Civil Engineering Reference
In-Depth Information
time-stepping techniques where series of coupled equations of motion are solved as static equilibrium
systems, but including inertia and damping effects. Time-stepping or response-history analysis is the
most natural and intuitive approach. It, however, requires signifi cantly more computing resources than
modal and spectral methods. It is noteworthy that the individual modes equations of motion derived
by decoupling in Section 4.6.1.1 may be solved in the time domain using the methods outlined in the
current section, thus providing a link between frequency- domain and time - domain solutions.
When subjected to strong ground motion, structures generally undergo deformations in the inelastic
range, as discussed in Section 2.3.3. Since their deformation is also relatively large, geometric non-
linearity may be signifi cant. Analysis of non-linear and inelastic systems subjected to seismic loads
involves continuously changing temporal solution characteristics. This is due to changes in stiffness,
and hence periods of vibration. To compute the response history of inelastic structures, it is necessary
to integrate directly the coupled equations of dynamic equilibrium [given by equations (4.9.1) and
(4.9.2)] as the principle of superposition is not applicable. Many numerical integration schemes are
available in the literature. A review is provided by Dokainish and Subbaraj (1989) , Subbaraj and
Dokainish (1989) and Wood (1990). Time-marching schemes are either conditionally stable (explicit)
or unconditionally stable (implicit). The response history is divided into time increments Δ
t
and the
structure subjected to a sequence of individual time-dependent force pulses Δ
F
(
t
). During each Δ
t
, the
structure is assumed to be linear and elastic. Between intervals, the material and geometry components
of the system stiffness matrix are modifi ed to refl ect the current state of deformation. The non-linear
response is thus approximated by a series of piece-wise linear systems.
The steps required to perform response history analysis of MDOF structures subjected to seismic
loads are as follows:
(a) Formulate the equation of motion for the discretized structure in incremental form as given
below:
()
()
MxCx K
D
+
D
+
t
D
x
=
D
F
t
(4.26)
t
where
K
t
(
t
) is the stiffness matrix for the time increment beginning at time
t
, and Δ
x
is the dis-
placement increment during the time interval Δ
t
.
(b) Integrate the incremental form given in equation (4.26) for each time step by using one of the
numerical integration schemes available in the literature.
(c) Evaluate the increments of displacement, velocity and displacement at the given time step.
(d) Update the displacement, velocity and acceleration at the beginning of the interval to derive the
corresponding quantities at the end of the time step interval.
(e) Evaluate stress states corresponding to the total displacements at the end of the given time step.
(f)
Update the tangent stiffness matrix
K
t
(
t
), if necessary.
The above steps show that the determination of the matrix K
t
(
t
) for each increment is the most
demanding part in the response history analysis. All individual member stiffnesses are re- computed
within each time increment and iteration within the time increment. This requires considerable comput-
ing resources for large structural systems.
The selection of the integration scheme to solve equation (4.26) and the value of integration operators
have signifi cant effects on the results. Manipulating algorithmic damping (intentionally) or falling
victim to it (inadvertently) could lead to 50% or more variation in force response (Broderick
et al
.,
1994). The selection of damping parameters in the presence of hysteretic (material) damping is also a
serious consideration that affects the results of the analysis.
In the modal and spectral analysis described in Section 4.6.1.1, the damping matrix
C
in the equation
of motion is often represented as a linear combination of the mass and stiffness matrices, e.g. equation
(4.16.1). Similarly, for non- linear systems,
C
can be assumed as a linear combination of the mass and
