Isolator2spring Section: Model to include buckling behavior of an elastomeric bearing

This command is used to construct an Isolator2spring section object, which represents the buckling behavior of an elastomeric bearing for two-dimensional analysis in the lateral and vertical plane. An Isolator2spring section represents the resultant force-deformation behavior of the bearing, and should be used with a zeroLengthSection element. The bearing should be constrained against rotation.

section Iso2spring $matTag $tol $k1 $Fyo $k2o $kvo $hb $PE <$Po>

Model Characteristic:

This material model is based on a two-spring mechanical model of an elastomeric bearing, originally developed by Koh and Kelly [1987] (Figure 1). The model yields the approximate results as predicted by stability analysis of a multi-layer bearing. The axial flexibility of the bearing is modeled by an additional vertical spring in series (not shown in Figure 1). The original model included only linear material behavior. The rotational stiffness is given by where is the Euler buckling load as a function of bending stiffness EIs and bearing height hb. The nominal shear stiffness and the vertical stiffness, where G is the shear modulus, Ec is the compression modulus, A is the cross-sectional area, and tr is the total height of rubber. Ec and G can be related by the bearing shape factor S [Kelly 1997].

Figure 1: Two-spring model of an isolation bearing in the undeformed and deformed configuration.

In this implementation, the linear shear spring has been replaced by a bilinear spring to represent the nonlinear behavior observed in elastomeric and lead-rubber bearings [Ryan et. al 2005] (Figure 2). The nonlinear behavior is implemented by rate-independent plasticity with kinematic hardening [Simo and Hughes 1998]. The behavior of the nonlinear spring is controlled by the initial stiffness k1, yield strength Fyo, and postyield stiffness k2o.

Figure 2: Bilinear shear spring and parameters.

Also included is an optional variation of strength with axial load, to represent the inability of lead-plug bearings to achieve their full strength when lightly loaded, as has been experimentally observed. An empirical equation for the yield strength as a function of compressive load has been developed from experimental data:

where P is the compressive load on the bearing and Po is the axial load at which approximately 63% of the nominal strength is achieved (Figure 3). The bearing is assumed to have an acting yield strength of zero in tension. If not specified, Po = 0, which means that the strength equals the nominal yield strength and no strength degradation occurs.

Figure 3: Empirical model for yield strength degradation.

The equilibrium equations and kinematic constraints for the two-spring model form a system of five nonlinear equations (below) which are solved by an internal iterative Newton algorithm (this is iteration within the material object). The stiffness matrix is formed by taking differentials of the equilibrium and kinematic equations. The return mapping algorithm is implemented at each iteration to determine the state of the shear spring.

Not defined previously, v is the deformation due to the vertical spring and ubv is the total vertical deformation, including the geometric effect of tilting.

Analysis of the linear two-spring model leads to the following approximate coupled lateral force-deformation and vertical force-deformation equations:

whereis the critical buckling load for the bearing, and

That is, lateral stiffness decreases as the axial load on the bearing approaches the critical load, and vertical flexibility increases in the laterally deformed configuration.

Example:

The following example demonstrates a simple cyclic lateral load test, and was used to produce the lateral force deformation behavior shown in Figure 4 by variation of the parameters P/Pcr and P/Po in the script file.

Figure 4: Cyclic lateral force deformation behavior of a bearing as a function of axial load: (a) postyield stiffness degrades as axial load P approaches the critical load Pcr, (b) yield strength degrades as axial load decreases relative to Po.

References:

Ryan, Keri L., James M. Kelly and Anil K. Chopra (2005). "Nonlinear model for lead-rubber bearings including axial-load effects" Journal of Engineering Mechanics, ASCE, 131(12).

Kelly, James M. (1997). Earthquake-Resistant Design with Rubber. Springer-Verlag.

Koh, C.-G. and Kelly, J. M. (1987). "Effects of axial load on elastomeric isolation bearings", Rep. No. UCB/EERC-86/12. Earthquake Engineering Research Center, University of California, Berkeley.

Kelly, James M. (2003). "Tension buckling in multilayer elastomeric bearings", Journal of Engineering Mechanics, ASCE, 129(12):1363-1368.

Simo, J. C and T. J. R. Hughes (1998). Computational Inelasticity. New York, NY, Springer.