Civil Engineering Reference
In-Depth Information
response of a structure is nonlinear prior to collapse, a general eigenvalue
or linear buckling analysis can provide useful estimates of collapse mode
shapes. Generally speaking, eigenvalue analysis is a straightforward prob-
lem. However, some structures have many buckling modes with closely
spaced eigenvalues, which can cause numerical problems. In these cases it
often helps to apply enough preload, just below the buckling load, before
performing the eigenvalue extraction. In many cases a series of closely
spaced eigenvalues indicate that the structure is imperfection sensitive.
In mathematics, an eigenvalue of Equation 14.4 indicates that at least
one diagonal element in the sum matrix becomes zero when
K
σ
is amplified
by that time. In structures, it means the critical point has been reached if
applied load has been multiplied by a factor of eigenvalue. In engineering, it
is meaningful only when its associated load is clearly defined. For example,
when
K
σ
is due to all structural weights, the first eigenvalue (λ) predicts
that the structure will lose its stability if all structural weights are equally
multiplied by a factor of λ. If an analysis is to know how many times a live
load will cause buckling,
K
0
and
K
σ
in Equation 14.4 should be adjusted
accordingly. To accurately predict the buckling load, a special-purpose finite
element analysis (FEA) package, which can sum
K
σ
at one stage due to cer-
tain loads into
K
0
and compute
K
σ
at another stage due to another load,
should be employed. Taking a cable-stayed bridge as an example,
K
0
in
Equation 14.4 should be able to include all the initial stresses accumulated
from the first construction stage until the deck is superimposed, and
K
σ
in
Equation 14.4 counts for only one particularly extreme live load. Therefore,
the eigenvalue may predict a meaningful engineering safety factor.
14.2.1 linear buckling of a steel plate
14.2.1.1 Formulation of plate buckling
In this section, plate buckling theory is discussed. The von Karman large
deflection equations for flat isotropic plates with in-plane loading were
modified to account for anisotropy by Rostovtsev (1940), and later the
effect of initial imperfections were included resulting in the following
simultaneous equations, which are considered the most general equa-
tions currently available for solving plate buckling problems (Murray
1984):
4
4
4
2
2
2
2
∂
∂
ω
∂
∂ ∂
ω
∂
∂
ω
∂
∂
φ
∂
(
y
x
+
ω
)
∂
∂
φ
∂
(
y
z
+
ω
)
D
+
2
H
x z
+
D
=
+
x
z
4
2
2
4
2
2
2
2
x
z
z
∂
x
∂
2
2
∂
∂ ∂
φ
∂
(
y
x z
+
ω
)
−
2
+
q
(14.5)
x z
∂ ∂
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