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2.5 COMPARISONS OF SEISMIC RESPONSES OF LINEAR AND BILINEAR ISOLATION SYSTEMS ISOLATION SYSTEMS
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2.5 COMPARISONS OF SEISMIC RESPONSES OF LINEAR AND BILINEAR
The modal responses were obtained from the overall response histories at all masses in the structures by sweeping with the free-free mode shapes, except for the unisolated structure, where the modal responses are in terms of the true unisolated modes.
The 'un-isolated' structure (case (i)) is a uniform linear chain system, with 4 equal masses and 4 springs of equal stiffness, the lowest being anchored to the ground. It has a first-mode undamped natural period of 0.5s, and 5% damping in all its modes.
Most of the 'isolated' cases represent systems obtained simply by adding below this structure an isolation system modelled as a base mass, a linear or bilinear-hysteretic base spring and a linear viscous base damper. However, two of the 'isolated' cases involve stiffer structures, with unisolated periods of 0.25 seconds, in order to show the effects of high elastic-phase isolation factors. In all the isolated cases, the added base mass is of the same value as the other masses, comprising 0.2 of the total isolated mass.
The viscous damping of the isolated structures is 5% of critical for all the free-free modes, with the nonlinear isolation systems having linear viscous base dampings b2 which are 5% of critical in the post-yield phases, as well as hysteretic damping. The table shows values of b for the linear isolators, and values of b, b1 and b2 for the bilinear isolators, where b = b2 TB/Tb2.
The cases were chosen to represent a wide variety of isolation systems, with various degrees of nonlinearity and pre- and post-yield isolation ratios. In calculating the isolation factors, I=Tb/T1(U) and I(Kb1)=Tb1/T1(U), the unisolated period T1(U) corresponds to that of the structure when the isolator is rigid, while the isolator periods Tb and Tb1 are calculated for the 5 masses, from the structure and the isolator, with all their interconnecting springs treated as rigid, mounted on the isolator spring.
Cases (ii) and (iii) represent medium-period structures with a high degree of linear isolation (T1(U)=0.5s, Tb=2.0s, I=4), and with low (b=5%) and high (b=20%) values for the viscous damping of the isolator, respectively.
Case (iv) is a bilinear hysteretic system with similar characteristics to that of the William Clayton Building (Section 6.2(d)), which was the first building isolated on lead rubber bearings. The parameter values are typical for structures with this type of isolation system.
The unisolated period of the structure is 0.25s (the William Clayton Building has a nominal unisolated period of 0.3s), with a pre-yield isolator period Tb1 of 0.8s and a post-yield isolator period Tb2=2.0s. The yield force ratio Qy/W is 0.05, less than the William Clayton Building's value of 0.07. However, the latter value was chosen to give a near-optimal base shear response (see section 4.3.2) in 1.5 El Centro, so scaling down the yield-force/weight ratio by approximately 2/3 is appropriate for a system with El Centro as the design motion. The post-yield isolator period is equal to the isolator period of the linear systems of cases (ii) and (iii). The equivalent viscous damping from the combined hysteretic and viscous base damping at the amplitude of its maximum response to El Centro is 24% (Table 2.1), comparable with the viscous damping of 20%
for the linear system (iii).
Case (v) represents bilinear systems with elastic- and yielding-phase isolation factors towards the low ends of their practical ranges. The unisolated period is 0.5s, with the isolator periods Tb1=0.3s and Tb2=1.5s, giving isolation factors of 0.6 and 3 in the two phases. The yield force ratio Qy/W is 0.05, as for all the nonlinear cases. This system has a moderate nonlinearity factor which is virtually identical to that of case (iv) (0.33 compared to 0.32), but considerably reduced isolation factors, most importantly in the elastic phase where it is 0.6. The low elastic-phase isolation gives response characteristics similar to those for a system with an isolator which is rigid before it yields.
In case (vi), the post-yield period of the isolator has been doubled from that of case (v), to Tb2=3.0s, but the other parameter values are the same. This change produces a considerably
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The response characteristics are similar to those for what is sometimes referred to as a 'resilient- friction base isolator' (Fan & Ahmadi (1990, 1992); Mostaghel & Khodaverdian (1987)).
The final example, case (vii), is a strongly nonlinear system, with a nonlinearity factor of 0.71, but unlike case (vi) it has high isolation factors in both phases of the response. The force- displacement characteristics of the isolator are almost elasto-plastic, with a post-yield period of 6.0s. The unisolated period of the structure (T1 (U)=0.25s) and the yield-force ratio (Qy/W=0.05) are identical to case (iv), and the pre-yield isolator period (Tb1=0.8s) and hence the elastic-phase isolation factor are very similar to those in case (iv). This represents a system with high hysteretic damping, high isolation in both phases of the response, and a maximum base shear closely controlled by the isolator yield force because of the nearly perfectly plastic characteristic in the yielding phase.
The response characteristics of this wide range of examples are illustrated in Figure 2.7, and demonstrate many of the key features of the response characteristics of base-isolated structures.
Comparisons can be made between features of the responses of unisolated and isolated structures, and between those of various isolated structures. Systematic variations in response quantities can be seen as the equivalent viscous damping, the nonlinearity factor and the elastic-phase isolation factor are varied.
The first point to note in Figure 2.7 is that the response scales for the unisolated structure of case (i), as emphasised by heavy axis lines, are five times larger than those for all the isolated cases shown in the other parts of the figure.
The next general comment regarding Figure 2.7 is that the force-displacement hysteresis loops have been drawn for cyclic displacements of +0.4 Xb. This has been done in order to show the relative slopes.
Direct comparisons of various response quantities can be made for the unisolated structure and the four cases (ii), (iii), (v) and (vi) involving the same structure on various isolation systems. Cases (iv) and (vii) involve shorter-period structures on the isolators, so direct comparisons of these with case (i) are not appropriate. The base shears of the isolated systems with the 0.5s structure are reduced by factors of 4.6 (for the lightly damped linear isolator of case (ii)) to over 10 (for case (vi) with high hysteretic damping). Base displacements, which contribute most of the total displacement at the top of the isolated structures, range from 0.7 to 2.5 times the top displacement of the unisolated structure. Inter-storey deformations in the isolated structures are much reduced from those in the unisolated structures, since they are proportional to the shears.
Since large deformations are responsible for some types of damage, the reduction in structural deformation is a beneficial consequence of isolation. First-mode contributions to the top-mass accelerations in the isolated structures are reduced by factors of about 6 to 14 compared to the values in the unisolated structure. The linear isolation systems show marked reductions in the higher-frequency responses as well, but the second-mode responses for the systems with the greatest nonlinearities are only slightly reduced from those in the unisolated structure. These effects are most evident in the top-floor response spectra.
Figure 2.7 shows several important characteristics of the response of isolated structures in general. In isolated systems, increased damping reduces the first-mode responses, but generally increases the ratio of higher-mode to first-mode responses, particularly where the damping results from nonlinearity. The elastic-phase isolation factor I(Kb1) has a marked effect on higher- mode responses, which increase strongly as I(Kb1) reduces from about 1.0 towards zero. The effects of these parameters are demonstrated by considering each of the isolated cases in turn.
The lightly damped linear isolation system of case (ii) reduces the base shear by a factor of 4.6 from the unisolated value, but requires an isolator displacement of 180mm.
The response is concentrated almost entirely in the first mode, as shown by the comparison of the first-mode and total acceleration and shear distributions and by the top floor spectra. The differences between the first-mode and total distributions largely arise from the difference between the free-free first-mode shape which was used in the sweeping procedure and the actual first-mode shape with base stiffness Kb. The maximum second-mode acceleration calculated by sweeping with the second free-free mode shape is only about 1/6th that found by sweeping with the first free-free mode shape.
By increasing the base viscous damping from b=5% to 20% of critical, as in case (iii), the maximum base displacement is reduced from 180mm to 124mm, with less percentage reduction in the base shear. The mode-2 acceleration more than doubles, showing the effects of increased base impedance from the increased base damping and modal coupling from the nonclassical nature of the true damped modes. The first-mode response still dominates, however. The floor response spectra reflect the reduction in first-mode response, but show increases in the second- and third-mode responses compared to case (ii).
Case (iv) has an effective base damping similar to case (iii), but with the main contribution coming from hysteretic damping. All first-mode response quantities, and those dominated by the first-mode contribution, including the base shear and the base displacement, are reduced from the values for the linear isolation systems. The nonlinearity of this system is only moderate (0.32), and there is a high elastic-phase isolation factor of 3.2, but the second-mode response is much more evident than for the linear isolation systems, particularly in the floor response spectrum.
Case (v) has the same degree of nonlinearity as the previous case, but a much reduced elastic-phase isolation factor of 0.6. The low elastic-phase isolation factor has produced a much increased second-mode acceleration response, which is 50% greater than the first-mode response on the top floor.
The distribution of maximum accelerations is much different from the uniform distribution obtained for a structure with a large linear isolation factor. The accelerations are much increased from the first-mode values near the top and near the base, while the shear distribution shows a marked bulge away from the triangular first-mode distribution at mid-height. Strong high-frequency components are evident in the top floor acceleration response spectrum, with prominent peaks corresponding to the second and third post-yield isolated periods.
Case (vi) is an exaggerated version of case (v). The post-yield isolator period has been increased to 3.0s, giving a high nonlinearity factor as well as a low elastic-phase isolation factor, both conditions contributing to strong higher-mode response. The nearly plastic behaviour of the isolator in its yielding phase produces a more than 40% reduction in the base shear from case (v), at the expense of a 33% increase in the base displacement. The maximum second- mode acceleration response at the top floor is 2.5 times the first-mode response, being the highest value of this ratio for any of the seven cases. The acceleration at the peak of the top- floor response spectrum at the second-mode post-yield period has the greatest value of any of the isolated cases, almost identical to the second-mode value in the unisolated structure, which, however, occurs at a shorter period.
Case (vii) demonstrates that high elastic-phase isolation can much reduce the relative contribution of the higher modes for highly nonlinear systems. The nonlinearity factor of 0.71 is the highest of any of the cases, but the second-mode response is less than 40% that of cases (v) and (vi), which have poor elastic-phase isolation. The high nonlinearity has reduced the base shear to 70% of that of case (iv). The mode-2 acceleration response has been reduced by 13%
from that of case (iv), but its ratio with respect to mode 1 has increased from 0.85 to 1.25.
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Maximum base shears and displacements of isolated structures are dominated by first-mode responses. Maximum first-mode responses of bilinear hysteretic isolation systems can in turn be approximated by the maximum responses of equivalent linear systems, as discussed earlier in this chapter. The final section of Table 2.1 demonstrates the degree of validity of the equivalent linearisation approach. It gives effective dampings and periods calculated for the equivalent linearisation of the bilinear isolators, using equation (2.11b) for TB and equations (2.11c) to (2.13) for B. The response spectrum accelerations and displacements for these values of period and damping are listed. The spectral values for the base displacements give reasonable approximations to the actual values, with correction factors CF of approximately unity, except for case (vii), with the nearly plastic post-yield stiffness, for which the correction factor is 1.6.
However, the spectral accelerations SA(TB,B) provide much poorer estimates of either the first- mode or overall base-mass accelerationX b. Much improved estimates of the base shear Sb can be obtained from KBXb, which has a smaller relative error than the estimate of Xb from SD(TB,B).
This is the procedure we recommend when using the equivalent linearisation approach (Section 2.2).