SV = [ u′) ]max = Relative Velocity Response
(t
SA = [ y″(t) ]max = [ u″(t) + x″(t)]max =
Absolute Acceleration
Response
Then using the close approximation of ω = ω D for β ≤0.1,
the more commonly employed versions for engineering
purposes are:
Sv = ω ⋅SD = Pseudo-Relative-Velocity Response
(D-10)
(D-11)
For the common structural damping ratios, and the
earthquake type of input motion, there is essential equality
for the real and pseudo values,
S ≅ SV
(D-12)
v
S ≅ SA
(D-13)
a
Of course, for long period structures, the velocity equality
breaks down since S approaches zero, while SV
v
approaches peak ground velocity (PGV). The relationships
between SD and Sa can be justified by the following
physical behavior of the vibrating system. At maximum
relative displacement SD, the velocity is zero, and
maximum spring force equals maximum inertia force,
K ⋅SD = m ⋅Sa ,
where K is stiffness and m is mass, giving
Sa = K/m ⋅SD = ω 2 ⋅SD
(D-14)
Detailed discussions on response spectra and their
computation from accelerograms are given in Ebeling
1992, Chopra 1981, Clough and Penzien 1993, and
Newmark and Rosenblueth 1971. An example of a typical
acceleration response spectrum is shown in Figure D-5(b).
Also, because of the relation Sa = ω Sv = ω 2 SD, it is
possible to represent spectra on tripartite log paper (Figure
D-9).
D-15
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