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
2.2.3 Parameters of Linear and Bilinear Isolation Systems
A typical isolated structure is supported on mounts which are considerably more flexible under horizontal loads than the structure itself. It is assumed here that the isolator is at the base of the structure and that it does not contribute to rocking motions. As a first approximation, the structure is assumed to be rigid, swaying sideways with approximately constant displacement along its height, corresponding to the first isolated mode of vibration.
Some isolation systems used in practice are 'damped linear' systems such as presented in equations (2.1) and (2.5). However, an alternative approach, for the provision of high isolator flexibility and damping, is to use nonlinear hysteretic isolation systems, which also inhibit wind-sway.
Such nonlinearity is frequently introduced by hysteretic dampers, or by the introduction of sliding components to increase horizontal flexibility, as discussed in Chapter 3. These isolation systems can usually be modelled approximately by including a component which slides with friction, and gives a bilinear force-displacement loop when the model is cycled at constant amplitude.
Models of linear and bilinear isolation systems, with the structure modelled by its total mass M, are shown in Figures 2.2(a) and 2.3(a).
Figure 2.2: Schematic representation of a damped linear isolation system.
(a) Structure of mass M supported by linear isolator of shear stiffness Kb, with velocity damper (viscous damper) of coefficient Cb.
(b) Shear force S versus displacement X showing the hysteresis loop and defining the secant stiffness of the linear isolator: Kb = Sb/Xb.
(c) Linear isolator with high damping coefficient and higher-mode attenuator Kc. (a)
(b)
(c)
18
Figure 2.3: Schematic representation of a bilinear isolation system.
(a) Structure of mass M supported by bilinear isolator which has linear 'spring' components of stiffnesses Kb1 and Kb2, together with a sliding (Coulomb) damper component.
(b) Shear force versus displacement showing the bilinear hysteresis loop and defining the secant stiffness of the bilinear isolator: KB = Sb/Xb. The
individual stiffnesses Kb1 and Kb2 are the slopes (gradients) of the hysteresis loop as shown, and (Xy, Qy) is the yield point.
(c) Comparison of linear hysteresis loop with a circumscribed rectangle, to enable definition of the nonlinearity factor NL.
The linear isolation system (Figure 2.2) has shear stiffness Kb and its coefficient of (viscous-) velocity-damping is Cb, where the subscript b is used to denote parameters of the linear isolator.
These parameters may be related to the mass M or the weight W of the isolated structure using equations (2.3) and (2.4). This gives the natural period Tb and the velocity damping factor b:
and
Figure 2.2(b) shows the 'shear force' versus 'displacement' hysteresis loop of such a damped linear isolator, which is traversed in the clockwise direction as the shear force and displacement cycle between maximum values +Sb and +Xb respectively. The 'effective stiffness' of the isolator is then defined as
The design values chosen for Tb and b will usually be based on a compromise between seismic K )
M/
( 2
Tb=
b (2.8a)M) /(4 T
= C
b bb
(2.8b)/X
=S
Kb b b (2.9)
(a)
(c) (b)
forces, isolator displacements, their effects on seismic resistance and the overall costs of the isolated structure.
When the isolator velocity-damping is quite high, say b greater than 20%, higher-mode acceleration responses may become important, especially regarding floor acceleration spectra. Such an increase in higher-mode responses may be largely avoided by anchoring the velocity dampers by means of components of appropriate stiffness Kc, as modelled in Figure 2.2(c).
The bilinear isolator model (Figure 2.3(a)) has a stiffness Kb1 without sliding, (the 'elastic-phase stiffness'), and a lower stiffness Kb2 during sliding or yielding, (the 'plastic-phase stiffness'). By analogy with the linear case, these stiffnesses can be related to corresponding periods of vibration of the system:
Corresponding damping factors can also be defined:
An additional parameter required to define a bilinear isolator is the yield ratio Qy/W, relating the yield force Qy of the isolator, Figure 2.3(b), to the weight W of the structure. Yielding occurs at a displacement Xy given by Qy/Kb1. When the design earthquake has the severity and character of the El Centro NS 1940 accelerogram it has been found that a yield ratio Qy/W of approximately 5% usually gives suitable values for the isolator forces and displacements. In order to achieve corresponding results with a design accelerogram which is a scaled version of an El Centro-like accelerogram, it is necessary to scale Qy/W by the same factor, as described in Chapters 4 and 5.
It is found useful to describe the bilinear system using 'effective' values, namely an appropriately defined 'effective' period TB and 'effective' damping factor B. The subscript B is used for these effective values of a bilinear isolator.
The effective bilinear values TB and B are obtained with reference to the 'shear force' versus 'displacement' hysteresis loop shown in Figure 2.3(b). This balanced-displacement bilinear loop is a simplification used to define these parameters of bilinear isolators. In practice, the reverse displacements, immediately before and after the maximum displacement Xb will have lower values. In general, the concept of these 'effective' values is a gross approximation, but it works surprisingly well. Note also that the simplified bilinear loop shown does not include the effects of velocity-damping forces. The damping shown is 'hysteretic', depending on the area of the hysteresis loop.
The 'effective' stiffness KB (also known as the 'secant' stiffness) is defined as the diagonal slope of the simplified maximum response loop shown in Figure 2.3(b):
This gives the effective period
K ) M/
( 2 , K ) M/
( 2 T =
Tb1 , b2
b1
b2 (2.10a)M) /(4 T M), C /(4 T
= C
,
b2 b b1 b b21
b
(2.10b)/ X
=S
KB b b (2.11a)
M/K 2
TB=
B (2.11b)20
An equivalent viscous-damping factor h can be defined to account for the hysteretic damping of the base. Any actual viscous damping b of the base must be added to h to obtain the effective viscous-damping factor B for the bilinear system. In practice h is usually larger than b, i.e. the damping of a bilinear hysteretic isolator is usually dominated by the hysteretic energy dissipation rather than by the viscous damping b.
Thus
where, from equation (2.4),
and where h is obtained by relating the maximum bilinear loop area to the loop area of a velocity-damped linear isolator vibrating at the period TB with the same amplitude Xb, to give
where Ah = area of the hysteresis loop.
For nonlinear isolators, it is convenient to have a quantitative definition of nonlinearity. We have found it useful to define a nonlinearity factor, NL, in terms of Figures 2.3(b) and 2.3(c), as the ratio of the maximum loop off-set, from the secant line joining the points (Xb,Sb) and (-Xb,-Sb), to the maximum off-set of the axis-parallel rectangle through these points, i.e., P1/P2. Hence the nonlinearity factor increases from 0 to 1 as the loop changes from a zero-area shape to a rectangular shape. For a bilinear isolator this is equivalent to the ratio of the loop area Ah to that of the rectangle.
The nonlinearity factor NL is thus given by
From equations (2.13) and (2.14) it is seen that the hysteretic damping factor h is proportional to the non-linearity factor NL for bilinear hysteretic loops. However, re-entrant bilinear loops may have a much lower ratio of damping to nonlinearity.