Page 242 - Marine Structural Design

218 Part II Ultimate Strength

The pre-buckling deformations of cylinders are highly non-linear due to ovalization of the
cross-section.

Brazier (1927) was the first researcher who derived elastic bending moment and cross-
sectional ovalization as a hction of curvature in elasticity. He found that the maximum
moment is reached when critical stress is
Et
cJE 0.33 - (1 1.20)
=
r
However, in plastic region, the buckling strain for cylinders in pure bending may be
substantially higher than that given by plastic buckling theory for cylinders in pure
compression. Many researchers have been trying to derive mathematical solutions for inelastic
cylinders in pure bending (see Ades, 1957 and Gellin, 1980). Unfortunately no one has been
successful so far.
The effect of boundary conditions may also play an important role affecting buckling strength
of un-stiffened short shells under bending. The shorter the cylinder, the higher the buckling
strength is. This is because pre-buckling deformation, which is less for shorter cylinder, may
reduce shell buckling strength. When the length of the cylinder is long enough, the bending
strength may be close to those given by Beazier (1927), Ades (1957) and Gellin (1980).
11.2.4 External Lateral Pressure
In the pre-buckling state, the external pressure sets up compressive membrane stresses in the
meridian direction. Retaining only the linear terms in Eq. (1 1.3):
Ne = -Pr (11.21)
Introducing Eq.(l1.21) into Eq. (1 1.8) yields the following stability equation:
Et a'w 1
DV8~+F-+-pV4 (1 1.22)
r ax4 r
The displacement hction is of the same form as the axial compression. Introducing Eq.
(1 1.22) yields:

(1 1.23)

where one axial wave (m = 1) gives the lowest buckling load. The last term is interpreted to be
the buckling coefficient, ke . The smallest value of k, may be determined by trial. If Ti is
assumed large (>>1), analytically minimizing Eq. (1 1.23) gives:
ke=--& (11.24)
4&
3R
The approximate buckling coefficient valid for small and medium values of Znow reads:

(11.25)

The first term is identical to the buckling coefficient of a long plane plate. When l/r
approaches infinity, Eq. (1 1.23) reduces to: