Stability of Elastomeric Isolators:
Critical Load Tests and Computations
By
Satish Nagarajaiah and Ian G. Buckle
One of the most common seismic isolators in use today is the elastomeric
bearing. The combination of rubber layers and reinforcing steel shims
gives a device that is axially very stiff but soft laterally.
Flexibility may be increased, and large period shifts achieved, simply
by increasing the number and/or thickness of the rubber layers. But
increasing the shear flexibility of these short columns can lead to
relatively low buckling loads, which may be further reduced when high
shear strains are simultaneously imposed. As a consequence, many design
procedures require the axial load rating of a bearing to be reduced as
the shear displacement increases (e.g., AASHTO 1999). These reductions
are based on engineering judgment and very little science.
For example, for a rectangular bearing of width B, the critical load Pcr,
at shear displacement Δ, is approximated by
Pcr
= Pcro [1 - Δ/B]
where Pcro is the critical load at zero
shear displacement. With such an expression, the axial load capacity
becomes zero at a displacement Δ = B.
Experimental work undertaken by Buckle and Liu (1994) showed that this approach was
very conservative at high shear strain, and that substantial axial load
capacity remained even at displacements equal to the width of the
bearing. The purpose of this present study is to validate a new
theoretical model developed to numerically study the buckling of
elastomeric bearings at high shear strains. To do so, the method
explicitly includes large displacements in the formulation of the
critical limit state and allows post-buckling phenomena to be studied (Nagarajaiah
and Ferrell, 1999)
In the experimental work previously reported (Buckle and Liu, 1994), a
total of twelve bearings were tested. Nine of the square bearings were
five inches by five inches (127 mm x 127 mm) in plan. Three of the
square bearings were ten inches by ten inches (254 mm x 254mm) in plan.
Bearing properties are shown in Table 1. All bearings had bolted
connections at the top and bottom to prevent overturning. The rubber
shear modulus, G, was
estimated to be 0.2 ksi (1.38 MPa) at 0 % shear strain and 0.136 ksi
(0.938 MPa) at 100% shear strain (Nagarajaiah et al. 1999). The steel
shim thickness was varied in order to maintain the same overall height.
All bearings tested had one-inch (25.4 mm) thick end plates. The
elastomeric bearings were tested using the uniaxial single bearing test
facility at the Earthquake Engineering Research Center at the University
of California at Berkeley (see Koh and Kelly, 1986).
Table 1. Five- and
10-Inch Elastomeric Bearing Details
|
Bearings
Tested
|
Nominal
Size
B
x B' x H *
(In.
x In. x In.)
|
No.
of
Rubber
Layers
|
Thickness
of Rubber Layers
(Inch)
|
Thickness
of Steel Shim
(Inch)
|
Shape
Factor
|
|
101,102,103
|
5
x 5 x 4.375
|
3
|
0.75
|
0.0625
|
1.67
|
|
201,202,203
|
5
x 5 x 4.375
|
4
|
0.50
|
0.1250
|
2.50
|
|
301,302,303
|
5
x 5 x 4.385
|
8
|
0.25
|
0.0550
|
5.00
|
|
401
|
10
x 10 x 4.375
|
3
|
0.75
|
0.0625
|
3.33
|
|
501
|
10
x 10 x 4.375
|
4
|
0.50
|
0.1250
|
5.00
|
|
601
|
10
x 10 x 4.385
|
8
|
0.25
|
0.0550
|
10.00
|
*
B = Width of the Square Bearing, B' = Breadth; H = Height of the
Bearing
The axial load - horizontal displacement, P-u,
variation is shown in Fig. 1 as a function of shear force for bearing
302. The equilibrium path,
a smooth curve passing through discrete points, shown in Fig. 1 at each
shear force level, passes through a limit point, which is the critical
load. In Fig. 1 the equilibrium paths are unstable past the limit
point (Nagarajaiah et al. 1999); hence, the critical load must decrease
with increasing horizontal The critical load,
Pcr
, obtained from Fig. 1 and normalized with respect to critical load at
zero displacement, is shown in Fig. 2, as a function of horizontal
displacement normalized with respect to the width of the bearing, B. In
Fig. 2(a) it is evident that significant reduction in Pcr
occurs at horizontal
displacements equal to the width of the bearing, B. The results from the
nonlinear analytical model developed by Nagarajaiah et al. (1999) are
also shown in Figs. 2(a) and 2(b) for the 300 and 500 series bearings.
The critical load variation for bearing series 500, shown in Fig. 2(b),
decreases with increasing horizontal displacement; however, the decrease
in
Pcr
is not as significant as in
bearing 302.
|
 |
|
Fig.
1. Axial Load-Horizontal Displacement Variation as a Function of Shear, F
|
The stability of
the elastomeric bearings is studied, using the ADINA finite element
program. The Mooney-Rivlin material model suited for rubber undergoing
large strains was adopted. The stability of the bearings was determined
by the following procedure involving equilibrium paths (Nagarajaiah et
al. 1999). The bearings were first deformed in shear to a predetermined
shear displacement by means of a constant shear force. Then additional
shear displacements were monitored as the axial load, in the form of
vertical pressure at the top surface of the bearing, was monotonically
increased up to the limit point of the equilibrium path. The equilibrium
path past the limit point could not be traced as the incremental
solution failed. The critical load is the axial load at the limit point
of each equilibrium path (Nagarajaiah et al. 1999). This procedure was
repeated for increasing values of initial shear displacement; the
corresponding critical load - horizontal displacement values were
obtained.
Horizontal Displacement u/B |
|
|
2
(a) |
2
(b) |
|
Fig.
2 (a) and (b): Critical load as a function of Horizontal
Displacement
|
The variation of normalized critical load as a function of normalized horizontal
displacement computed using the ADINA finite element program and
experimental results are presented in Figs. 2(a) and 2(b), for 300 and
500 series bearings, respectively. The comparison in Figs. 2(a) and 2(b)
indicate good agreement for both 5 inch (127 mm) and 10 inch (254 mm)
bearings with different shape factors. The reduction in critical load
with increasing horizontal displacement is captured in both the
analytical model results and the ADINA results. The comparisons indicate
that the effect of large horizontal displacements on the critical load
can be reliably predicted. It is worth noting that a two-degree of
freedom nonlinear analytical model (Nagarajaiah et al. 1999) can capture
the complex nonlinear behavior adequately as compared to the finite
element model. It is evident from the results in Figs. 2(a) and 2(b)
that substantial critical load capacity exists at a horizontal
displacement equal to the width of the bearing and is not zero, as
predicted by the corrector factors used in design to account for large
shear displacements. These factors are not conservative at smaller
displacements and overly conservative at larger displacements.
For further details refer to Buckle et al. (2002) and Nagarajaiah
et al. (1999).
Conclusions
A substantial reserve of axial load capacity exists
in elastomeric bearings even when displaced in shear to a distance equal
to the width of the bearing. This capacity may be demonstrated
experimentally and theoretically using a new analytical model, which
captures the complex nonlinearities that occur in elastomeric bearings
at high shear strain.
Acknowledgements
This project was funded under Federal Highway
Administration Contract Number DTFH61-92-C-00106, which, in part, is
studying the use of earthquake protective systems for the seismic
retrofitting of highway bridges.
References
AASHTO
(1999) "Guide specifications for seismic isolation design", American
Association of State Highway and Transportation Officials, Washington
DC, 76 pp.
Buckle,I.G. and Kelly, J.M., (1986). "Properties of Slender Elastomeric
Isolation Bearings During Shake Table Studies of a Large-Scale Model
Bridge Deck," Joint Sealing and Bearing Systems for Concrete Structures (ACI),
Vol. 1, 247-269.
Buckle, I.G. and Liu, H., (1994). "Experimental
Determination of Critical Loads of Elastomeric Isolators at High Shear
Strain," NCEER Bulletin,
Vol. 8, No 3, 1-5.
Buckle,
I. G., Nagarajaiah, S., and Ferrell, K. (2002). "Stability of
elastomeric isolation bearings: Experimental study," Journal
of Structural Engineering, ASCE, Vol. 128, No. 1.
Koh,
C.G. and Kelly, J.M. (1986). "Effects
of Axial load on Elastomeric Bearings," UCB/EERC
- 86/12, Earthquake Engineering Research Center, University of
California, Berkeley.
Nagarajaiah, S., and Ferrell, K. (1999). "Stability of elastomeric
seismic isolation bearings," Journal
of Structural Engineering, ASCE, Vol. 125, No 9, 946-954. |
MCEER Bulletin Articles: Research
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