Civil Engineering Reference
In-Depth Information
Hilber, Hughes & Taylor [33] have suggested an extension of Newmark's method by
the introduction of the numerical damping coefficient
0
α ≤
into the dynamic
equilibrium condition
(
)
(
)
1
1
Mr
++
α
Cr
−
α
Cr
++
α
Kr
−
α
Kr R
=
(9.196)
k
+
1
net k
+
1
net k
net k
+
1
net k
dyn
α
and accordingly, evaluate the dynamic load at a
(
)
1
t
t
t
t
+
α
−
α
=
+
α
Δ
. I.e., if
k
+
1
k
k
+
1
load linearity within the time step is adopted, then
(
)
R
=+
1
α
R
−
α
R
(9.197)
k
1
k
α
+
Combining Eqs. 9.191 and 9.196 and solving for
r
will then again render
k
1
+
−
1
r
=
K R
, but now
⋅
K
and
R
are extended into
k
1
eff
eff
eff
k
eff
k
+
k
1
k
1
1
1
+
+
+
+
1
γ
⎫
(
)
(
)
1
1
K
=
M
+
+
α
C
+
+
α
K
⎪
eff
net
net
k
1
2
t
+
t
β
Δ
β
Δ
(9.198)
⎬
⎪
(
)
1
R
=+
α
R
−
α
R
+⋅
M a
+
C
⋅
b
+
⎭
α
c
eff
k
+
1
k
k
net
k
k
k
+
1
where
⎛
γ
⎞
γ
⎛
γ
⎞
1
t
(9.199)
c
=
C
+
K
⋅
r
+
C
r
+
C
−
Δ ⋅
r
⎜
⎟
⎜
⎟
k
net
net
k
net
k
net
k
t
2
β
Δ
β
β
⎝
⎠
⎝
⎠
For the numeric integration methods the establishment of initial conditions at
t
=
0
r
. Dynamic
before the iteration procedure can start requires the choice of
r
and
t
=
0
equilibrium at
renders the corresponding acceleration
(
)
1
−
=⋅
rM R
−
CrKr
−
(9.200)
0
dyn
net
0
net
0
0
and thus, iteration may commence. Stability may be evaluated from the properties of a
single degree of freedom system (or a modal approach) similar to that which has been
shown for the central difference method shown above. In general, the Newmark method
is unconditionally stable if
2
1
1
⎛
⎞
12
γ≥= and
ββ
≥=
γ
+
⎠
(9.201)
⎜
⎟
0
0
4
2
⎝
