Dynamic/earthquake analysis - Computational Analysis and Design of Bridge Structures - page 520

By using modal response history analysis (RHA), the modal coordinate
q n ( t ) is governed by

 

q 2 q

2

q  u t ( )

+ ζ ω + ω = − Γ (17.10)

n n n n n n n g

In which ω n is the natural vibration frequency and ζ n is the damping ratio
for the n th mode. The solution q n ( t ) can readily be obtained by comparing
Equation 17.10 to the equation of motion for the n th-mode elastic SDOF
system, an SDOF system with vibration properties — natural frequency ω n
and damping ration ζ n — of the n th mode of the MDOF system, subjected
to ü g ( t ) .
Besides RHA, modal response spectrum analysis (RSA) was also adopted
for linear seismic analysis where the peak modal response can be combined
by the conservative absolute sum (ABSSUM) modal combination rule:

N

∑ 0
1

r ≤ r (17.11)

n n

n =

or by the more reasonable square-root-of-sum-of-square (SRSS) rule:

½

N

∑ 2
1

r ≅ r (17.12)

n n

n =

or by the complete quadratic combination (CQC) rule to a system with
closely spaced natural frequencies:

½

N N

∑ ∑

r ≅ ρ r r (17.13)

n i 0 n 0

in

j = 1 n = 1

17.2.3.1 Linear and nonlinear seismic analyses
Four distinct analytical procedures, as shown in Figure 17.7, can be used in
systematic rehabilitation of structures (FEMA-273 1997): linear static, lin-
ear dynamic, nonlinear static (pushover), and nonlinear dynamic procedures
(NDPs). Linearly elastic procedures (linear static and linear dynamic) are
the most common procedures in seismic analysis and design of structures
due to their simplicity. On the other hand, adjustments to overall deforma-
tions and material acceptance criteria can be incorporated to consider the
inelastic response. Based on their importance, bridges can be classified as
either ordinary or important bridges where ordinary bridges can be further
defined as standard and nonstandard ordinary structures. In their Caltran
study, Aviram et al. (2008) described bridge seismic analysis types based on
the bridge classifications, which are also listed in Table 17.1.