Civil Engineering Reference
In-Depth Information
Stability
14.1 BaSicS of Structural StaBility
Structural stability is the ability of a structure to resist loading. Loss of
such ability, so-called instability, is a state in which the structure is no lon-
ger in equilibrium with change in the geometry of a structure or structural
component under loads. One phenomenon of structural failure led by insta-
bility is excessive structural displacements or component deformations.
The underlying causes are the loss of stiffness in some particular degrees
of freedom due to geometric and/or material constitutional reasons, that is,
geometric and material nonlinearities.
According to the principle of minimum total potential energy, a structure
is in equilibrium when the total energy no longer changes or the first-order
derivative of the total energy to displacements equals to zero. As illustrated
in Section 3.2.1, Equation 3.1 (or Equation 12.4 where ∂
Π δ
d
0), which
leads to the establishment of global equilibrium equation 3.3, reveals any
possible state that makes the total energy minimal or maximal (locally or
globally). Further, the value of the second-order derivative tells the trend of
total energy changes as shown in Figure 14.1 and Equations 14.1 through
14.3. The engineering purpose of stability analyses is to find any practical
solution for Equation 3.3, or a state, that meets Equation 14.2 or 14.3.
=
2
δ
δ
Π
d
>
0
The solution of Equation 3.3 is structurally stable
(14.1)
2
2
δ
δ
Π
d
=
0
The solution of Equation 3.3 is in a state of unknown (14.2)
2
2
δ
δ
Π
d
<
0
The solution of Equation 3.3 is structurally unstable (14.3)
2
From the perspective of stiffness matrix analysis in the global equilibrium
formulation (Equation 3.3 where
K K
[
]
0
+
+
K da
=
F
), Equations 14.2
σ
L
435
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