Civil Engineering Reference
In-Depth Information
where
u
i
,
u
i
, and
u
i
are approximations to the displacement, velocity, and acceleration at step
i
;
t
is the time interval.
Newmark (1959) proposed one of the most popular algorithms for the solution of structural
dynamics. The method is based on the following interpolations of displacements, velocities,
and accelerations from step
i
to step
i
+
1:
u
i
+
1
=
u
i
+
[(1
−
γ
)
t
]
u
i
+
(
γ
t
)
u
i
+
1
,
(9.51)
t
)
u
i
+
(0
t
)
2
u
i
+
β
t
)
2
u
i
+
1
u
i
+
1
=
u
i
+
(
.
5
−
β
)(
(
(9.52)
where
u
i
+
1
,
u
i
+
1
, and
u
i
+
1
are approximations to the displacement, velocity, and acceleration
at step
i
+
1;
β
and
γ
are the parameters that define the variation of the Newmark method.
1
2
and
6
≤
β
≤
1
1
1
4
,
the Newmark method becomes a special case called the 'average acceleration' method. The
average acceleration method is unconditionally stable. When
Typical selection for
γ
and
β
is
γ
=
4
, respectively. When
γ
=
2
and
β
=
1
6
the Newmark
method becomes a special case called the 'linear acceleration' method. The linear acceleration
method is stable if
1
2
γ
=
and
β
=
t
T
n
≤
0
.
551
where
T
n
is the shortest natural period of the structure.
For the incremental formulation required by the nonlinear system, Equations (9.51) and
(9.52) can be rewritten as
u
i
=
(
t
)
u
i
+
(
γ
t
)
u
i
(9.53)
t
)
2
2
(
t
)
2
u
i
=
(
t
)
u
i
+
u
i
+
β
(
u
i
(9.54)
Equation (9.54) can be solved as
1
1
1
2
u
i
=
t
)
2
u
i
−
t
)
u
i
−
u
i
(9.55)
β
(
β
β
(
Substituting Equation (9.55 into (9.53) gives
t
1
u
i
γ
β
u
i
−
γ
β
2
u
i
=
t
u
i
+
−
(9.56)
β
The incremental equation of motion is given by
m
u
i
+
c
u
i
+
k
u
i
=
p
i
(9.57)
Substituting Equations (9.55) and (9.56) into Equation (9.57) gives
k
i
u
i
=
p
i
(9.58)
where
γ
β
1
k
i
=
k
i
+
t
c
+
t
)
2
m
(9.59)
β
(
