Page 256 - Marine Structural Design

232 Pari N ultimate Strength

[H:i]=[H,!Ax H;A, /3 HIA, 13 HIZ, /3 Hi I, HIIz] (1 2.23)
Considering fin Eq. (12.7) as a plastic potential and applying the flow theory of plasticity, we
may obtain:
(12.24)

(1 2.25)
The increment of the parameter CY,, due to isotropic strain hardening is defined as:

&, = {v/W @CY) (12.26)
Substituting Eqs. (12.24) and (12.22) into Eq. (12.26), the isotropic cross-sectional strain
hardening rate defined in Eq. (12.15) is given as:

(12.27)

The kinematic hardening rate for the cross-section will be derived using a similar approach.
The yield surface translation increment {da}can be obtained by using Ziegler's rule and the
Mises yield criteria (Fulikubo et al, 1991):
= [H;kIbP I (12.28)

where [Hik] used in the present chapter is taken as:
[Hik]= \Hi A, H; A, /3Hl A, /3H; Z, /3H; I, Hi I, J (12.29)

Substituting Eqs. (12.24) and (12.28) into Eq. (12.14), we obtained:

(12.30)

Finally, we shall determine the strain-rate hardening rate for the cross-sections. The increment
of the parameter CY,, , due to strain-rate hardening is estimated by use of a constitutive equation
that expresses the relationship between g, and the equivalent plastic strain. For instance, the
Cowper-Symonds constitutive equation is expressed as (Jones, 1989):
Do, = 0, ti + (k:/Oy/'J (12.3 1)

where 0, is the yield stress and .F: denotes the plastic strain-rate for a point. Values of D and
q that are often used are the following:

for mild steel D40.4 sec-' q=5
and for aluminum alloy D= 6500 sec-' q-1