The Design of Modern Steel Bridges - TEDI

Figure 5.6 Buckling of plates in compression.
and
where
T ¼ p2 bF
8a d2 m 2
D ¼ flexural rigidity of the plate, equal to Et 3 /12(1 m 2 )
t ¼ thickness of the plate
E ¼ Young’s modulus
m ¼ Poisson’s ratio.
The critical value of F is thus given by
Fcr ¼ sx crt ¼ p2 a 2 D
m 2
Rolled Beam and Plate Girder Design 111
m2 n2
þ
a2 b2 2
ð5:18Þ
Another method for obtaining the critical value of F is to consider the
St Venant differential equation of equilibrium of the plate given by
q 4 o
qx4 þ 2q4o qx2qy2 þ q4o F q
¼
qy4 D
2 o
qx2 ð5:19Þ
The deflected shape given by equation (5.17) is one solution of this
differential equation. Substitution of equation (5.17) into equation (5.19) leads
directly to the critical value of F given by equation (5.18). The limitation of
this method is that closed-form expressions for the deflected shape of the plate
are found only for a few types of applied stress patterns.
The St Venant differential equation (5.19) for a plate in compression is
equivalent to the following well-known differential equation of equilibrium of
a strut
EI d4o dx4 ¼ P d2o dx2