17th Analysis and Computation Specialty Conference

1
Formulation and Implementation of a Lead-Rubber
Bearing Model Including Material and Geometric
Nonlinearities

• Keri L. Ryan, Assistant Professor, Civil and
  Environmental Engineering, Utah State University
• James M. Kelly, Professor Emeritus, Civil and
  Environmental Engineering, Univ. of Calif.,
  Berkeley
• Anil K. Chopra, Johnson Professor, Civil and
  Environmental Engineering, Univ. of Calif.,
  Berkeley

2
Lead-rubber bearings are composed of alternating
rubber and steel layers, and an energy
dissipating lead core.
3
Material nonlinearity is usual accounted for
through a bilinear force-deformation model.

• Can be implemented by rate-independent plasticity
  model or a Bouc-Wen model
• Model is defined by initial stiffness kI,
  postyield stiffness kb, and strength Q of lead
  core

4
Geometric nonlinearities (stability or axial-load
effects) of a multi-layer bearing are represented
by a linear two-spring model (Kelly, 1997).
and
h
where
is the conventional Euler buckling load
5
Explicit lateral and vertical bearing force
deformation solved from equilibrium in deformed
configuration.
Equilibrium Equations
P
ub
fb
duz
Kinematic Equations
6
Explicit lateral and vertical bearing force
deformation solved from equilibrium in deformed
configuration (cont).

• Solve for critical buckling load from Eqs. 1
  and 2
• Evaluate coupled Eqs. 1 and 2 for s and ?,
  substitute into Eq. 4 for lateral
  force-deformation
• Substitute values of v (Eq.3), s and ? into Eq.
  5 for vertical force-deformation

7
A numerical model was developed to combine the
nonlinear material behavior and the geometric
effects.

• The two-spring model was extended to follow a
  bilinear force-deformation relation in the shear
  spring.
• An explicit formulation was no longer possible.

fs
Q
kb
k1
s
8
The behavior of the shear spring was modified to
include strength degradation of the lead core as
observed experimentally.

• Strength Q of the lead core degrades according
  to
• Qo nominal strength at large loads Po selected
  to fit data from bearing tests

9
An element routine was developed for use with a
global stiffness procedure (1) compute the
bearing forces as a function of the deformations.

• Equilibrium and kinematic equations recast in
  root finding form.

System of 5 nonlinear equations
5 unknowns (Bearing forces and internal
deformations)
10
Element routine (1) Compute the bearing forces
as a function of the deformations (cont).

• Nonlinear system of equations solved
  iteratively using Newtons method. An improved
  solution is computed by

where the Jacobian J(x)
11
Element routine (2) Compute the bearing tangent
stiffness matrix.

• Take differentials of the equilibrium and
  kinematic equations. E.G. Equation 1

Equilibrium Equations
Element Forces Element Deformations Internal
Deformations
Kinematic Equations
Define
12
Element routine (2) Compute the bearing tangent
stiffness matrix (cont).

• Rearrange in matrix form
• (1) Differentials of Equilibrium Equations

or
(2) Differentials of Kinematic Equations
or
13
Element routine (2) Compute the bearing tangent
stiffness matrix (cont).

• Combine these two equations to derive the
  flexibility matrix

Invert to get stiffness matrix
14
Element implementation Element is now available
for use in OpenSees.

• Documentation at http//opensees.berkeley.edu/
  OpenSees/manuals/ usermanual/index.html
• Model is under material/section/Isolator2spring
  (for use with zeroLengthSection element)
• Use in dynamic analysis
• View source code

15
Verification Observed lateral stiffness from
cyclic tests matched theoretical stiffness
derived from two-spring model.

• Theoretical lateral stiffness

16
Verification Observed stiffness from cyclic
vertical tests at imposed lateral strains matched
theoretical vertical stiffness as a function of
lateral strain.

• Theoretical vertical stiffness
• Element behavior should be verified under
  realistic earthquake loading!

17
The significance of geometric nonlinearities
(stability or axial-load effects) was explored by
lateral-rocking analysis of a rigid block on
isolators.
Rigid block with mass m and radius of gyration r
Governing equations
18
CONCLUSION Axial-load effects can usually be
neglected in analysis to determine the peak
response of the isolation system.

• Percent error in neglecting axial-load effects
  usually less than 10.
• Recommendations for when to use the advanced
  model have been made (Ryan and Chopra, 2005).

19
Questions??

• References
• Kelly, J. M. (1997). Earthquake-Resistant Design
  with Rubber. Springer-Verlag.
• Ryan, K. L. and Chopra, A. K. (2005). Estimating
  the seismic response of base-isolated buildings
  including torsion, rocking, and axial-load
  effects. Rep. No. UCB/EERC-2005-01. Earthquake
  Engineering Research Center, Univ. of Calif.,
  Berkeley.
• Ryan, K. L., Kelly, J. M., and Chopra, A. K.
  (2005). Nonlinear model for lead-rubber
  bearings including axial-load effects. Journal
  of Engineering Mechanics (ASCE), 131(12),
  1270-1278.