...
Direct-integration
...
...
...
analysis
...
is
...
a
...
...
,
...
dynamic
...
analysis
...
method
...
in
...
which
...
the
...
equilibrium
...
equations
...
of
...
motion
...
are
...
fully
...
integrated
...
as
...
a
...
structure
...
is
...
subjected
...
to
...
dynamic
...
loading.
...
Analysis
...
involves
...
the
...
integration
...
of
...
structural
...
properties
...
and
...
behaviors
...
at
...
a
...
series
...
of
...
time
...
steps
...
which
...
are
...
small
...
relative
...
to
...
loading
...
duration.
...
The
...
equation
...
of
...
motion
...
under
...
evaluation
...
is
...
given
...
as
...
follows:
| Info |
|---|
NOTE: Integration is performed at every time step of the input record, regardless of the output increment. |
Stability conditions
For well-behaved problems, we recommend using the Newmark method with
| Wiki Markup |
|---|
{math}beta{math} |
...
...
1
...
/
...
4
...
,
...
which
...
yields
...
the
...
constant
...
average
...
acceleration
...
method,
...
and
...
for
...
poorly
...
converging
...
nonlinear
...
...
...
cases,
...
the
...
Hilber-Hughes-Taylor
...
(
...
...
)
...
method
...
with
...
0
...
<
| Wiki Markup |
|---|
{math}alpha{math} |
...
...
-
...
1
...
/
...
3
...
.
...
During
...
HHT
...
application,
...
when
| Wiki Markup |
|---|
{math}alpha{math} |
...
...
0,
...
formulation
...
is
...
identical
...
to
...
the
...
average
...
acceleration
...
method,
...
so
...
HHT
...
will
...
actually
...
suffice
...
for
...
all
...
problems.
...
Note
...
that
...
all
...
nodes
...
should
...
have
...
...
,
...
and
...
rotational
...
inertia
...
may
...
be
...
added
...
to
...
improve
...
dynamic
...
response.
...
The
...
Newmark
...
method
...
is
...
unconditionally
...
stable
...
only
...
when
...
2
| Wiki Markup |
|---|
{math}beta{math} |
...
| Wiki Markup |
|---|
{math}gamma{math} |
...
...
1
...
/
...
2
...
,
...
which
...
is
...
why
...
we
...
recommend
...
using
| Wiki Markup |
|---|
{math}beta{math} |
...
...
1
...
/
...
4
...
.
...
When
| Wiki Markup |
|---|
{math}beta{math} |
...
...
1
...
/
...
6
...
,
...
which
...
yields
...
the
...
linear
...
acceleration
...
method,
...
formulation
...
is
...
only
...
conditionally
...
stable,
...
and
...
can
...
become
...
unstable
...
when
...
dt
...
/
...
T
...
>
...
2√
...
3
...
/
...
(2
...
)Wiki Markup {math}pi{math}
...
,
...
where
...
dt
...
is
...
the
...
time
...
step
...
and
...
T
...
is
...
the
...
shortest
...
structural
...
period
...
which
...
is
...
excited
...
by
...
loading.
...
The
...
shortest
...
period
...
may
...
be
...
obtained
...
during
...
...
analysis
...
such
...
that
...
time
...
step
...
may
...
be
...
coordinated
...
to
...
ensure
...
convergence.
See Also
- Mass and stiffness-proportional damping – Damping of stiff elements with inelastic softening
- Output accuracy – Time-history output-acceleration accuracy
- Ground motion at specific supports – Manual multi-support excitation