Page 254 - Marine Structural Design

230 Part 11 Ultimate Strength

the plastic node method in an international journal, and Fujikubo (1987) published his Ph.D.
thesis on this simplified plastic analysis method.
Fujikubo (1991) further extended the theory of plastic node method to account for the effect of
strain hardening. In the following sections, the existing theory is further extended to account
for the effects of strain-rate hardening.
12.3.2 Consistency Condition and Hardening Rates for Beam Cross-Sections
For a beam-column element with strain hardening and strain-rate hardening, the yield
condition of its cross-section is expressed as:
f=Y({a-~a))-a,(FP,~P)=O (1 2.7)
where Y is the yield (fill plastic) function, {a } represents the translation of the yield surface
due to kinematic hardening, and 0, is a parameter expressing the size of the yield surface. The
vector {a } has the same dimension as the generalized stress and is expressed as:

(a>= 6, afi “3 am a, aJ (12.8)
Due to isotropic hardening, the yield surface is expanding as the plastic deformations increase.
This expansion of the yield surface is expressed by the stress parameter a, , which is a
function of the generalized equivalent plastic strain EPand of the plastic strain-rate 2P. The
equivalent strain EPis evaluated as a summation of its increments, which are defined as:

a,dlP ={a-a)T[dsP) (12.9)
where the increments of the generalized plastic strain are taken to be:

(12.10)
The equivalent plastic strain-rate, Bp is defined as:

(12.11)

where dt is an increment of time t.
The increment of the parameter a, due to isotropic strain hardening and that due to strain-rate
hardening are de-coupled to the simplest form.

da, = dg, (2’)+ dg,($”) (12.12)
where dg, (F‘) expresses the increment of the parameter a,, for a beam cross-section due to
isotropic strain hardening and dg,(gP)denotes the increment of the parameter a, due to
strain-rate hardening. Similar equations were used by Yoshimura et a1 (1987), and Mosquera
et al, (1985).
The consistency condition for a yielded cross-section satisfylng the yield condition Eq.( 12.7)
is expressed as:

(12.13)