Page 258 - Marine Structural Design

234 Part II Ultimate Strength

where,

substituting Eq. (12.41) into Eqs. (12.40) and (12.42) is obtained:

(12.42)

Equating the plastic work increments mv,P in Eq. (12.39) and dwip in Eq. (12.42), Eq. (12.43)
is obtained:
dcp = h,dil, (12.43)

where,

(12.44)
A simpler alternative approach for determining the strain-hardening rate at a plastic node is to
establish relationships between the plastic nodal displacements and the generalized plastic
strain vector at the node in the form of

{dqp = @up ]/Ldi = {+i Idai /L, (12.45)
where Ldi denotes an equivalent length of the plastic region.
The increment of the equivalent plastic strain at the node can be evaluated by substituting Eq.
(12.45) into Eq. (12.9) and obtain Eq. (12.43):

hi = (ai -ai Y {+i 14Ldiooi) (12.46)
Integration along the axial axis of the element becomes unnecessary when Eq. (12.46) is
applied to calculate ds,! instead of Eq. (12.44). This results in an extremely simple numerical
procedure. Unfortunately, the actual regions where the plastic flow occurs, causes a change in
shape, size, and may disappearhe-appear. Evidently, the equivalent length of the plastic region
for each stress component should be different and considered to be a function of time.
However, for simplicity, we would like to find a constant value that will provide adequate
approximations. Then length Ldi can simply be approximated as:
L, = a,H (12.47)
or

L, = aLL (12.48)
where aD and aL are coefficients, H is the diameter for a circular cross-section or a width (or
height) for a rectangular cross-section, etc. This approach will be used in the case where a
structural member is modeled by only one element. Substitution of Eq. (12.43) into Eq.
(12.35) gives:
de. = {+i >’ {&}- H:,dA, + a, = 0 (1 2.49)