5.1 Basics of the Layered Approach
107
Fig. 5.5 Difference between the a strain and b stress distribution in the elasto-plastic range for an
ideal plastic material. The strain and stress state is uniaxial where only a single normal strain and
stress in acting
Introducing the normalized displacements ˆ
u Y in Eq. (5.9), ones obtains
(ε X ) k,i = −
Z
k
c ×
ˆ
u i+1 − 2 ˆ
u i + ˆ
u i−1
2
×
M
pl
lim L
2
E I Y
i
.
(5.11)
Based on the relation M
pl
lim = 3k I Y / h (see [3]) and Hooke’s law, the normalizing
factor can be expressed as:
M
pl
lim L
2
E I Y
=
3k I Z
E I Z h
=
3L
2
h
×
k
E
Hooke
=
3L
2
h
× ε
init
.
(5.12)
Finally, one can state the condition for an elastic layer k at node i as:
(ε X ) k
ε init
i
=
Z
k
c ×
ˆ
u i+1 − 2 ˆ
u i + ˆ
u i−1
X 2
×
3L
2
h
i
≤ 1 (elastic) .
(5.13)