Civil Engineering Reference
In-Depth Information
It is seen that the conditions at
t
may be defined by
1
1
1
⎫
⎛
⎞
a
r
r
1
r
=
+
+
−
⋅
⎜
⎟
⎪
k
k
k
k
2
β
Δ
t
2
β
t
β
Δ
⎝
⎠
⎪
⎬
(9.190)
γ
⎛
γ
⎞ ⎛
γ
⎞
⎪
1
1
t
b
=
r
+
−
⋅
r
+
−
Δ ⋅
r
⎜
⎟ ⎜
⎟
k
k
k
k
⎪
t
2
β
Δ
β
β
⎝
⎠ ⎝
⎠
⎭
r
r
in which case
and
simplifies into
k
1
k
1
+
+
1
⎫
r
r
a
=
−
⎪
k
1
k
1
k
+
+
2
β
t
Δ
⎪
⎬
⎪
(9.191)
1
r
=
r
−
b
k
+
1
k
+
1
k
⎪
t
β
Δ
⎭
t
Introducing this into the dynamic equilibrium equation at
+
1
k
Mr
+
C
r
+
K
r
=
R
(9.192)
1
net k
1
net k
1
dyn
k
+
+
+
+
1
will then render
⎛
1
⎞
γ
MCKr
R
Ma
Cb
+
+
⋅
=
+
⋅
+
⋅
(9.193)
⎜
⎟
⎜
⎟
net
net
k
+
1
dyn
k
net
k
2
k
1
β
Δ
t
+
t
β
Δ
⎝
⎠
Defining
1
γ
⎫
K
=
M
+
C
+
K
⎪
eff
net
net
k
+
1
2
t
t
β
Δ
β
Δ
(9.194)
⎬
⎪
R
=
R
+
Ma
⋅
+
C
⋅
b
eff
dyn
k
net
k
⎭
k
+
1
k
+
1
and thus
1
−
r
=
K R
(9.195)
⋅
k
+
1
eff
eff
k
+
1
k
+
1
t
t
It is seen that the response at time step
as well as
the displacement, velocity and acceleration response at
t
. If the system is entirely
linear, then
is calculated from the load at
+
1
+
1
K
is constant throughout the calculations.
eff
