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∂
∂
4
φ
ν
ν
∂
∂ ∂
4
φ
∂
∂
4
φ
∂
∂
2
y
∂
∂
2
ω
1
1
1
x
z
+
2
−
−
+
=
t E x
4
K
t E
t E
x z
2
2
t E z
4
z
2
x
2
z
z
xz
x
x
z
z
x
x
2
∂
∂ ∂
y
x z x z
2
∂
∂ ∂
2
ω
∂
∂
y
x
2
∂
∂
2
∂
∂
2
ω ω
+
∂
∂
2
2
∂
∂ ∂
ω
x z
(14.6)
+
2
−
−
2
z
2
z
2
x
2
where:
ω
is the lateral deflection
ϕ is the stress function
q
is the load intensity on the plate
D
x
,
D
z
are plate stiffnesses
E
x
,
E
z
are moduli of elasticity
t
x
,
t
z
are the thicknesses of plate
v
x
,
v
z
are the Poisson's ratios
14.2.1.2 Solving plate and box girder buckling problem
The high bending moments and shearing forces for long-span bridges may
consider the use of fabricated plate and box girders. In their simplest form,
plate and box girders can be considered as an assemblage of webs and
flanges. To reduce the self-weight of these girders, slender plate sections are
employed. Hence the local buckling and postbuckling reserve the strength
of plates, they are important design criteria. For the efficient use of thin
plates, flanges and webs in a box girder are often reinforced with stiffeners.
However, there are some difficulties that are usually encountered by the
designers of plated structures (Ryall et al. 2000):
• The engineer's simple “plane sections remain plane” theory of bend-
ing is no longer adequate, even for linear elastic analysis.
• Nonlinear elastic behavior caused by the buckling of plates can be of
great importance and must be allowed for.
• Because of this complex nonlinear elastic behavior as well as stress
concentration problems, some yielding may occur at loads that are
quite low in relation to ultimate collapse loads. While such yielding
may not be of great significance with regard to rigidity and strength,
it means that simple maximum stress criteria are no longer sufficient.
• Because of the buckling problem in plates and stiffened panels, com-
plete plastification is far from being realized at collapse. Hence simple
plastic criteria are also not sufficient.
• Complex interactions occur between langes, webs, and diaphragms, and
the pattern of this interaction can change as the level of load increases.
To demonstrate the linear buckling problem, a rectangular plate is com-
pressed in its middle plane by forces uniformly distributed along the sides
x
= 0 and
x
=
a
, as shown in Figure 14.4.
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