Stability Analysis - Mechanics of Structures: Variational and Computational Methods - page 704

FIGURE 11.38
The value of det K using the analytical stiffness matrix.

Thus, the lowest eigenvalue, which is the critical load factor, is 2.031. The lowest eigenvalue
identifies the critical load, in the sense that the bifurcation of equilibrium is reached when
the load combination is increased by a factor of 2.031.
If the same procedure is performed with the linear solution as the fundamental state, a
load factor of 2.024 is obtained. From this, we can conclude that for frameworks it is sufficient
to investigate the lowest level of bifurcations of equilibrium using a linear fundamental
state. This conclusion cannot necessarily be drawn for higher eigenvalues, which for a
global investigation of the system may also be of practical interest. For the determination
of higher eigenvalues, we choose here the second order solution computed using analytical
functions or higher power series. In this case, the root search of the determinant has to be
performed with more sophisticated tools [Zurm uhl, 1963].
For this framework, the first eight eigenvalues were obtained as the roots of the analytical
system matrix the values of which are given in Fig. 11.39. Comparison of the second root
obtained by using the geometric matrix only (Fig. 11.37) and the one resulting from the
solution with analytical functions as provided in Fig. 11.38, shows that the approximate
approach is only appropriate for the first eigenvalue, i.e., for the lowest critical load of the
system. Both of these figures diagram the determinant det K versus .

Whereas the eigenvalues are associated with the critical loads, the corresponding eigen-
vector describes the buckling mode associated with it. The amplitudes of a buckling shape
are determined relative to an appropriately chosen value. These shapes (Fig. 11.39) can
provide insight into the buckling behavior of the framework. We observe that the first
buckling shape is connected with a horizontal displacement of the whole frame, while
the second eigenvalue is associated with the local buckling of the bar-column element 3,
and the next eigenvalue with element 3 interacting with element 1, etc. It is note-
worthy that the investigation of bar 3 as a single member (local buckling) yields the same
value.

λ