Civil Engineering Reference
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Π
Π
Π
d
d
d
0
0
0
(a)
Stable state
(b)
Unknown state
(c)
Unstable state
Figure 14.1
(a-c) States of structural equilibrium.
and 14.3 are equivalent to any diagonal element being zero and being less
than zero, respectively. Based on the making of the global stiffness and its
changing from positive to zero or even negative, the instability of a struc-
ture can be in the following three categories:
1. Buckling. Scenarios where the stiffness change due to the large dis-
placement is ignored
(
K
L
=
0
, and when evaluating elastic matrix
D
in Equation 3.12, a constant Young's modulus
E
is assumed, that
is, small displacements and elastic material. Only the stiffness of ini-
tial stress
K
σ
is considered. Therefore, buckling is an elastic stability
problem in which the stiffness due to geometric change is ignored.
When buckling happens, the structure suddenly changes to an unsta-
ble or unknown state. As a point clearly divides the structural states
from stable to unstable, buckling is often referred to as bifurcation
buckling and the loads to reach this point are called critical loads.
A column or beam under compression as shown in Figure 14.2a is a
typical buckling problem. By solving general eigenproblem as shown
)
P
Cable in
tension
Column in
compression
Beam in deflection
P
P
(a)
Buckling
W
(c)
Material entering plastic
(b)
Large displacements
Figure 14.2
(a-c) Categories of structural instability.
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