You will need Aladdin 2.0 to run this problem.

### PROBLEM STATEMENT

Before a new type of lead-rubber bearing can be used in a large structural system, design codes require that its hysteretic force-displacement capabilities be validated by experiment. In a typical experimental procedure the bearing displacements will be measured for a pre-determined combination of axial and cyclic shear forces.

In this example we demonstrate that the FRAME_3DS finite element can replicate the cyclic shear behavior of lead-rubber bearings derived from a force-controled experiment. Analysis procedures that incorporate nonlinear finite elements of this type are an important part of performance-prediction for base-isolated structures.

A lead-rubber bearing is a laminated rubber bearing with a lead plug down its center, as depicted in Figure 1.

#### Figure 1 : Lead-Rubber Laminated Bearing

The rubber/steel laminated bearing is designed to carry the weight of the structure and provide post-yield elasticity. The lead core deforms plastically under shear deformations. Its size can be selected to produce the required amount of damping.

In the design of highway bridges that must resist severe lateral loads, there are several benefits in using lead-rubber bearings for base-isolation:

1. Lead-rubber bearing are sufficiently stiff so that vertical loads and small lateral loads can be carried without excessive deformations occuring. Indeed, for small lateral loads (e.g., less than 0.1g ground acceleration) base-isolated and fixed-base structures should exhibit very similar behavior.

The size of the lead plug is proportional to the yield strength of the isolator. The post-yielding stiffness is proportional to the rubber bearing stiffness, and increases with the plan size of the rubber bearing and reductions in the isolator height.

2. The material properties of lead include:

• A low yield shear strength [about 10 MPa],

• Sufficiently high initial shear stiffness [G is approximately equal to 130 MPa],

• Post yielding behavior is essentially elastic-plastic. The capability of a lead plug to dissipate energy is proportional to the plastic displacements.

• Good fatigue properties for plastic cycles.

During severe seismic attacks, however, the lead plug is capable of deforming through many low-cycle plastic deformations without a loss of strength occuring.

Experimental studies indicate that lead responds essentially with elastic-perfectly plastic loops. Hence, for practical purposes the post-yielding isolator stiffness is equal to the stiffness of the rubber bearing alone. The global hysteresis loop is a bi-linear solid with an initial elastic stiffness

```    k_1 = 10 . k_r
```

where k_r is the stiffness of rubber, then followed by a post yield stiffness k_2 = k_r [2,3].

3. These devices allow a bridge to expand thermally without causing excessive forces in the bridge structure.

4. These devices also protect a structure with period shifting and increased damping in a single device [1].

### EXPERIMENTAL PROCEDURE

In this section we will briefly outline the experimental procedure for testing a single lead-rubber bearing in cyclic shear under compressive axial loads of 3125 kN.

Geometry and Material Properties

Figure 2 is a plan and elevation view of the lead-rubber laminated bearing we will consider in the analysis.

#### Figure 2 : Plan and Elevation of Lead-Rubber Laminated Bearing

These dimensions correspond to the largest lead-rubber isolator tested in an experimental program conducted by Robinson and co-workers [2].

In Aladdin the geometry of the lead-rubber bearing is represented by the variables:

```    iso_h = 197 mm;
r_dia = 650 mm;
l_dia = 170 mm;
```

The material properties of the lead are:

```    Kv = 600 kN/mm;
Ea = Kv*iso_h/(PI/4*r_dia*r_dia);

```

And the material properties of rubber are:

```    Kr = 1.75 kN/mm;
E_rubber = Ea;
G_rubber = Kr*iso_h/(PI/4*(r_dia*r_dia-l_dia*l_dia));
```

Here "E_lead" and "E_rubber" are the modulus of elasticity in the isolator's axial direction. The stiffness of the lead and rubber in the lateral direction(s) are given by "G_lead" and "G_rubber," respectively. "fv_lead" represents the yield stress of lead.

Test Procedure

The following test procedure will be used to assess the force-displacement capabilities of the lead-rubber laminated bearing under combined axials loads and cyclic shear loadings:

1. Apply a compressive axial force of 3125 kN to the bearing.

2. Apply a controled shear loading that varies between -200 kN and 200 kN.

### FINITE ELEMENT MODEL

We now consider the problem of modeling the lead-rubber isolator with the fiber element in ALADDIN. Because the lead-rubber isolator is composed of two different materials, each with its own shear modulus, the isolator behavior is modeled with a FIBER_3DS element.

Nodes and Elements

The block of code:

```    AddNode( 1, [ 0 m, 0 m, iso_h ] );
AddNode( 2, [ 0 m, 0 m,   0 m ] );

```

generates a finite element mesh containing two nodes a one finite element having the attribute label "isolator_attr".

After the boundary condition

```    FixNode( 2, [1,1,1,1,1,1] );
```

has been applied, the isolator finite element model has six d.o.f. See Figure 3.

#### Figure 3 : Elevation Views of Finite Element Mesh

Flexural Behavior

Figure 4 is a plan view of a lead-rubber isolator, showing the section dimensions and positions of the fiber elements.

#### Figure 4 : Fiber Model of Lead-Rubber Laminated Bearing

We model the flexural behavior of the isolator by partitioning the lead and rubber sections into quadrants -- each quadrant of the isolator is modeled with two fibers, one for the lead, and a second for the rubber.

The fibers are positioned at the centroid of the material quadrant. The material properties are as mentioned in the previous section.

Shear Behavior

Shear deformations in the lead and rubber are modeled with springs having equivalent shear stiffness (i.e., G_lead and G_rubber), one for each fiber.

#### Figure 5 : Elevation View of Isolator with Shear Spring Fibers

Finite Element Attributes

For simplicity the three-dimensional behavior of the base-isolation pad will be modeled with only one fiber finite element containing 8 fibers.

```    ElementAttr("isolator_attr") { type     = "FIBER_3DS";
section  = "iso_sec";
material = "iso_mat";
fiber    = "iso_fib";
}

```

Material and Section Properties

```    SectionAttr("iso_sec") { area  = PI/4*r_dia*r_dia;
depth = r_dia;
width = r_dia;
unit_weight  = 0.0001 kN/m;
shear_factor = 1.0;
J            = PI/32*r_dia^4;
}

MaterialAttr("iso_mat") { poisson  = 0.4;
E        = 3.562e8 N/m^2;
}
```

Fiber Geometry

The block of code:

```    no_fiber_type = 2;
divid_no      = 4;
theta = PI/divid_no;
l_r = l_dia*sin(theta)/theta/3;
r_r = r_dia*sin(theta)/theta/3;
no_fiber = 2*divid_no;

fcoord = Matrix([ 2, no_fiber ]);
farea  = Matrix([ 1, no_fiber ]);
fmap   = Matrix([ 1, no_fiber ]);
for( i=1 ; i<=divid_no ; i=i+1 ) {
fcoord[1][i] = l_r*cos((2*i-1)*theta);
fcoord[2][i] = l_r*sin((2*i-1)*theta);
farea[1][i]  = theta*l_dia*l_dia/4;
fmap[1][i]   = 1;

fcoord[1][i+divid_no] = r_r*cos((2*i-1)*theta);
fcoord[2][i+divid_no] = r_r*sin((2*i-1)*theta);
farea[1][i+divid_no]  = theta*(r_dia*r_dia - l_dia*l_dia)/4;
fmap[1][i+divid_no]   = 2;
}
```

defines three matrices, "fcoord", "farea", and "fmap" which store the coordinates, equivalent areas and material mappings for the eight fibers shown in the lower half of Figure 4.

Material Properties of Individual Fibers

The block of code:

```    fattr  = Matrix([ 6, no_fiber_type ]);
for( j=1 ; j <= no_fiber_type ; j=j+1 ) {
if( j == 1 ) {  /* lead */
fattr[3][j] = 10e6 MPa;
}
if( j == 2 ) {  /* rubber */
fattr[1][j] = E_rubber;
fattr[2][j] = E_rubber;
fattr[3][j] = 10e6 MPa;
fattr[4][j] = G_rubber;
fattr[5][j] = G_rubber;
fattr[6][j] = 10e6 MPa;
}
}

FiberAttr( no_fiber, "iso_fib" ) { FiberMaterialAttr = fattr;
FiberCoordinate   = fcoord;
FiberArea         = farea;
FiberMaterialMap  = fmap;
}
```

allocates memory for a 6-by-2 array fattr, and initializes its contents with the material properties shown in Table 2.

 Fiber Attribute Lead Rubber Young's Modulus E_lead E_rubber Tangent Young's Modulus E_lead E_rubber Yield Stress 10e06 MPa 10e06 MPa Shear Modulus G_lead G_rubber Tangent Shear Modulus G_lead/1000000 G_rubber Shear Yield Stress fv_lead 10e06 MPa

Table 1 : Fiber Attributes for Lead-Rubber Isolator

### HYSTERETIC BEHAVIOR OF LEAD-RUBBER ISOLATOR

The test-procedure is as follows:

First we apply a compressive force of 3.25 MN to the isolator:

```    stiff = Stiff();
NodeLoad( 1, [ 0 kN, 0 kN, -3150 kN, 0 kN*m, 0 kN*m, 0 kN*m] );
displ = Solve( stiff, P_old );
ElmtStateDet( displ );
stiff = Stiff();
```

and compute the element state determination.

```    if( step == 1 ) then {
NodeLoad( 1, [ 200 kN, 0 kN, 0 kN, 0 kN*m, 0 kN*m, 0 kN*m] );
} else {
if( step <= 21 ) {
NodeLoad( 1, [  10 kN, 0 kN, 0 kN, 0 kN*m, 0 kN*m, 0 kN*m] );
}
if( step > 21  && step <= 101 ) {
NodeLoad( 1, [ -10 kN, 0 kN, 0 kN, 0 kN*m, 0 kN*m, 0 kN*m] );
}
if( step > 101 ) {
NodeLoad( 1, [  10 kN, 0 kN, 0 kN, 0 kN*m, 0 kN*m, 0 kN*m] );
}
}
```

At each step of the analysis we compute the nonlinear displacements corresponding to the force-controlled shear loading. The Aladdin code is really quite straight forward:

```   no_step = 180;
for( step=1 ; step <= no_step+1 ; step=step+1 ) {

/* Force-controled shear load increment */

..... details shown above ....

dP    = P_new - P_old;
P_old = P_new;

/* Newton-Raphson Iteration for State Determination */

while( L2Norm(dP) > 0.001 ) {

dp    = Solve( stiff, dP );
displ = displ + dp;

ElmtStateDet( dp );

stiff = Stiff();
dP    = P_new - PR;        /* Compute unbalanced force */
}

UpdateResponse();
print step,"\t",P_new[max_dof][1](kN),"\t",displ[max_dof][1](mm),"\n";
}
```

Comparison of Finite Element and Experimental Behavior

Figure 6 compares the force-displacement relationship for the FIBER_3DS element with the experimental data generated by Robinson [2].

#### Figure 6 : Hysteresis Loops for Lead-Rubber Isolator

Since there is a good match between these curves, we conclude that the bi-linear fiber element will provide sufficient modeling accuracy for this study.

References

1. Priestley M.J.N., Seible F., and Calvi G.M., Seismic Design and Retrofit of Bridges, John Wiley and Sons, Inc., 1996.
2. Robinson W.H., Lead-Rubber Hysteretic Bearings Suitable for Protecting Structures During Earthquake, Earthquake Engineering and Structural Dynamics, Vol 10, pp. 593-602, 1982.
3. Skinner R.I., Robinson W.H., McVerry G.H., An Introduction to Seismic Isolation, John-Wiley and Sons, Inc, 1993.