before performing the eigenvalue analysis. In many cases, a
series of closely spaced eigenvalues indicate that the structure is
imperfection-sensitive. An eigenvalue buckling analysis will not give accu-
rate predictions of the buckling load for imperfection-sensitive structures. In
this case, the static Riks procedure, used by ABAQUS [1.29], which will be
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