Civil Engineering Reference
In-Depth Information
m
d
=
L
m
(6.61)
1
where
d
is the imposed input displacement and
L
m
is the change in length of an
element. This expression states that the summation of the element displacements is
equal to the overall imposed displacement.
Knowing
L
o
,
m
+
L
m
L
o
,
m
ε
L
,
m
=
ln
(6.62)
or
L
m
=
L
o
,
m
e
ε
L
,
m
−
1
(6.63)
Hence,
m
e
ε
L
,
m
−
1
d
=
L
o
,
m
(6.64)
1
Finally,
m
L
o
,
m
(
e
ε
L
,
m
−
1
)
−
d
=
0
(6.65)
1
It should be noted that
L
o
,
m
can vary along the length of the specimen per each ele-
ment and it is the initial length of the element from the prior time step (
i
−
1) as a
result of
d
being defined as an increment (i.e., constant).
Therefore, using Eqs. (
6.60
) and (
6.65
), the strain in each element in the axial
direction can be determined by solving the system of equations. These strains
can then be used to determine the strain in the other directions for each element
using an associated flow rule, and the new dimensions of the elements can be
determined. Since the corresponding stress and area in each element relate to an
overall equivalent force, this force can be determined and used with the displace-
ment given to produce a force and displacement profile (just as would occur for an
experimental test).
6.2.2 Deformation/Strength Model Solution Method
There are several numerical methods to find the roots of this nonlinear implicit
system of equations. The Newton-Raphson method is efficient and converges
quickly given good initial approximations of the variables. The method is given by
Chapra and Canale [
7
] for a system of nonlinear equations:
