Page 287 - Marine Structural Design

Chapter 13 Collapse Analysis of Ship Hulls 263

13.3.3 Proposed Ultimate Strength Equations

The ultimate moment capacity obtained from the modified Smith is the maximum value on the
bending moment-curvature curve. Therefore, it is time consuming when reliability analysis
relative to the ultimate strength failure mode is carried out by means of the modified Smith
method. Some ultimate strength equations have been proposed based on various assumptions
of stress distribution over the cross-section. For instance, a moment capacity equation may be
derived based on the assumption that the midship section is filly plastic (elastic-perfectly
plastic) for the tensile side and is in ultimate strength condition for the compressive side. This
assumption gives generally good agreement with more exact predictions by correctly
estimating the position of the neutral axis. Successful experience using this approach has been
described in a study involving the ultimate strength of corroded pipes under combined
(intemaVextema1) pressure, axial force and bending. See Bai, (2001).
Several other assumed stress distributions are available in the literature, including a stress
distribution that assumes the middle of the hull depth is elastic while the rest of the hull depth
is plastidultimate strength. Xu and Cui (2000) assumed a stress distribution in which the
middle 1/3 of the hull depth is elastic while the rest of the hull depth is plastichltimate
strength. The present authors suggest that the ultimate moment capacity Mu can be predicted
by the following equation:
(13.38)
Mu =Eff:, ApsiZi +zQ:,Aps,Zj +zQ:Ap$kzk
i j
where A, is the area of stiffened panels/hard comers and z is the distance to the neutral axis.
Figure 13.6 shows a schematic diagram of stress distribution under sagging condition. The
stress distribution used in Eq.(13.38) does account for the ultimate strength of individual
stiffened panels and hard comers, e.g. it is not uniformly distributed in Figure 13.6.
In Eq.(13.38), the compressive ultimate strength region, tensile ultimate strength region and
elastic region are denoted by i, j, k respectively. afis the ultimate compressive strength for
stiffened panels or yield stress for hard comers. 0.' is the ultimate tensile strength (yield
stress). Elastic stress ge has a linear distribution around the neutral axis. Based on
observations of stress distribution from more comprehensive numerical analysis, it is
suggested by the present authors that the total height of the elastic region may be taken as half
of the hull depth, e.g. g, + g, = 012 The height of the compressive region g, , and the height
of the tensile region g,, may be estimated based on beam theory, which assumes plane
remains plane after bending.
Based on Eq.(13.38), the ultimate moment capacity of a hull girder may be estimated by the
following steps:
- Subdivide the cross-section into stiffened panels and hard comers;
- Estimate the ultimate strength of each stiffened panel using recognized formulas;
- Calculate the distance "H from the bottom of the ship to the neutral axis by assuming the
total force from the stress integration over the cross-section is zero;
- Calculate the ultimate moment capacity of the hull girder using Eq.(13.38).