Page 284 - Marine Structural Design

260 Part II Ultimate Strength

Each plastified node is checked for unloading. For nodes in non-linear spring elements as well
as for shear panel elements, unloading is detected when the load increment and the load have
different signs. For all elements with unloading nodes, the stifmess is changed and the
procedure is continued from point (0.

When no further unloading is detected, the increments of displacement are obtained. Then the
internal forces for each element are calculated. For each elastic element, a check may be made
to determine whether yielding occurs during the step. If this is the case, the increment for that
element is divided into a part that is treated elastically and a part that is treated like plastic.
The increment in the internal force for the element is calculated from:
(uk} = factor [KE ldu) + (1 - fuctorpc, Xdu) (13.26)
wherefactor is the elastic fraction of the increment.
Unloading is again checked.
If either loading or unloading takes place, and no kind of iteration is carried out, the change of
state gives rise to unbalanced forces, which should be added to the load in the next step. This
unbalanced force is calculated to be the difference in internal forces due to changes in the
elastic-plastic state. This gives place to yielding (dx) = (1 - fuctorNKE]- [Kp Ndu} and to
unloading {q) = ([K, ] - [KE Ndu}.
Note that the global set of equations remains unchanged due to plastification in elements, this
means that the influence on the global situation from one node changing state, is disregarded.
A revision is made to determine whether any elements have tom. If this is the case, these
elements are removed and their internal forces are added as unbalanced loads in the next step.
When the step is accepted, a new increment begins at point (a).

13.3 Analytical Equations for Hull Girder Ultimate Strength

Buckling and collapse strength of hull girders under bending may be predicted as the fully
plastic moment, the initial yield moment, and the progressive collapse moment. The last
includes buckling and post-buckling strength of individual components of the hull girder. The
fully plastic mode provides an upper bound of the ultimate strength, which is never attained in
a hull of normal configurations. The initial yield mode assumes that buckling does not occur
prior to yielding. The initial yield strength is a function of the elastic section modulus of the
hull girder and yield strength of the material.
In this section, an ultimate strength equation is proposed to account for the effects of lateral
pressures, bi-axial loading, and shear stress using analytical solutions. The ultimate strength
equation is then compared to the sophisticated approach described in Section 13.4. The
ultimate strength equation may be applied for the quantification of structural risks of aging
ships with corrosion and fatigue defects, see Parts IV and I of this book.
13.3.1 Ultimate Moment Capacity Based on Elastic Section Modulus

In the initial yield moment approach, it is assumed that the ultimate strength of the hull girder
is reached when the deck (alone) has yielded. Premature buckling is assumed not to occur. In
this approach, the elastic section modulus is the primary factor for measuring the longitudinal
bending strength of the hull. With these assumptions, the initial yield moment can be written
as: