Civil Engineering Reference
In-Depth Information
T
IME
-D
OMAIN
S
OLUTIONS
Step-By-Step Integration
Explicit Schemes
Implicit Schemes
E
LASTIC
/I
NELASTIC
E
LASTIC
Spectral Analysis
F
REQUENCY-
D
OMAIN
S
OLUTIONS
Complex Frequency Analysis
Modal Analysis
Power Spectra
E
LASTIC
Figure 4.32
Methods of dynamic analysis of structures
Note: The nature of modal and modal-spectra analysis is considered herein as spanning between time and frequency
domains
The matrix form of the dynamic equilibrium of motion given in equation (4.9.2) is identical to
the equation of motion for single-degree of freedom (SDOF) systems given by equation (3.14) in
Chapter 3. However, mass, damping and restoring forces (or stiffness for linearly elastic structures)
for MDOF systems are expressed by matrices of coeffi cients representing the additional degrees of
freedom.
Several methods of dynamic analysis of structures exist as shown in Figures 4.31 and 4.32 . These
methods can be employed either in the time or the frequency domain. The most commonly used methods
for dynamic analysis of structures subjected to earthquake loads are modal, spectral and response
history. These methods are presented hereafter. It is, however, beyond the scope of this topic to provide
a comprehensive discussion of numerical algorithms used for each method. The reader is referred to
the existing extensive literature (e.g. Bathe, 1996, among many others). It is important to note the
special nature of modal analysis. Modal decomposition of the coupled equations of motion leads to a
number of equations describing the motion of individual modes in the time domain. Once the maxima
of response quantities are evaluated without response history analysis, the method crosses the boundary
between time domain, where the individual modes are described, and the frequency domain, where the
maxima may be considered with no reference to their time of occurrence. It is the writer's opinion that
modal analysis has time-domain as well as frequency-domain features, but this view is not universally
accepted.
