Page 172 - Introduction to Naval Architecture

158 STRENGTH

supporting members, k will be proportional to the plate thickness. Thus
the slenderness ratio can be expressed as the ratio of the plate span to
thickness.
When a panel of plating is supported on its four edges, the support
along the edges parallel to the load application has a marked influence
on the buckling stress. For a long, longitudinally stiffened panel,
breadth b and thickness t, the buckling stress is approximately;

where v is the Poisson's ratio for the material.
For a broad panel, length S, with transverse stiffening, the buckling
stress is:

2
12(1 -V )S 2
The ratio of the buckling stresses in the two cases, for plates of equal
thickness and the same stiffener spacing is: 4[1 + (S/b) ]~ 2
Assuming the transversely stiffened panel has a breadth five times its
length, this ratio becomes 3.69. Thus the critical buckling stress in a
longitudinally stiffened panel is almost four times that of the
transversely stiffened panel, demonstrating the advantage of longitudi-
nal stiffening.
The above formulae assume initially straight members, axially
loaded. In practice there is likely to be some initial curvature. Whilst
not affecting the elastic buckling stress this increases the stress in the
member due to the bending moment imposed. The total stress on the
concave side may reach yield before instability occurs. On unloading
there will be a permanent set. Practical formulae attempt to allow for
this and one is the Rankine-Gordon formula. This gives the buckling
load on a column as:

where
f c and C are constants depending on the material
C depends upon the fixing conditions
A is the cross-sectional area
l/k is the slenderness ratio.