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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