Page 240 - Marine Structural Design

216 Part II Ultimate Strength

The plate sti&ess, D, is given by:
D=w] (11.5)
Et3

The pressure, p, is positive outwards. Note the similarity between Eq. (11.3) and the
corresponding expression for plate equilibrium by substituting:

(11.6)

The only new term is N,/r , which represents the lateral component of the circumferential
stress. Thus, unlike plates, cylindrical shells can carry lateral loads by pure membrane action
and no bending. This is a very efficient property, but at the same time, this makes shells
sensitive to buckling.
Eqs. (1 1 .I) thru (1 1.3) form a coupled set of three non-linear equations with four variables--
N, N,, No, and w. By introducing the kinematic and constitutive relationships, and
applying the operator V, Eq. (1 1.3) may also be written as,

(11.7)
which is also known as Donnel's equation.

11.2.2 Axial Compression
Consider a cylinder subjected to an axial compressive load, P. If the end effects are neglected,
the following assumptions apply:
P
N =- * N,e=Ne=O (11.8)
I 2m
Introduction of these values into Eq. (1 1.7) gives:

(11.9)

The solution to this differential equation takes the form:

(11.10)

where m is the number of half waves in the longitudinal direction and n is the number of entire
waves in the circumferential direction, which gives:

(11.11)

where 2 is the Batdorf parameter,

(11.12)
rt