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Chapter 3 ISOLATOR DEVICES AND SYSTEMS
3.5 LAMINATED-RUBBER BEARINGS FOR SEISMIC ISOLATORS
3.5.1 Rubber Bearings for Bridges and Isolators
Another method of seismically isolating structures is by mounting them on laminated rubber bearings (elastomeric bearings). These bearings are a fully developed commercial product whose main application has been for bridge superstructures, which often undergo substantial dimensional and shape changes due to changes in temperature. More recently their use has been extended to the seismic isolation of buildings and other structures (Chapter 6).
These bearings are designed to support large weights while providing only small resistance to large horizontal displacements, and to moderate tilts, of the upper surfaces of the bearings. A typical bridge bearing consists of a stack of horizontal rubber layers vulcanised to interleaved steel plates, as shown schematically in Figure 3.11 for a cylindrical bearing.
Figure 3.11: Sketch of laminated elastomeric bearing, of area A and circumference C, in which rubber layers, of thickness t, are bonded to thin steel plates.
For a given bearing area and rubber composition, the load capacity is increased by reducing the thickness of each rubber layer, while the resistance to horizontal and tilting movements is reduced by increasing the total height of the rubber.
Rubber bearings, of the types used for bridges, can be dimensioned to provide the support capacity and the horizontal flexibility required for seismic isolation mounts. Of particular importance is the ratio of bearing weight capacity to horizontal flexibility, which determines the maximum achievable value for the rigid-structure period Tb. Of equal importance is the maximum acceptable horizontal displacement Xb, which is set either by the allowable rubber strain or by the allowable offset between the plan areas of the top and bottom of the bearing.
Rubber bearings also provide adequate isolator centring forces during large seismic displacements.
Rubber bearings have a considerable range of applications in seismic isolators, as described later in this chapter. In their basic form, rubber bearings may be used to provide support, horizontal flexibility and centring forces. Isolator damping may then be increased by separate components. Alternatively, lead plugs may be inserted in rubber bearings to add high hysteretic damping to the features of the basic bearings, as described in 3.6 below. Again, rubber bearings may be surmounted by horizontal slides which provide increased horizontal flexibility and frictional damping. Additional isolation roles for rubber bearings include tilting supports for rocking structures and elastic components in displacement-limiting buffers.
The detailed design and the manufacture of rubber bearings call for some technical sophistication. However, the approximate features of rubber bearings may be derived using simple well-known approaches, as described below. An understanding of the factors influencing the features of elastomeric bearings is useful when developing isolation systems, and may assist during preliminary design studies.
3.5.2 Rubber Bearing, Weight Capacity Wmax
The principal features of rubber bearings can be seen from the behaviour of a thin rubber disc, with rigid plates bonded (vulcanised) to its plane surfaces, when subjected to normal (axial) and to parallel (or shearing) loads. The relationship between the load W and the maximum engineering shear strain in the disc has been derived by Gent & Lindley (1959) as outlined below in modified form. (Following Borg (1962), = xz = w/x + u/z = 2xz where xz is the tensor shear strain.)
When the rubber is assumed incompressible, a vertical compressive strain z causes the rubber to bulge by an amount proportional to its distance from the centre of the disk. When the bulge profile at any radius r is approximated by a parabola, constant rubber volume gives the maximum shear strain xz as:
where the vertical strain z = t/t, the thickness of the rubber layers is denoted t, and the shape factor S = (loaded area)/(force-free area). For example, for a circular disc of unstrained diameter D and thickness t, S = D/4t.
The rubber shear forces cause a pressure gradient within the disc which is proportional to the distance from the centre. This gives a parabolic pressure distribution, as shown in Figure 3.12.
Figure 3.12: Sketch of circular layer of rubber, diameter D, thickness t, and of the parabolic pressure distribution p.
The maximum pressure po is given by:
where G = shear modulus of rubber.
The corresponding load W may be obtained by summing the pressure over a disc area A to give:
Now consider a basic rubber bearing consisting of n equal rubber layers of any compact shape.
Also let the top of the bearing be displaced by Xb to give an overlap area A between the top and bottom of the bearing, as shown in Figure 3.13.
xz= 6 S
z (3.8a)
xzo
= 2 GS
p
(3.8b)
xzAGS
=
W
(3.8c)68
Figure 3.13: Sketch of rubber cylinder of diameter D, with a shear displacement Xb and overlap A'.
Then experiment and analysis show that equation (3.8c) may be generalised approximately as follows:
where
Wmax = allowable weight
w = allowable shear strain due to weight A = overlap of bearing top and bottom
The use of A in equation (3.8d) is a somewhat arbitrary simplification and is probably conservative.
3.5.3 Rubber-Bearing Isolation: Stiffness, Period and Damping
If an isolator consists of a set of equal rubber bearings, each supporting an equal weight, then the isolator period can be calculated directly from the weight and stiffness for a single bearing.
In practice the average weight per bearing may be reduced because the weight on some bearings has been reduced to offset vertical seismic loads, or for structural or architectural convenience. However, such weight reductions are neglected here and the isolator parameters are expressed in terms of those for a single bearing.
Bearing Horizontal Stiffness Kb
A rubber bearing may be approximated as a vertical shear beam, since the steel laminations severely inhibit flexural deformations while providing no impediment to shear deformations. The approximate horizontal stiffness Kb is therefore given by
where
A = rubber layer area h = total rubber height.
There will be some reduction in bearing height with large displacements, partly due to flexural beam action and partly due to increased compression of the reduced overlap area A.
wmax
= A GS
W (3.8d)
GA/h
=
Kb (3.9)
The resulting inverted pendulum action, under structural weight, reduces the horizontal stiffness Kb and in extreme cases might cause serious reductions in the centring forces. However, the inverted pendulum forces are reduced by increasing the layer shape factor S, and these forces are unlikely to be serious for S values in the range from 10 to 20, a range appropriate for isolator mounts.
Bearing Period Tb
The bearing weight capacity, Wmax, from equation (3.8d), and the horizontal stiffness, Kb, from equation (3.9), can be combined to give the bearing and isolator period Tb, when the bearing is supporting its maximum weight, as
where w is the allowable shear strain due to the weight W.
For example let S = 16, h = 0.15 m, A/A = 0.6, and w,max = 0.2 L/L, where the breaking tensile strain L/L = 5, (typically 4.5 to 7.0). Then Tb = 2.4 seconds.
Bearing Damping zb
Energy losses in the deforming rubber layers provide damping which is predominantly velocity-dependent. Typical bridge bearings provide bearing and isolator damping factors in the range from 5% to 10% of critical. However, acceptable bearing rubbers have been manufactured which increase the bearing and isolator damping to about 15%, and development aimed at higher damping values continues.
Bearing Vertical Stiffness Kz
Some isolator applications of rubber bearings are influenced by their vertical stiffness, and some by their related bending stiffness.
The vertical deflection of a bearing is the sum of the deflections due to rubber shear strain and to rubber volume change, and these two respective stiffnesses are added in series. Thus the overall vertical stiffness is
where Kz(), the vertical stiffness of the bearing without volume change, is given by equations (3.8a) and (3.8c) as
and where Kz(V), the vertical stiffness due to volume change without shear strain, is simply
where = rubber compression modulus.
Thus
/Ag) A (Sh 2
=
Tb
w ‰ (3.10)
K (V) /
K
+ K
V
K
=
Kz z
z z
z (3.11a) = 6 G S A / h ,
K
z
2 (3.11b)
V = A / h ,Kz
(3.11c)h ) + S G /(6 A GS 6
K
z=
2
2
(3.11d)70
Equations (3.11) show that a small shape factor S gives a moderate vertical stiffness which is controlled by shear strain, while a sufficiently large value of S gives a very high vertical stiffness which is controlled by volume change. For a typical bridge-bearing rubber, with G = 1 MPa and
= 2000 MPa, shear strain and volume change make equal contributions to vertical stiffness when S 18. The above discussion neglects the usually small reduction in Kz() which occurs, due to a pressure redistribution in the layers, when rubber compressibility is introduced. When the S-value is high, rubber compressibility reduces considerably the bearing vertical stiffness and the related bending stiffness. However, rubber compressibility causes little change in the other bearing parameters described.
3.5.4 Allowable Seismic Displacement Xb
Displacement Limited by Seismic Shear Strain gs
When the rubber shear strain w, due to the vertical load W, is below its maximum allowable value there is a reserve shear strain capacity, say s, to accommodate a horizontal displacement Xb, which is given by
where s = allowable shear strain due to horizontal seismic displacement.
If this displacement is inadequate it may be increased by increasing the rubber height h. In addition, or alternatively, s may be increased if the strain due to weight w is reduced.
Displacement Limited by Overlap Factor A¢/A
For an isolator bearing, a lower limit to the overlap factor A/A is set by the reducing weight capacity, equation (3.8d), and sometimes by the increasing end moments. Typical lower limits for the overlap factor may be 0.8 for a sustained horizontal displacement and 0.6 for design earthquake displacements. Where possible, such overlap limits should be based onlaboratory tests and field experience. The relationship between the overlap factor A/A, the bearing displacement Xb and the bearing dimensions depends somewhat on the shape of the horizontal section of the bearing.
For a cylindrical bearing with rubber discs of area A and diameter D:
where sin = Xb/D .
Hence for moderate values of Xb/D
Similarly, for a rectangular bearing
where Xb(B) and Xb(C) are the bearing displacements parallel to the sides of lengths B and C respectively. Hence, for displacements parallel to side B,
sb
= h
X
(3.12)
2/
+ sin
cos
- 1
= /A
A (3.13a)
1 - A /A
D 0.8
Xb (3.13b)
B / B-X
C / C-X 1 /A
A b b (3.13c)
1 - A /A
B
Xb(B) (3.13d)
When the displacement Xb may be in any direction, a more appropriate displacement limit is
where B is the shorter side of the bearing.
From equations (3.13b) and (3.13e) it is seen that, for a seismic overlap factor A/A = 0.6, the allowable values of Xb are D/3 and B/3 respectively.
When the weight per bearing is low, the bearing diameter D or side B may be too short to accommodate the required seismic displacement Xb. If the discrepancy is not great it might be met by increasing the bearing area A and/or by reducing the design-earthquake displacement Xb. The bearing area may be increased, without changing the bearing stiffness ratio Kb/W, if there is a compensating reduction in the rubber shear modulus G and/or an increase in the rubber height h, as required by equation (3.9). Again, the bearing area may be increased if it is possible to design the isolator with fewer bearings and hence with a greater weight W per bearing. Alternatively, the design earthquake displacement Xb may be reduced by increasing the effective isolator damping.
If the weight per bearing is so low that the allowable displacement falls well short of the design earthquake displacement, then the allowable displacement may be increased as required, by segmenting the bearing and introducing stabilising plates, as described below.
Segmented Bearing for a Low Weight/Displacement Ratio W/Xb
When a rubber bearing supports a small weight W it has a small area A, and hence its displacement capacity, as given by equation (3.13b) or (3.13e), is also small. Such a simple bearing may be replaced by an equivalent segmented bearing, as shown in Figure 3.14, which increases the displacement capacity.
Figure 3.14: Segmented bearing formed by rubber segments placed at the corners of common stabilising plates, illustrated by 6 stabilising plates and 20 (multilayer) segments.
Consider the replacement of a simple bearing by an equivalent segmented bearing in which sets of 4 segments are located near the corners of rectangular stabilisation platforms or plates,
1 - A /A
B 0.8
Xb (3.13e)
72
If all the linear dimensions (including the thickness) of the segment rubber layers are half those of the simple bearing layers, and if the number of layers is increased so that the rubber height is unaltered, then both bearings have the same values for the rubber area A and the rubber height h, and the same shape factor S, resulting in the same load capacity and the same horizontal stiffness Kb. For a given rubber and operating conditions, a shape factor which is suitable for a non-segmented bearing is also suitable for the equivalent segmented bearing.
Typically each of the cylindrical segments shown in Figure 3.14 will be multilayer, to give the small layer thickness required without the use of more stabilising plates than are necessary to retain the overlap factor required for overall bearing stability.
When, as here, the segments have half the horizontal dimensions of the corresponding non-segmented bearing, and there are n segments in each vertical stack (eg, n=5 in Figure 3.14), then a required overlap factor is retained with an increased allowable displacement given by
Where
Xb(1) = allowable displacement for the corresponding non-segmented bearing.
3.5.5 Allowable Maximum Rubber Strains Allowable Shear Strains gw and gs
The allowable rubber shear strains for various loads and displacements are important factors in the performance of rubber bearings, as discussed above. When bearings are used as isolation mounts for compact structures, they must withstand the combined rubber shear strains due to structural weight and seismic displacements. When bearings isolate bridge superstructures, some provision must be made for additional shear strains due to traffic loads and to thermal displacements. In addition to their seismic design, rubber bearing mounts must be checked for their capacity to withstand the more sustained non-seismic loads and displacements.
The damaging effect of a given rubber strain increases with its total duration and with the number of times it is reduced or reversed. In particular, rubber strains due to frequent and fluctuating traffic loads are found to be more severe than a corresponding steady strain applied for the life of a bearing. On the other hand, laboratory tests show that the cyclic strains due to seismic displacements are much less severe than corresponding long-duration steady strains, evidently because they involve so few cycles and have such a short duration.
The sustainable steady shear strain in a rubber bearing is sometimes given as (Bridge Engineering Standards, 1976)
where t = short-duration failure strain in simple tension.
Experiments suggest that corresponding factors for shear strain during earthquakes are 0.4 or more for design earthquakes and say 0.7 for extreme earthquakes.
Allowable Negative Pressure
Under the combined action of uplift forces and end moments the rubber within isolator bearings may be subjected to large negative pressures. Consider a rubber bearing subject to an uplift force of -Wmax. From equation (3.8) it is found that this gives a small increase in bearing height of
h = hw/(6S), and a large central negative pressure of po = -2GSw. For a typical bridge bearing, with G = 1 MPa, h = 0.15 m, S = 10, and -w = -1.0, it follows that h = 2.5 mm and po = -20 MPa.
2 / X (1) n
=
Xb(n) b (3.14)
w= 0.2
t (3.15)Negative pressures may also arise from bearing end moments, which are generated by relative displacement and tilting of the ends of a bearing. These end moments cause local increases and decreases of the pressure within the bearing discs.
A large negative pressure evidently causes a set of small cavities within the bearing rubber, which grow progressively during sustained and cyclic negative pressures. The cavities cause a large reduction in axial stiffness, which may be regarded as resulting from a reduction in the effective shape factor S, but there is little reduction in the horizontal shear stiffness.
Figures 3.15(a) and (b) show a vulcanised laminated rubber bearing before and during vertical loading, while Figure 3.15(c) is a stress-strain plot showing both compression and tension. This bearing failed in the rubber at a tensile strain of 350%, though small internal cracks were most probably formed before this strain was reached.
Figure 3.15: (a) Vulcanised laminated rubber bearing before loading. (Tyler, 1991.) (b) Vulcanised laminated rubber bearing under vertical tension. (Tyler, 1991.)
(c) Stress-strain curve for the vulcanised laminated rubber bearing under both compression and tension. (Tyler, 1991.)
(a) (b)
(c)
74
It is normal practice to design bridge bearing installations so that negative pressures do not occur in the rubber under the combined action of non-seismic loads and motions. It is also appropriate to design isolated structures so that non-seismic actions do not cause negative pressures.
However, when seismic actions cause negative pressures in isolator mounts, their duration and frequency are so low that considerable negative pressures might be tolerated (Tyler, 1991). In general, an isolator design should be adopted which avoids very high negative pressures during seismic action. In the particular case of high uplift forces under the corner columns of two-way frame structures, high negative pressures in corner rubber bearings may be avoided by attaching the bearing tops to the bottom beams of the frames designed to allow corner uplift, for example as described by Huckelbridge (1977).
3.5.6 Other Factors in Rubber Bearing Design
In practice the application of laminated-rubber bearings to seismic isolation calls for sophisticated design and specialised manufacturing technology. The rubber must be formulated for long-term stability and resistance to environmental factors, particularly deterioration due to ozone and ultraviolet light. The bonds (vulcanising) between the rubber and the interleaved metal plates must resist the large and varying operating stresses. Bearings must be provided with end and side rubber cover to inhibit corrosion of the metal plates and to remove rubber-surface deterioration from regions of high operating strains. The rubber cover and additional surface materials may be used to increase fire resistance. Interleaved steel plates must have adequate strength to resist rubber shear forces. However, some plate bending may reduce the build-up of rubber tension when large displacements give high end moments. Bearing end-plates must provide for dowels or for other means of preventing end slip under high shear forces. Such shear connections must operate despite end moments and in some cases when uplift occurs.
The effect of a fire on the performance of rubber elastomeric bearings and lead rubber bearings has been checked by Miyazaki (1991) in Japan, by heating the outside of bearings to greater than 800oC for more than 100 minutes while carrying a vertical load.
After this heating the rubber elastomeric bearings and the lead rubber bearings performed in a satisfactory way without any appreciable change in their force-displacement loops or load bearing capacities.
3.5.7 Summary of Laminated Rubber Bearings
Laminated rubber bearings are already in use in bridges, in order to accommodate thermal expansion. Their modification for the seismic isolation of buildings and bridges is a fairly simple engineering concept, but in practice it requires sophisticated design and specialised manufacturing technology.