Page 278 - Marine Structural Design

254 Part II Utimate Strength

In order to apply the fracture mechanics criteria, the increment of plastic strain fd: ] at every
node is evaluated as:
1 = - (dup 1 (13.7)
1
'd
where I,, is the equivalent length of the plastic region. The value of 1, is evaluated to be half
the partial-yielded region 1, as show in Figure 13.1.

Before the local stiffness matrix is added to the global stiffness matrix, several transformations
are necessary. It may be convenient that the local axes do not coincide with the neutral axes.
Furthermore, the neutral axis moves when the effective width of the plating changes during
loading. Finally, the shear center may differ from the neutral axis of bending. A transfornation
matrix [q that accounts for this can be found in standard textbooks (Pedersen, P. Temdrup &
Jensen, J. Juncher 1983). This matrix transforms the stiffness matrix to:
E' I = [SI '[kP lis1 (13.8)

where [kp] is the local stiffness matrix with respect to the neutral axes, and [k*] is the local
stiffness matrix with respect to the nodal axis. This matrix is transformed to obtain the global
coordinate:
[k,l= [ml[Tl= [~lT[~lT[~Pl~~I[~l (13.9)
where [kgw 1 is the stiffness matrix in global coordinates:

(1 3.1 0)

- -
1 0 0 0 E, -Eyi
0 1 0 -e, 0 E,
0 0 1 eyi Exi 0 (13.11)
0 0 0 1 0 0
000 0 1 0
- 000 0 0 1 -

where (eyi , e,) are the coordinates of the shear-center and (Eyi, Ezi) are the coordinates of
the neutral-axis in the local system to the beam end for node '5". This transformation for a
neutral axis offset is extremely convenient when only part of the hull is analyzed.
13.2.2 Attached Plating Element
The stiffened plate element is an extension of the beam-column in which an effective width is
added to the beam. For a long plate, see Figure 13.2, the effective width is obtained by
assuming that Carlsen's ultimate stress equation (Carlsen, 1977) is valid for the region up to,
and beyond, the ultimate state: