Stability of Elastomeric Isolators: Critical Load

Stability of Elastomeric Isolators: Critical Load Tests and Computations By Satish Nagarajaiah and Ian 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 ∆, is approximated by Pcr = Pcro [1 - ∆/B] where Pcro is the critical load at zero shear displacement. With such an expression, the axial load capacity becomes zero at a displacement ∆ = 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).

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MCEER Bulletin, Vol. 16, No.1, Spring/Summer 2002

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

101,102,103 201,202,203 301,302,303 401 501 601

5 x 5 x 4.375 5 x 5 x 4.375 5 x 5 x 4.385 10 x 10 x 4.375 10 x 10 x 4.375 10 x 10 x 4.385

3 4 8 3 4 8

0.75 0.50 0.25 0.75 0.50 0.25

0.0625 0.1250 0.0550 0.0625 0.1250 0.0550

1.67 2.50 5.00 3.33 5.00 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 displacement. 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.

Bearing 302: Experimental

100

Axial Load: P (kip)

80

F = 0.5

60 F = 1.0 F = 1.5 F = 2.0 F = 2.5 F = 3.0 20 F = 3.5 40

0 0

1

2 3 4 5 Horizontal Displacement: u (in)

6

Fig. 1. Axial Load—Horizontal Displacement Variation as a Function of Shear, F

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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. 500 Series Bearings

300 Series Bearings 0.8 0.6

o .. *

Experimental Analytical Approximate ADINA

0.4 0.2 0 0

0.5 1 Horizontal Displacement: u/B

2 (a)

1 Critical Load: Pcr/Pcro

Critical Load: Pcr/Pcro

1

0.8 0.6

- Analytical .. Approximate o ADINA

0.4 0.2 0 0

0.5 1 Horizontal Displacement: u/B

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).

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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.

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