Chapters 1, 2, 3 and 8)
Chapter 2 GENERAL FEATURES OF STRUCTURES WITH SEISMIC ISOLATION SEISMIC ISOLATION
2.3 NATURAL PERIODS AND MODE SHAPES OF LINEAR STRUCTURES - UNISOLATED AND ISOLATED UNISOLATED AND ISOLATED
2.3.1 Introduction
It has been stated above that most or all of the important seismic responses of a structure with linear isolation, and many of the seismic responses with nonlinear isolation, can be approximated using a rigid-structure model.
However, more detailed information is often sought, such as the effects of higher modes of vibration on floor spectra, especially for special-purpose structures for which seismic isolation is often the most appropriate design approach. Such higher-mode effects are conveniently studied by modelling the superstructure as a linear multi-mass system mounted on the isolators.
Linear models and linear analysis can be used for unisolated structures and also when the structure is provided with linear isolation, except that high isolator damping may complicate responses. Simplified system models may be adopted to approximate the isolated natural periods and mode shapes when there is a high degree of modal isolation, namely when the effective isolator flexibility is high in comparison with the effective structural flexibility. The 'degree of modal isolation' is a useful concept.
When a structure is provided with a bilinear isolator, it is found that the distribution of the maximum seismic responses of higher modes can be interpreted conveniently in terms of the natural periods and mode shapes which prevail during plastic motions of the isolator. This approach is effective for the usual case in which the yield displacement is much less than the maximum displacement. These mode shapes and periods are given by a linear isolator model which has an elastic stiffness equal to the plastic stiffness Kb2 of the bilinear isolator. These mode shapes explain the distribution of maximum responses through the structure, but in general the amplitudes of the responses will be different to those of a linear system with base stiffness Kb2. The elastic-phase isolation factor I(Kb1)=Tb1/T1(U) and the nonlinearity factor NL are important parameters affecting the strengths of the higher-mode responses.
2.3.2 Structural Model and Controlling Equations
The earthquake-generated motions and loads throughout non-yielding structures has been studied extensively, e.g., Newmark & Rosenblueth, (1971); Clough & Penzien, (1975). The structures are usually approximated by linear models with a moderate number N of point masses mr, as illustrated in Figure 2.4(a) for a simple 1-dimensional model.
Figure 2.4: (a) Linear shear structure with concentrated masses. The seismic displacements of the ground and of the r-th mass mr are ug and (ur+ug) respectively. The relative
displacement of the r-th mass is ur. Here k(r,s) and c(r,s) are, respectively, the stiffness and the velocity-damping coefficient of the connection between masses
r and s.
(b) Uniform shear structure with total mass M and overall unisolated shear stiffness K, such that the level mass mr=M/N and the intermass shear stiffness kr=KN. If N tends to infinity, the overall height l=hN, the mass per unit height m=M/l and the stiffness per unit height k=Kl.
In general, each pair of masses mr, ms is interconnected by a component with a stiffness k(r,s) and a velocity damping coefficient c(r,s). In Figure 2.4(a), each mass mr has a single horizontal degree of freedom, ur with respect to the supporting ground, or ur + ug with respect to the pre- earthquake ground position, where the horizontal displacement of the ground is ug.
At each point r, the mass exerts an inertia force -(ür+üg)mr, while each interconnection exerts an elastic force -(ur - us)k(r,s) and a damping force -(ur -u s)c(r,s). The N equations which give the balance of forces at each mass can be expressed in matrix form:
where [M], [C] and [K] are the mass, damping and stiffness matrices, and where the matrix elements crs and krs are simply related to the damping coefficients and the stiffnesses, c(r,s) and k(r,s) respectively.
Here [M], [C] and [K] are N x N matrices since the model has N degrees of freedom, and u is an N-element displacement vector.
The model in Figure 2.4(a) and the force-balance equation (2.17) can be extended readily to a 3-dimensional model with 3N translational degrees of freedom (and 3N rotational degrees of freedom if the masses have significant angular momenta).
u [M]1 -
= u [K]
+ u C]
[ + M]Å
[ g (2.17)
24 2.3.3 Natural Periods and Mode Shapes
The seismic responses of the N-mass linear system, defined by Figure 2.4(a) and equation (2.17), can be obtained conveniently as the sum of the responses of N independent modes of vibration.
Each mode n has a fixed modal shape
n29 (provided the damping matrix satisfies an orthogonality condition as discussed below), and a fixed natural frequency n and damping n. These modal parameters depend on M, C and K. Other features of modal responses follow from their frequency, damping, shape and mass distribution, and the frequency characteristics of the earthquake excitation.Modal responses are developed here in outline, with attention drawn to features which clarify the mechanisms involved. Important steps in the analysis parallel those for a simpler single-mass structure.
The natural frequencies of the undamped modes are obtained by assuming that there are free vibrations in which each mass moves sinusoidally with a frequency . Let
where the displaced shape
31 varies with position in the structure and with , but is independent of t. Substitute equation (2.18) in equation (2.17), with the damping and ground acceleration terms removed:Applying Cramer's rule it may be shown that non-trivial solutions are given by the roots of an Nth-order equation in 2:
For a general stable structure, equation (2.20) is satisfied by N positive frequencies n, termed the undamped natural or modal frequencies of the structure. The N natural frequencies are usually separate, although repeated natural frequencies can occur. The shape
n34 of mode n is now found by substituting n in equation (2.19) to give N linear homogeneous equations:Since the scale factor of each mode shape
n36 is arbitrary it is here assumed, unless otherwise stated, that the top displacement of each mode is unity: Nn=1. A mode-shape matrix may then be defined as:At each natural frequency n, the undamped structure can exhibit free vibrations with a normal mode shape
n38 which is classical; that is, all masses move in phase (or antiphase where rn is negative).) + t ( sin
=
u
(2.18)0
= ) + t ( sin ) [M]
- K]
[
2
(
(2.19)0
= ) [M]
- [K]
det(
2 (2.20)0
= ) M]
[ - K]
[
2n
n(
(2.21)] ., . . , , . . . , [
= ]
[
1
n
N (2.22)2.3.4 Example - Modal Periods and Shapes
Natural periods and mode shapes for unisolated and well-isolated structures may be illustrated using a continuous uniform shear structure, hereafter referred to as the standard structure. If a frame building has equal-mass rigid floors, and if the columns at each level are inextensible and have the same shear stiffness, the building can be approximated as a uniform shear structure.
This may be modelled as shown in Figure 2.4(b) with mr = M/N and k(r, r-1) = KN for r = 1 to N, and with all other stiffnesses removed. The model is given linear isolation by letting k(1,0) = Kb, where Kb is typically considerably less than the overall shear stiffness K. It is given base velocity damping by letting c(1,0) = Cb.
The structural model is made continuous by letting N .
From the partial differential form of equations (2.17) which arises in the limit of N , or
otherwise, it may be shown that the mode shapes
n39 have a sinusoidal profile, and that the modal frequencies n are proportional to the number of quarter-wavelengths in the modal profile. Unisolated modes have (2n-1) quarter-wavelengths and isolated modes have just over (2n-2) quarter-wavelengths, as shown in Figure 2.5. If the stiffnesses K and Kb are chosen to give first unisolated and isolated periods of 0.6 and 2.0 seconds respectively, the periods of other modes follow from the number of quarter-wavelengths as shown in Figure 2.5. Moreover, there are 0.6/2.1 quarter-wavelengths in isolated mode 1, so that the first-mode shape value b1 at the base of the structure, above the isolator, is given by b1 = cos (0.29 x 90o) = 0.90, as shown. Higher isolated modes rapidly converge towards (2n-2) quarter-wavelengths with increasing n, and corresponding periods occur.Figure 2.5: Variation, with height hr, of rn, which is the approximate shape of the n-th mode at the r-th level of the continuous uniform shear structure obtained by letting N tend to infinity in the structural model of Figure 2.4(b), when T1(U)=0.6 s and Tb=2.0 s.
The modal shapes and periods are shown when the structure is unisolated (U) and isolated (I). Note that the responses interleave, with periods Tn(I) and Tn(U)
alternating between 2.09, 0.6, 0.29, 0.2, 0.15 and 0.12 seconds respectively.
26
Modal acceleration profiles have the same shapes as the corresponding displacement profiles but are of opposite sign, and hence, for a uniform mass distribution, the modal force profiles also have the same shapes as the displacement profiles. The shear at a given level may be
obtained by summing the forces above that level, so it is evident from Figure 2.5 that the shear profiles for the higher modes (n > 1) of the isolated structures have small near-nodal values at the base level, because of the cancelling effects of the positive and negative half-cycles of the profile.
The unisolated and isolated natural periods and modal profiles of Figure 2.5 may be expressed as follows:
For structures which are non-shear and non-uniform, and have inter-mass stiffnesses in addition to k(r, r-1), period ratios are less simple but retain the general features given by Figure 2.5. For a well-isolated structure, the first-mode period is controlled by the isolator stiffness. All other isolated and unisolated periods are controlled by the structure and are interleaved in the order given by Figure 2.5. The isolated mode-1 profile is still approximately rectangular. Higher-mode profiles are no longer sinusoidal but have the same sequences of nodes and antinodes.
Moreover the shear profiles of higher isolated modes still have small near-nodal values at the isolator level.
For all well-isolated structures, the damping of mode 1 is controlled by the isolator damping. The damping of all higher modes is controlled by structural damping, provided the velocity damping of the isolator is not much greater than that of the structure. It is commonly assumed that the structural damping is approximately equal for all significant modes.
2.3.5 Natural Periods and Mode Shapes with Bilinear Isolation
When a structure is provided with a bilinear isolator there are two sets of natural periods and two corresponding sets of mode shapes; one set is given by a system model which includes a linear isolator which has the elastic stiffness Kb1 of Figure 2.3, while the other set is given when the linear isolator has the plastic stiffness Kb2.
The yield level of a bilinear isolator is normally chosen to ensure that the maximum seismic displacement response, for a design-level excitation, is much larger than the isolator yield displacement. With such isolators the distribution of the maximum seismic motions and loads, and the floor spectra, can be expressed effectively in terms of the set of modes for which the shapes, and the higher-mode periods, are those of the normal modes which arise when the structure has a linear isolator of stiffness Kb2. An approximate effective period for mode 1 is derived from the secant stiffness KB at maximum displacement, as given by equation (2.11a) and illustrated in Figure 2.3(b). The relevance of the normal modes arising with stiffness Kb2 is to be expected, since maximum or near-maximum seismic responses should normally occur when the isolator is moving in its plastic phase, with an incremental stiffness Kb2.
seconds -
1) n- /(2 0.6
= T (U)
n (2.23a)
seconds -
1
>
n for , 2) - n /(2 0.6 T (I)
; 2.1
= T (I)
1 n (2.23b)