Civil Engineering Reference
In-Depth Information
Thus, the linear system can be expressed in the standard form:
(6.73)
Ax
=
b
or
[
J
]
{
X
i
+
1
}
T
=−{
F
i
}
T
+
[
J
]
{
X
i
}
T
(6.74)
This system can be solved using a technique such as Gauss elimination. The process can
be repeated iteratively to obtain refined estimates of the unknown variables. For the sim-
ulations performed in this work, the Newton-Raphson method was repeated until the
norm (i.e., vector length) of the root vector was very small (i.e., less than 1
×
10
−
10
).
Once the strains in each element are known in the axial direction, the strain in
the width and thickness direction can be determined using a material flow rule.
Presently, it is proposed to assume the von Mises yield criterion and material isot-
ropy (note: other yield criterion or anisotropy could be applied here as well). This
results in the Levy-Mises equations:
d
ε
1
2
σ
1
=
d
ε
2
−
σ
1
=
d
ε
3
(6.75)
−
σ
1
where
d
ε
p
is the incremental strain in the primary three directions and
σ
1
is the
stress applied along the length axis.
This results in
ε
w
,
m
= ε
t
,
m
=−
1
2
ε
L
,
m
(6.76)
After determining the strain in each direction (length, width, and thickness), the
new element geometry can be determined by
L
i
+
1
m
=
L
o
,
m
e
ε
L
,
m
w
i
+
1
m
=
w
o
,
m
e
ε
w
,
m
(6.77)
t
i
+
1
m
=
t
o
,
m
e
ε
t
,
m
where
L
i
+
1
m
m
,
and
t
i
+
m
are the new length, width, and thickness of the element,
respectively. The new cross-sectional area
,
w
i
+
1
A
i
+
1
m
of the elements can be calculated
using the new width and thickness by
A
i
+
1
m
=
w
i
+
1
t
i
+
1
m
(6.78)
m
The stress for each element can be given by
σ
m
ε
L
,
m
,total
,
T
m
=
K
m
ε
n
L
,
m
,total
e
ε
L
,
m
,total
s
m
(6.79)
where the constants are calculated from Eqs. (
6.55
)-(
6.57
) and the total strain in
the axial direction of the element is used. The force is calculated (equal for each
element) as follows:
F
=
A
i
+
1
m
σ
m
(6.80)
