Civil Engineering Reference
In-Depth Information
other continuity condition is that the summation of the individual element dis-
placements adds to the total imposed displacement.
6.2.1 Deformation/Strength Model Derivation
This section contains the derivation of the deformation/strength model. The deri-
vation is written for a time step (
i
). Thus, at a given time step
i
, force equivalency
gives
F
=
F
m
=
F
m
+
1
for
m
= {
1
...(
m
−
1
)}
(6.38)
where
F
is the force and
m
is the number of nodes/elements in the model.
The force can be written in terms of stress and area by
σ
m
A
m
= σ
m
+
1
A
m
+
1
(6.39)
or
(6.40)
σ
m
t
m
w
m
= σ
m
+
1
t
m
+
1
w
m
+
1
where
t
and
w
are the thickness and width, respectively.
Knowing that
t
=
t
o
e
ε
t
(6.41)
w
=
w
o
e
ε
w
(6.42)
where
t
o
is the initial thickness,
ε
t
is the thickness strain,
w
o
is the initial width, and
ε
w
is the width strain. This yields
σ
m
t
o
,
m
w
o
,
m
e
ε
t
,
m
e
ε
w
,
m
=
σ
m
+
1
t
o
,
m
+
1
w
o
,
m
+
1
e
ε
t
,
m
+
1
e
ε
w
,
m
+
1
(6.43)
or
σ
m
t
o
,
m
w
o
,
m
e
ε
t
,
m
+
ε
w
,
m
=
σ
m
+
1
t
o
,
m
+
1
w
o
,
m
+
1
e
ε
t
,
m
+
1
+
ε
w
,
m
+
1
(6.44)
Since volume is conserved,
ε
t
+ ε
w
+ ε
L
=
0
or
ε
t
+ ε
w
=−ε
L
(6.45)
where
ε
L
is the incremental strain in the length direction (i.e., along axial length).
Incremental strain is the strain in each individual element over one time step. The
incremental strain accrues over time to the accumulative strain of each element.
This gives
σ
m
t
o
,
m
w
o
,
m
e
−
ε
L
,
m
=
σ
m
+
1
t
o
,
m
+
1
w
o
,
m
+
1
e
−
ε
L
,
m
+
1
(6.46)
or
σ
m
A
o
,
m
e
−
ε
L
,
m
=
σ
m
+
1
A
o
,
m
+
1
e
−
ε
L
,
m
+
1
(6.47)
