LeadRubberX

This command is used to construct an LeadRubberX bearing element object in three-dimension. The 3D continuum geometry of an lead rubber bearing is modeled as a 2-node, 12 DOF discrete element. This elements extends the formulation of Elastomeric_Bearing_(Plasticity)_Element (or Elastomeric_Bearing_(Bouc-Wen)_Element) elements, however, instead of an user providing material models as input arguments, it only takes geometric and material properties of an elastomeric bearing as arguments. The material models in six direction are formulated within the element from input arguments, which also includes a new material model of behavior of elastomeric bearing under cyclic tension. In two shear directions, it uses a special case of the Bouc-Wen hysteresis model extended by Nagarajaiah et al.(1991) for seismic isolation bearings. In addition to the behavior captured by existing bearing elements, this element can capture the following:

1. Strength degradation in cyclic shear loading due to heating of lead core
2. Coupled bidirectional motion in horizontal directions
3. Coupling of vertical and horizontal motion
4. Cavitation and post-cavitation behavior in tension
5. Strength degradation in cyclic tensile loading due to cavitation
6. Variation in critical buckling load capacity due to lateral displacement

For the full capabilities of this element, users are referred to: EESD Article

The syntax of command to use this element in a 3D problem:

element ElastomericX $eleTag $Nd1 $Nd2 $qRubber $uh $Gr $Kbulk $D1 $D2 $ts $tr $n <<$x1 $x2 $x3> $y1 $y2 $y3> <$kc> <$PhiM> <$ac> <$sDratio> <$m> <$cd> <$tc> <$qL> <$cL> <$kS> <$aS>

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$eleTag	unique element object tag
$Nd1 $Nd2	end nodes
$qRubber	yield strength
$uh	yield deformation
$Gr	shear modulus of elastomeric bearing
$Kbulk	bulk modulus of rubber
$D1	internal diameter
$D2	outer diameter (excluding cover thickness)
$ts	single steel shim layer thickness
$tr	single rubber layer thickness
$n	number of rubber layers
$x1 $x2 $x3	vector components in global coordinates defining local x-axis (optional)
$y1 $y2 $y3	vector components in global coordinates defining local y-axis (optional)
$kc	cavitation parameter (optional, default = 10.0)
$PhiM	damage parameter (optional, default = 0.5)
$ac	strength reduction parameter (optional, default = 1.0)
$sDratio	shear distance from iNode as a fraction of the element length (optional, default = 0.5)
$m	element mass (optional, default = 0.0)
$cd	viscous damping parameter (optional, default = 0.0)
$tc	cover thickness (optional, default = 0.0)
$qL	density of lead (optional, default = 11200 kg/m3)
$cL	specific head of lead (optional, default = 130 N-m/kg oC)
$kS	thermal conductivity of steel (optional, default = 50 W/m oC)
$aS	thermal diffusivity of steel (optional, default = 1.41e-05 m2/s)

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An example use of this element can be found here: to be updated!

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NOTE:

Because default values of heating parameters are in SI units, user must provide the heating parameters values if using Imperial units

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References

1. Kumar, M., Whittaker, A., and Constantinou, M. (2014). "An advanced numerical model of elastomeric seismic isolation bearings." Earthquake Engineering & Structural Dynamics, Published online, DOI: 10.1002/eqe.2431. Link
2. Kalpakidis, I. V., Constantinou, M. C., and Whittaker, A. S. (2010). "Modeling strength degradation in lead-rubber bearings under earthquake shaking." Earthquake Engineering and Structural Dynamics, 39(13), 1533-1549.