Numerical methods in bridge structure analysis - Computational Analysis and Design of Bridge Structures - page 78

3.2.8 elastic stability
As shown in Equation 3.39, stiffness may be enhanced or reduced by K σ —
the initial stress stiffness due to existing stresses when large displacements
are considered. When total stiffness is reduced by the initial stress stiff-
ness, as in columns or plates under compression, there will be a critical
point in which stiffness in one or many degrees of freedom reaches 0 (i.e.,
K

0 + σ becomes singular). This phenomenon is the so-called elastic stabil-
ity problem, in which a critical point clearly defines the entry to an unstable
state. In addition to the elastic problem, stability problems can further be
classified as plastic stability and excessive displacements according to the
reason of singularity of the total stiffness (tangential stiffness K

K

0 + + σ L ).
For example, if the stability problem is due to the elastic matrix D , it is
plastic stability problem; if it is due to large displacements, it is the exces-
sive displacements problem. It is obvious that both are nonlinear problems
and are the same in a mathematical view. When nonlinear stability is of a
concern, both plastic and large displacements should be considered together.
When excessive displacements happen, some components may have entered
plastic range, and when some components enter plastic range, displacements
may become large. The approach to nonlinear stability solutions is the same
as normal nonlinear problems as illustrated in the previous section. In this
section, only the elastic stability is discussed, as it gives the upper limits of
critical loads and is more essential to structural analyses. For instance, dur-
ing preliminary designs of bridges in which compression and bending are
dominating (i.e., arch bridges and cable-stayed bridges), elastic stability is
usually analyzed first. The upper limit will guide the adjustment to structure
dimensions and component sizes. Further discussion and application on sta-
bility is discussed in Chapter 14.
The solution to an elastic stability problem can be categorized into an
eigenvalue problem. When only initial stress is considered, the following
equilibrium equation can be derived from either the global equilibrium
equation 3.33 or the tangential equilibrium equation 3.37:

K K

( ) =

K

0 +

K a f (3.49)

σ

K σ is proportional to the current axial tension/compression stress as shown
in Equation 3.48. The search for critical loads in elastic stability can be
simplified by amplifying K σ until the total stiffness matrix in Equation 3.49
becomes singular, which is equivalent to the following general eigenvalue
problem:

K + λ σ K = 0 (3.50)

0