Page 241 - Marine Structural Design

Chapter I1 Ultimate Strength of Cylindrical Shells 217

and
- izl (11.13)
n =-
M
The solution to Eq. (1 1.1 1) may be expressed as:

(1 1.14)

For cylinders of intermediate length, a close estimate of the smallest critical load may be
obtained by analytically minimizing Eq. (1 1.1 1) with respect to the following quantity:

Then, the minimum is found to be:

( m y 2 - ;fz (11.15)

which gives the following critical load,

(1 1.16)

This is the classical solution for an axially compressed cylinder. It should be noted that m and
are treated as continuous variables (for diamond-shaped bulges) in the minimization process
while they are actually discrete quantities. The correct values can be found by trial and error.
For short cylinders, the buckling mode will be asymmetric with m=l and n=O, which is plate-
like buckling. The following buckling coefficient may be obtained:
1222
k, =l+- (11.17)
n4
is valid for:
?r2
Z < - 2.85 (11.18)
=
2Js
For long cylinders, column buckling is a potential collapse mode, and the buckling stress is
expressed by:

(11.19)

11.2.3 Bending

In elastic region, studies carried out in this field indicate that the buckling stress in bending is
close to that for buckling in axial compression for all practical purposes, see Timoshenko and
Gere (1961). It is more complicated to analyze cylinders subjected to bending because,
The initial stress distribution is no longer constant around the circumference.