Åu=
Chapter 3 ISOLATOR DEVICES AND SYSTEMS
3.6 LEAD RUBBER BEARINGS
3.6.2 Properties of the Lead Rubber Bearing
78 Thus to a very good approximation
where the shear stress at which the lead yields (Pb) = 10.5 MPa, A(Pb) is the cross-sectional area of the lead, K(r) is the stiffness of the rubber in a horizontal plane, and X is the displacement of the top of the bearing with respect to its base. This fact is illustrated in Figure 3.18 where the maximum shearing force, minus the force due to the elastic stiffness of the rubber, is plotted against the cross-sectional area of the lead insert. The slope of this line is the yield stress of lead, 10.5 MPa (Robinson (1982). Note Qy of a hysteretic damper is given approximately by
(Pb)A(Pb).
Figure 3.18: Force due to the lead, F(b) - F(r), as a function of the cross-sectional area of the lead insert. (Robinson, 1982.)
Figure 3.19: (a) Force-displacement hysteresis loops for a lead rubber bearing used in the William Clayton Building, at 45 and 110mm strokes, with a vertical force of 3.15MN at 0.8Hz. (From
Robinson, 1982.).
(b) Force-displacement curves for the bearings used in the Wellington Press Building (Chapter 8). (From Robinson & Cousins, 1987 & 1988).
Figure 3.19 contains the force-displacement hysteresis loops for two recent examples, namely the lead rubber bearings for the seismic isolation for (a) the William Clayton Building and (b) the Wellington Press Building. For both of these examples the initial stiffness Kb1~10K(r) while the post-yield stiffness is approximately K(r).
X K(r) + A(Pb) (Pb)
=
F
(3.17)Rate Dependence
For a number of applications it is necessary to know the behaviour of the lead rubber bearing under creep conditions. For example, if a bridge deck is mounted on the bearings then, during the normal 24 hour cycle of temperature, the bearings will have to accommodate several displacements of ~+3 mm without producing large forces. In order to determine the effect of creep rates of ~1 mm/h, the second lead rubber bearing made, (that is, one with dimensions of 356 x 356 x 140 mm with a 100 mm lead plug) was mounted in the back-to-back reaction frame in the Instron testing machine. The first result was obtained at 6 mm/h, with the force due to the lead alone reaching a maximum after 2.5 h before decreasing slowly.
After 6 hours the displacement was held constant and the force due to the lead decreased to one half in about one hour, and continued to fall with time, giving a relaxation time of 1 to 2 h.
Another creep test was carried out at 1 mm/h for six hours, when the direction was reversed, giving the hysteresis shown in Figure 3.20. For completeness the force F(R), due to the rubber, is included with its +20 per cent error bar. The shear stress in the lead plug reached a maximum of 3.2 MPa, which is ~30 per cent of the stress of 10.5 MPa for the dynamic tests. The force due to the rubber is great enough to drive the deformed lead, and the structure, back to its original position.
Figure 3.20: Force due to lead during creep of 356 mm2 bearing with 100 mm lead plug, at vertical force of 400 kN. Open points are 6 mm/h, filled points are 1 mm/h
and dashed line is F(r). (Robinson, 1982.)
Because of the large errors caused by F(r), it was not possible to determine accurately the rate-dependence of the lead in the lead rubber bearing. To overcome this problem three lead hysteretic dampers, which had been developed earlier to operate in shear without a rubber bearing (Robinson (1982), were tested at various strain rates. These dampers consisted of lead cylinders whose diameters varied parabolically as shown in the insert to Figure 3.21, and whose ends were soldered to two brass plates. The parabolic variation was designed to minimise the effect of bending stresses, which occur away from the neutral axis of the lead, during the application of shearing displacements: in fact, the shear stress near the parabolic surface of the lead remained constant to a first approximation. The rate-dependence of these dampers, with their shear stress normalised to that at = 1 s-1, is shown in Figure .21, by the circled points. This figure also denotes, with the symbol (x), the values obtained for the second lead rubber bearing made, at rates of =10-5 and 3 x 10-1 s-1. These results have a rate-dependence
where below = 3 x 10-4 s-1, b = 0.15 and above, b = 0.035. For the lead extrusion damper (Figure 3.10) it was found that, for the two regions, b = 0.14 and 0.03. For slow creep other authors conclude that b = 0.13 (Birchenall (1959), Pugh (1970)). When the experimental errors are taken into account, all of these results are in reasonable agreement.
(Pb) = a
b (3.18)80
Figure 3.21: Rate dependence of lead cylinders of parabolic section (see insert) in shear, as indicated by the circled points. The crosses indicate the rate dependence of the lead
plug in a lead rubber bearing. (Robinson, 1982.)
These results indicate that the lead rubber bearing has little rate-dependence at strain rates of 3 x 10-4 s-1 to 10 s-1, which includes typical earthquake frequencies of 10-1 to 1 s-1. For this range of strain rates, an increase in rate by a factor of ten causes an increase in force of only 8 per cent.
Below strain rates of 3 x 10-4 s-1, the dependence of the shear stress on creep rate is greater, with a 40 per cent change in force for each decade change in rate. However, this means that at creep displacements of ~1 mm/h for a typical bearing 100 mm high (that is, at ~ 3 x 10-6 s-1), the shear stress has dropped to 35 per cent of its value at typical earthquake rates, ~ 1 s-1.
Fatigue and Temperature
The lead rubber bearing can be expected to survive a large number of earthquakes, each with an energy input corresponding to 3 to 5 strokes of +100 mm. For example, the results for a series of dynamic tests on the 650 mm diameter bearing with a 140 mm diameter lead plug are shown in Figure 3.22. The symbols F(a) and F(b) correspond to points such as a and b on Figure 3.17.
F(a) and F(b) decreased by 10 and 25 per cent over the first five cycles but recovered some of this decrease in the five-minute breaks between tests.
Figure 3.22: Dynamic tests on lead rubber bearing over seven simulated earthquakes. (Robinson, 1982.)
An interval of 12 days between the last two tests did not give a greater recovery than that obtained in 5 minutes. The effect of the 24 cycles is shown more clearly by Figure 3.23, where the outer hysteresis loop is the first, and the inner loop is the twenty-fourth. The area of the twenty-fourth loop is 80 per cent of the first, indicating that the bearing has retained most of its damping capacity over these seven simulated earthquakes.
Figure 3.23: First and 24th hysteresis loops for lead rubber bearing shown in Figure 3.22. The outer loop is the first and the inner loop is the 24th. (Robinson, 1982.)
As a further check on the fatigue performance, the 356 mm bearing was dynamically tested at a shear strain of 0.5 for a total of 215 cycles in a two-day period. This bearing was also subject to 11,000 strokes at +3 mm (0.9 Hz), to demonstrate that it could withstand the daily cycles of thermal expansion which occur in a bridge deck over a period of 30 years. It performed satisfactorily.
The 356 mm bearing was also studied with dynamic tests ( ~ 0.5, 0.9 Hz) at temperatures of -35, -15 and +45oC, to ensure its performance in extreme temperature environments. The ratio of the force F(b) to that at 18oC for the first cycle was 1.4, 1.2 and 0.9 at -35, -15 and +45oC respectively, showing that the lead rubber bearing is not strongly temperature-dependent (Robinson (1982).
Effect of Vertical Load on Hysteresis
As can be seen from the results of Figure 3.20, it is possible to design lead rubber bearings which have little change in their hysteresis loops over a wide range of vertical loads (Tyler & Robinson (1984). On the basis of a simple model, the nominal upper limit of hysteretic force, y(Pb)A(Pb), should be achieved if there is no vertical slippage of the plug sides and no horizontal slippage of the plug ends. Side slip can be made small by using a small spacing between the plates and by ensuring a large confining pressure po.
Satisfactory results are achieved with a spacing t less than d/10, and with a pressure po, as given approximately by equation (3.8b) when S is greater than 10. The effect of end slip can be made small by using a lead plug with an adequate height to diameter ratio h/d, say not less than 1.5.
Complicating factors include the hysteretic forces due to the lead which is extruded small distances into the spaces between the plates, additional forces which may increase overall hysteretic forces beyond their nominal upper limit. Again the confining pressure is enhanced, beyond that given by the vertical load, by inserting a lead plug whose volume exceeds that of the undeformed cavity in the bearing.
82 Bilinear Parameters for Small Earthquakes
When the isolator motions arise from small earthquakes, with displacement spectra reduced by a factor of 2 or more, the bilinear loop parameters change in the same general way as the bilinear loop parameters for an isolator consisting of laminated rubber bearings mounted beside steel-beam dampers, with the same beneficial results. Reduced displacements cause considerable reductions in Qy and considerable increases in Kb2, as shown in Figure 3.24. As a net result, the effective (secant) period, and sometimes the hysteretic damping, falls more slowly, with decreasing earthquake severity, than they would with a fixed-parameter bilinear loop.
Figure 3.24: (a) Difference in bilinear
loop parameters corresponding to small and large displacements.
(b) Load-displacement loops for various strokes of lead rubber bearing used in Press Hall, Petone (see Chapter 6). (Robinson & Cousins, 1987, 1988.)
Summary of Lead Rubber Bearings
For strain rates of ~1 s-1, the lead-rubber hysteretic bearing can be treated as a bilinear solid with an initial shear stiffness of ~10 Kb(r) and a post-yield shear stiffness of Kb(r). The yield force of the lead insert can be readily determined from the yield stress of the lead in the bearing, i.e.
y(Pb) ~10.5 MPa. Thus the maximum shear force for a given displacement is the sum of the elastic force of the elastomeric bearing and the plastic force required to deform the lead. The actual post-yield stiffness is likely to vary by up to + 40 per cent from Kb(r) but will probably be within + 20 percent of this value. The initial elastic stiffness has only been estimated from the experimental results and may in fact be in the range of 9 Kb(r) to 16 Kb(r). The prediction for the maximum force, F(b), is more accurate and has instead an uncertainty of + 20 percent which is the same as expected for the uncertainty in the shear stiffness of manufactured elastomeric bearings. The actual area of the hysteresis loop formed by this bilinear model is approximately 20 per cent greater than the area of the measured hysteresis loop.
The lead-rubber hysteretic bearing provides an economic solution to the problem of seismically isolating structures, in that the one unit incorporates the three functions of vertical support and horizontal flexibility (via the rubber) and hysteretic damping (by the plastic deformation of the lead). Further discussion on lead rubber bearings is contained in Robinson & Cousins (1987, 1988); Skinner et al (1980); Skinner, Robinson & McVerry (1991); Cousins, Robinson & McVerry (1991).