Civil Engineering Reference
In-Depth Information
For
β
-values below
β
it is only conditionally stable. The stability limit is then given
0
by
1
Δ≤Δ
t
t
=
(9.202)
cr
ω
ββ
−
i
0
For
γγ
=
there is no numeric (artificial) damping in the system. With
γγ
>
or
0
0
positive or negative numeric damping is introduced into the system. Positive
numeric damping may be used as an effective tool to dampen out undesirable effects of
higher modes in the system (which may also be obtained by adopting Hilber, Hughes &
Taylor method with
γγ
<
0
13
0
12
0
Newmark's method
becomes identical to the second central difference method, in which case the stability
limit is given by
−<<
α
). With
γ =
and
β =
Δ=
t
2
ω
, where
ω
is the largest eigen-frequency contained in the
cr
i
i
12
14
system. If
then Newmark's method becomes identical to a
numerical integration method based on the assumption of a constant average
acceleration, which is unconditionally stable. If
γ =
and
β =
then Newmark's
method becomes identical to a numerical integration method based on the assumption of
a linear variation of the acceleration, in which case the stability limit is given by
12
γ =
12
and
β =
16
t
Δ=
ω
.
cr
i
(
)
t
12
R
As previously mentioned
Δ
should never be chosen larger than about
ω
max
where
ω
is the largest frequency contained in the load.
R
max
Tangent-stiffness approach:
For heavily non-linear displacement or material problems the stiffness may change
considerably throughout the response process, in which case necessary accuracy may
only be obtained by updating the stiffness from one time step to the next. In such cases a
tangent-stiffness approach may be adopted. Assuming sufficiently short time steps and
linearity within each step, then the change of internal forces from
t
to
t
is given by
1
+
int
tan
Δ= ⋅ Δ
RK r
(9.203)
k
k
where
tan
k
is the updated tangent stiffness at
t
and
K
Δ=
rr
−
r
. Thus, the internal
k
1
k
+
t
force vector at
is
1
+
int
int
tan
RRKr
(9.204)
=
+
⋅ Δ
k
1
k
k
+
