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TABLE 10.3
Critical Time Step Estimates for the Central Difference Method for Various Elements
Type of
Critical
Element
Mass Matrix
Time Step ∆
t
Two-node bar
Lumped
/
c
/
√
3
c
Two-node bar
Consistent
2
]
−
1
/
2
Two-node beam
Lumped
min
{
/
c,
(/
c
s
)
[1
+
A
/
I
(/
2
)
}
c
d
g
1
/
2
Four-node quadrilateral
Consistent
2
/
(Chapter 6, Section 6.7.2)
where
=
El
eme
nt length
=
√
E
c
/γ
, the bar wave velocity
γ
=
Mass density
c
s
=
Gk
s
/γ
, the beam shear wave velocity
I
=
Moment of inertia
A
=
Area of the beam cross-section
c
d
=
G
(
4
G
−
E
)/
[
(
3
G
−
E
)/γ
]
A
2
i
=
1
j
=
1
B
ij
B
ij
in which
B
ij
is an entry in
g
=
4
/
(
1
2
y
2
−
y
4
)(
y
3
−
y
1
)(
y
4
−
y
2
)(
y
1
−
y
3
)
B
=
(
x
4
−
x
2
)(
x
1
−
x
3
)(
x
2
−
x
4
)(
x
3
−
x
1
)
and
A
is the area of the quadrilateral element.
In order to have a stable solution using the central difference explicit integration algo-
rithm, the time step
t
used for integration must be smaller than the critical time step as
given by Eq. (10.109). When the frequency gets higher, the allowable time step will de-
crease. The conditionally stable characteristic is the main disadvantage in using an explicit
integration method.
For different types of finite elements, the critical time step is different. Table 10.3 lists
these time steps corresponding to different elements. These time step estimates are derived
in Hughes (1987).
Equation (10.109) is often employed for multi-DOF systems in which
t
crit
=
2
/ω
max
,
where
max
is the largest natural frequency of the system, i.e., the
n
d
th natural frequency
of the
n
d
DOF model of the structure.
ω
10.5.2
Houbolt Method
Houbolt (1950) used the following finite difference equations for velocity and accelera-
tions at time
t
=
(
n
+
1
)
t
1
(
V
n
+
1
=
2
[2
V
n
+
1
−
5
V
n
+
4
V
n
−
1
−
V
n
−
2
]
t
)
(10.110)
1
V
n
+
1
=
t
[11
V
n
+
1
−
18
V
n
+
9
V
n
−
1
−
2
V
n
−
2
]
6
These equations were obtained from consideration of a cubic curve that passes through
four successive ordinates.
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