Civil Engineering Reference
In-Depth Information
The dynamic equilibrium condition at
t
is then given by (see Eq. 9.192)
1
+
int
tan
Mr
k
+
C
r
+
R
+
K
Δ
r
=
R
(9.205)
1
net k
1
k
net
dyn
+
+
k
k
1
+
Introducing the Newmark iteration scheme given in Eqs. 9.188 and 9.189 (and that
r
−=Δ
r
r
), then the following is obtained
k
+
1
k
⎛
1
1
⎞
tan
int
MCK r
+
+
⋅ Δ
=
R
−
R
⎜
⎟
⎜
net
net
⎟
dyn
k
2
t
k
k
1
t
β
Δ
+
β
Δ
⎝
⎠
(9.206)
⎡
1
1
⎤
⎡
1
⎤
⎛
⎞
⎛
γ
⎞ ⎛
⎞
+⋅
Mr
+
−⋅
1
r
+
C
⋅
−⋅
1
r
+
−⋅ Δ⋅
1
t
r
⎢
⎜
⎟
⎥
⎢
⎜
⎟ ⎜
⎟
⎥
k
k
net
k
k
t
2
2
β
Δ
β
β
β
⎝
⎠
⎝
⎠ ⎝
⎠
⎣
⎦
⎣
⎦
Thus
(
)
1
int
−
Δ=
rK
⋅
R
−
R
+
Ma
⋅
+
C b
⋅
(9.207)
eff
dyn
k
eff
net
eff
k
k
+
1
k
k
where
1
1
⎫
tan
K
=
M
+
C
+
K
⎪
eff
net
net
k
2
k
β
Δ
t
β
Δ
t
⎪
⎪
1
1
⎛
⎞
⎪
1
a
=
r
+
−
⋅
r
(9.208)
⎬
⎜
⎟
eff
k
k
k
t
2
1
β
Δ
β
⎝
⎠
⎪
⎪
⎛
γ
β
⎞ ⎛
⎞
⎪
b
=−⋅
1
r
+ −⋅ Δ⋅
1
t
r
⎜
⎟ ⎜
⎟
eff
k
k
k
2
β
⎪
⎝
⎠ ⎝
⎠
⎭
Such a procedure will generally require error control. This may be obtained by
minimising the estimated external load error
err
k
Δ
R
, defined as the difference between
+
1
t
the actual load at
and the corresponding load which can be calculated from the
estimated displacements
+
1
(
)
err
k
int
Δ=
RR
−
Mr
+
Cr
+
R
(9.209)
1
dyn
k
1
net k
1
k
+
+
+
k
1
+
est
Thus, iterations until
err
k
R
is less than a specified limit will be required within each
time step. Initial conditions and stability criteria are identical to those presented above
for the numeric integration methods.
Δ
1
+
