Keywords: base isolation, LRB systems, optimal design.

A method is proposed that determines the optimal design of Lead Rubber Bearing (LRB) systems based on a systematic use of the interrelations that govern the behaviour of a single LRB. This is based on the assumption of constant velocity spectra and the analysis of a single degree of freedom system adopted by many design codes. The optimal cost of the bearing that consists of the cost of rubber and lead is determined by varying the external and the internal diameters together with the total height of rubber. The constraints refer to the damping ratio and the allowable shear strain as well as performance and capacity constraints. The optimal solution corresponds to the maximum values for the damping ratio and shear strain and is deduced systematically on a curve derived for every specific problem for the critical capacity constraint that controls buckling. This simplification is extended to problems having different groups of LRBs and the optimal solution is deduced accordingly. Numerical results are presented which are compared with those determined following a mathematical programming approach and their agreement is established.

Seismic isolation constitutes a design strategy that attempts to separate the structure from the damaging effects of the earthquake ground motions [1]. One practical seismic isolation system that provides the necessary requirements in the form of a single elastomeric bearing unit is the Lead Rubber Bearing (LRB) [2].

The procedures suggested by various codes of practice for the evaluation of isolated structures are in increasing order of complexity, a) the equivalent static analysis, b) the response spectrum analysis and c) time-history analysis. The equivalent static method is based on the assumption of a rigid superstructure above the isolators and is suitable also for the cost optimization of the isolation system and its performance.

The constant velocity spectrum assumption implies that the displacement is proportional to the period and the acceleration is inversely proportional to period . In addition, the base shear coefficient that represents the design forces on the superstructure is also inversely proportional to the period .

The material cost is proportional to a constant and the vertical load . This implies that the cost curves on the and plane have the same form as the and constraint curves.

The proposed algorithm for the optimal design of a single LRB system consists of the following steps:

1. Specify input data , , and .
2. Set and to their maximum allowable values.
3. Calculate and using the appropriate equations.
4. Calculate the diameter ratio .
5. Iterate on the critical capacity buckling constraint for different values of and .

The simplified formulation of the Single LRB optimization problem guarantees optimum cost and the maximum possible isolation performance associated with the optimum cost.

The Single LRB optimization problem can be directly extended to cover the design of LRB base isolation systems that consist of bearings having the same total rubber thickness and diameter ratio , but may have one or more different external diameters .

From the analysis and results it becomes evident that the optimal material cost solution of a single LRB system can be deduced systematically once the damping coefficient and the allowable shear strain are set to their maximum values. This is performed through the establishment of the curve of optimal cost in the and plane which is intersected by the critical capacity buckling constraint [4] to determine the unique solution. The problem is reduced to the solution of the critical non-linear capacity constraint and can be programmed easily into a spread sheet type of application. The proposed method offers an insight to the interrelations that govern the problem and in that respect is superior to a standard optimization formulation. Further more it can be effectively used for the optimal design of base isolation systems with several groups of isolators simply by solving for the equivalent single bearing and treating the various capacity constraints individually for every different type. This is computationally very efficient as compared to a mathematical programming type of formulation.

1

Naeim, F., and Kelly, J.M. (1999). Design of seismic isolated structures: From theory to practice, Wiley, Chichester, England.

2

Kelly, J.M. (1997). Earthquake resistant design with rubber, 2nd Ed., Springer Verlag, New York.

3

Skinner, R.I., Robinson, W.H., and McVerry, G.H. (1993). An introduction to seismic isolation, Wiley, Chichester, England.

4

Buckle, I.G., Nagarajaiah, S., and Ferrell, K. (2002). "Stability of elastomeric Isolation Bearings: Experimental Study."J. Struct. Engrg., ASCE, Vol. 128,No 1, January 1, 2002. doi:10.1061/(ASCE)0733-9445(2002)128:1(3)

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