Page 257 - Marine Structural Design

Chapter 12 A Theory of Nonlinear Finite Element Analysis 233

Eq. (12.31) was obtained for uni-axial stress tests. It is assumed that this equation is still a
valid approximation when it is applied to multi-axially loaded beam cross-sections. Eq.
(1 2.3 1) becomes:

(12.32)
where N, = ApY
Using Eq. (1 2.32), the strain-rate hardening rate defined in Eq. (1 2.16) may be given as:

H:, = N,,(gP/Dj+-') /q (12.33)

12.3.3 Plastic Displacement and Strain at Nodes
The plastic deformations of the element are concentrated at the node in a mechanism similar to
the plastic hinge. Referring to Eq. (12.7), the yield condition at the node is expressed as:

F; = ?(bj ai})- a0,(q!, $)= 0 (12.34)
-
where the subscript "i", denotes values at the node No. i. From Eq. (12.21), the consistency
condition for node i is expressed as:

-
dc. {+; IT {h) [Hi, + Hi, + H:, /(&ft)Ij dqp + a, = 0 (12.35)
where,
{4i I = {W /&I (1 2.36)

a, = [H:, /e], ep (12.37)
where {x>is the nodal force vector.
Applying the plastic flow theory, the increments of plastic nodal displacement of the element
due to plasticity at node i, are estimated as (Ueda and Yao, 1982):
{du p } = dai {4, } (12.38)

where dR, is a measure of the magnitude of plastic deformation.
In the following paragraphs, we shall establish a relationship between dqp and d;li using a
plastic work procedure (Ueda and Fujikubo, 1986 and Fujikubo et al, 1991). The increment of
the plastic work done at the plastic node "i" is expressed as:

dwip = {xy bu,! ] = {x}' {4,}dAj (12.39)
The increment of the plastic work done in the actual plastic region around node i is evaluated
as:

(12.40)

From Eq. (12.21) the increment of the equivalent plastic strain at a coordinate s, can be
expressed as a fimction of the value at node i in the form:
dFP = g(s)dFp (12.41)