Stability - Computational Analysis and Design of Bridge Structures - page 438

P P

Primary path

Bifurcation point

Limit point
(on primary path)

Limit point

P cr

Postbuckling
(secondary path) Secondary path

(postbuckling)

P cr

Bifurcation point

Actual (imperfect) structure

D D

(a) (b)

Figure 14.3 Possible load versus displacement behavior of thin-walled structures. (a) Linear
prebuckling path and rise postbuckling path. (b) Nonlinear prebuckling path
and drop postbuckling path. (From Cook, R.D. et al.: Concepts and Applications
of Finite Element Analysis , 4th edition, New York, 2002. Copyright Wiley-VCH
Verlag GmbH & Co. KGaA. Reproduced with permission.)

is the alternative path that originates when the critical load is reached. The
two paths intersect at the bifurcation point . Once past the bifurcation point,
the primary path is unstable. It is possible that mathematically the structure
follows the primary path, whereas the real structure follows the secondary
path. If the secondary path has a positive derivative (rises), the structure has
postbuckling strength (Figure 14.3a). A limit point is a maximum on a load-
displacement curve, but this point is not a bifurcation point because there is
no immediate adjacent equilibrium configuration. When a limit-point load
is reached under increasing load, snap-through buckling occurs, as the struc-
ture assumes a new configuration. A collapse load is the maximum load a
structure can sustain without gross deformation. It may be greater or less
than the computed bifurcation buckling load as shown in Figure 14.3.
Linear perturbation analyses can be performed from time to time during
a fully nonlinear analysis by including the linear perturbation steps between
the general response steps. The linear perturbation response has no effect
as the general analysis is continued. If geometric nonlinearity is included
in the general analysis on which a linear perturbation study is based, stress
stiffening or softening effects and load stiffening effects are included in the
linear perturbation analysis.
The loads for which the stiffness matrix becomes singular are searched
by an eigenvalue buckling problem. Equation 14.4 has nontrivial solutions
where K 0 is the tangent stiffness matrix when the loads are applied, and K σ
is the initial stress stiffness. Eigenvalue buckling is generally used to esti-
mate the critical buckling loads of stiff structures, for example, structures
carrying their loads primarily by axial or membrane action. Even when the