Chapters 1, 2, 3 and 8)
Chapter 2 GENERAL FEATURES OF STRUCTURES WITH SEISMIC ISOLATION SEISMIC ISOLATION
2.2 ROLE OF EARTHQUAKE RESPONSE SPECTRA AND VIBRATIONAL MODES IN THE PERFORMANCE OF ISOLATED STRUCTURES THE PERFORMANCE OF ISOLATED STRUCTURES
2.2.1 Earthquake Response Spectra
The horizontal forces generated by typical design-level earthquakes are greatest on structures with low flexibility and low vibration damping. The seismic forces on such structures can be reduced greatly by supporting the structure on mounts which provide high horizontal flexibility and high vibration damping. This is the essential basis of seismic isolation. It can be illustrated most clearly in terms of the response spectra of design earthquakes.
The main seismic attack on most structures is the set of horizontal inertia forces on the structural masses, these forces being generated as a result of horizontal ground accelerations. For most structures, vertical seismic loads are relatively unimportant in comparison with horizontal seismic loads. For typical design earthquakes, the horizontal accelerations of the masses of simple shorter-period structures are controlled primarily by the period and damping of the first
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The dominance of the first mode occurs in isolated structures, and in unisolated structures with first-mode periods up to about 1.0 seconds, a period range which includes most structures for which isolation may be appropriate.
Neglecting the less important factors of mode shape and the contribution of higher modes of vibration, the seismic acceleration responses of the isolated and unisolated structures may be compared broadly by representing them as single-mass oscillators which have the periods and dampings of the first vibrational modes of the isolated and unisolated structures respectively.
The natural (fundamental) period T, natural frequency and damping factor of such a single-mass oscillator, of mass m, are obtained by considering its equation of motion:
where u is the displacement of the single-mass oscillator relative to the ground, ug is the ground displacement, k is the 'spring stiffness' and c is the 'damping coefficient'.
The natural (fundamental) frequency of undamped, unforced oscillations (c=0 and üg=0) is
or
The solution for damped, unforced oscillations is
where
and where A and are constants representing the initial displacement amplitude and initial phase of the motion.
The damped, unforced oscillation has thus a lower frequency d than the natural frequency, and d decreases as the value of the damping coefficient c is increased. If c is increased to a 'critical value' ccr such that d=0, the system will not oscillate. The critical damping is given by
A 'damping factor' can then be defined which expresses the damping as a fraction of critical damping:
The equation of motion can then be divided by m to give
or
mÅ -
= ku + u c +
mÅ g (2.1)
k/m
2=
(2.2)m/k 2
=
T
(2.3)) + t ( cos e
A
=
u
dt m) 2
-(c/
m ) 2 (c/
- (k/m)
=
22
dmk 2 c
cr=
m 4 cT/
= m 2 c/
= ) mk c/(2 c =
c/
=
cr
(2.4)- u
= m u + k u m + c
u
g- u
= u + u 2 +
u
2
g (2.5)For this (damped, forced) dynamic system, the displacement response to ground accelerations may be given in closed form as a Duhamel integral, obtained by expressing üg(t) as a series of impulses and summing the impulse responses of thesystem. When the system starts from rest at time t = 0, this gives the relative displacement response as:
By successive differentiation, similar expressions may be obtained for the relative velocity response u and the total acceleration response ü + üg. For particular values of and , the responses to the ground accelerations of a given earthquake may be obtained from step-by- step evaluation of equation (2.6) or from other evaluation procedures.
Since structural designs are normally based on maximum responses, a convenient summary of the seismic responses of single-mass oscillators is obtained by recording only the maximum responses for a set of values of the oscillator parameters (or T) and . These maximum responses are the earthquake response spectra. They may be defined as follows:
Such spectra are routinely calculated and published for important accelerograms, e.g. EERL Reports (1972.5).
Figure 2.1 shows response spectra for various damping factors (0, 2, 5, 10 and 20 percent of critical) for a range of earthquakes.
) Å ( ) exp[- (t - )] sin (t - ) d /
(1 -
=
u(t)
g dt
d
0 (2.6)) u(t
= ) , S (T
; ) (t u
= ) , S (T
; ) Å )(t + (Å
= ) ,
S A(T
g max V
max D
max (2.7)(a)
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Figure 2.1: Response spectra for various damping factors. In each figure, the curve with the largest values has 0% damping and successively lower curves are for damping factors of 2, 5, 10 and 20% of critical.
(a) Acceleration response spectrum for El Centro NS 1940
(b) Acceleration response spectrum for the weighted average of eight
accelerograms (El Centro 1934, El Centro 1940, Olympia 1949, Taft 1952). The symbols U and I refer to unisolated and isolated structures respectively.
(c) Displacement spectra corresponding to Figure 2.1(b).
Figure 2.1(a) shows acceleration response spectra for the accelerogram recorded in the S0oE direction at El Centro, California, during the 18 May 1940 earthquake (often referred to as 'El Centro NS 1940'). This accelerogram is typical of those to be expected on ground of moderate flexibility during a major earthquake. The El Centro accelerogram is used extensively in the following discussions because it is typical of a wide range of design accelerograms, and because it is used widely in the literature as a sample design accelerogram.
Seismic structural designs are frequently based on a set of weighted accelerograms, which are selected because they are typical of site accelerations to be expected during design-level earthquakes.
(b)
(c)
The average acceleration response spectra for such a set of 8 weighted horizontal acceleration components are given in Figure 2.1(b). Each of the 8 accelerograms has been weighted to give the same area under the acceleration spectral curve, for 2% damping over the period range from 0.1 to 2.5 seconds, as the area for the El Centro NS 1940 accelerogram (Skinner, 1964).
Corresponding response spectra can be presented for maximum displacements relative to the ground, as given in Figure 2.1(c). These displacement spectra show that, for this type of earthquake, displacement responses increase steadily with period for values up to about 3.0 seconds. As in the case of acceleration spectra, the displacement spectral values decrease as the damping increases from zero. The spectra shown in Figures 2.1(b) and 2.1(c) are more exact presentations of the concept illustrated in Figure 1.1.
While the overall seismic responses of a structure can be described well in terms of ground response spectra, the seismic responses of a light-weight sub-structure can be described more easily in terms of the response spectra of its supporting floor. Floor response spectra are derived from the accelerations of a point or 'floor' in the structure, in the same way that earthquake response spectra are derived from ground accelerations. Thus they give the maximum response of light-weight single-degree-of-freedom oscillators located at a particular position in the structure, assuming that the presence of the oscillator does not change the floor motion. It is also possible to derive floor spectra which include interaction effects. Floor response spectra tend to have peaks in the vicinity of the periods of modes which contribute substantial acceleration to that floor.
The response spectrum approach is used throughout this book to increase understanding of the factors which influence the seismic responses of isolated structures. The response spectrum approach also assists in the seismic design of isolated structures, since it allows separate consideration of the character of design earthquakes and of earthquake-resistant structures. A technique which is given some emphasis is the extension of the usual response spectrum approach for linear isolators to the case of bilinear isolators.