Modeling Triple Fp in Sap2000 6-27-2010

MODELING TRIPLE FRICTION PENDULUM ISOLATORS IN PROGRAM SAP2000 A. A. S. Sarlis1 and M.C. Constantinou2

June 27, 2010 1

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Graduate Student, Department of Civil, Structural and Environmental Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260 Professor, Department of Civil, Structural and Environmental Engineering, University at Buffalo, State University of New York, Buffalo, NY 14260.

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1. INTRODUCTION This document presents guidelines on modeling the Triple Friction Pendulum isolator in program SAP2000. Particularly, this document presents the following: 1) 2) 3) 4) 5) 6)

A summary description of the Triple Friction Pendulum isolator behavior. Presentation of the parallel model for modeling Triple Friction Pendulum isolators. Implementation of parallel model in SAP2000. Use of direct integration and fast nonlinear analysis in SAP2000. Modeling of global damping in seismically isolated structures in SAP2000. Significance of Ritz vector modes in capturing correctly the response of seismically isolated structures in SAP2000. 7) Consideration of isolator P- effects in direct integration and in fast nonlinear analysis in SAP2000. 8) Verification examples including input files for modeling Triple Friction Pendulum isolators using the parallel and the series models. While the document concentrates on the Triple Friction Pendulum isolator, the issues discussed on damping specification, Ritz vector modes, and P- analysis also apply to other isolation systems. Moreover, while the presentation is restricted to program SAP2000, the concepts are transferable to program ETABS that shares the same features as SAP2000 but limited to fast nonlinear analysis.

2. DESCRIPTION OF TRIPLE FRICTION PENDULUM ISOLATOR The Triple Friction Pendulum (FP) isolator exhibits multiple changes in stiffness and strength with increasing amplitude of displacement. Details on the behavior of the isolator, including test data, shake table testing and comparison of analytical and experimental data may be found in Morgan (2007) and Fenz and Constantinou (2008a to 2008e) . The construction of the force-displacement loop is complex as it may contain several transition points which depend on the geometric and frictional properties. Figure 1 shows the geometry of a Triple FP bearing. Its behavior is characterized by radii R1, R2, R3 and R4 (typically R1=R4 and R2=R3), heights h1, h2, h3 and h4 (typically h1=h4 and h2=h3), distances (related to displacement capacities) d1, d2, d3 and d4 (typically d2=d3 and d1=d4) and friction coefficients 1 , 2 , 3 and 4 (typically 2 = 3 ). The actual displacement capacities of each sliding interface are given by:

d i* 

Reffi d , i  1...4 Ri i

(1)

Quantity Reffi is the effective radius for surface i given by:

Reffi  Ri  hi , i  1...4

(2)

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Figure 1 Triple FP Bearing Definition of Dimensional and Frictional Properties The lateral force-displacement relation of the Triple FP isolator is illustrated in Figure 2. Five different loops are shown in Figure 2, each one valid in one of five different regimes of displacement. The parameters in the loops relate to the geometry of the bearing, the friction coefficient values and the gravity load W carried by the isolator as described in Fenz and Constantinou (2008a and 2008b). Triple FP isolators are typically designed to operate in regimes I to IV, whereas regime V is reserved to act as a displacement restrainer. In regime V the isolator has consumed its displacement capacities d1 and d4 and only slides on surfaces 2 and 3 (see Figure 1).

Figure 2 Force-Displacement Loops of Triple FP Bearing 3

Table 1 (adopted from Fenz and Constantinou, 2008a) presents a summary of the forcedisplacement relationships of the Triple FP bearing in the five regimes of operation. Note that

Ffi  iW is the friction force at interface i and W is the axial compressive load on the bearing.

* * * * Consider the special case in which Reff 1  Reff 4 , Reff 2  Reff 3 , d1  d 4 , d 2  d 4 , 1   4 and

 2   3 . Furthermore, consider that the bearing does not reach regime V. The result is an isolator with tri-linear hysteretic behavior as illustrated in Figure 3. Note this special case represents a typical case of configuration of Triple FP isolators. The force-displacement relation of the special Triple FP isolator is described by:

F

 R  W u  1  (1   2 ) eff 2  W Reff 1  2 Reff 1 

(3)

Force 2μ2W

μ1W μW

W/2Reff1 W/2Reff2

μ2W

u*=2(μ1-μ2 )Reff2

Displacement 2u*

Figure 3 Force-Displacement Loop of Special Triple FP Isolator The relation given by equation (3) is valid until the lateral force and displacement reach the values given by the following equations:

F

W * d1  F f 1 Reff 1

(4)

u  u*  2d1*  2(1  2 ) Reff 2  2d1* 4

(5)

u*  u**  2( 1  2 ) Reff 2

(6)

The force at zero displacement is given by 

W   1  ( 1  2 ) 

Reff 2  W Reff 1 

(7)

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Table 1 Summary of Triple FP Bearing Behavior (Nomenclature Refers to Figure 1) Regime I

II

III

IV

V

Description Sliding on surfaces 2 and 3 only Motion stops on surface 2; Sliding on surfaces 1 and 3 Motion is stopped on surfaces 2 and 3; Sliding on surfaces 1 and 4 Slider contacts restrainer on surface 1; Motion remains stopped on surface 3; Sliding on surface 2 and 4 Slider bears on restrainer of surface 1 and 4; Sliding on surfaces 2 and 3

Force-Displacement Relationship

F

Ff 2 Reff 2  Ff 3 Reff 3 W u Reff 2  Reff 3 Reff 2  Reff 3

Valid until:

F  Ff 1 , u  u    1   2  Reff 2   1  3  Reff 3

Valid until:

F  Ff 4 , u  u  u   4  1  Reff 1  Reff 3

F

F  Fdr1 

 Reff 4  W * d1  F f 1 , u  udr1  u  d1*  1     4  1   Reff 1  Reff 4   Reff 1  Reff 1 

W W * d1  F f 1  u  udr1   Reff 2  Reff 4 Reff 1

Valid until:

F



Ff 1  Reff 1  Reff 2   Ff 2 Reff 2  Ff 3 Reff 3  Ff 4  Reff 4  Reff 3  W u Reff 1  Reff 4 Reff 1  Reff 4

Valid until:

F



F  Fdr 4 

 d *   d*  W * d 4  F f 4 , u  udr 4  udr1   4   4    1  1    Reff 2  Reff 4     Reff 1  Reff 4  Reff 4    

W W * d 4  Ff 4  u  udr 4   Reff 2  Reff 3 Reff 4

Assumptions: (1) Reff 1  Reff 4  Reff 2  Reff 3 , (2)  2  3  1   4 , (3) d1*    4  1  Reff 1 , (4), d 2*   1   2  Reff 2 (5) d 3*    4   3  Reff 3

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3. MODELING OF TRIPLE FP ISOLATOR IN SAP2000 3.1 Introduction The series model has been developed by Fenz and Constantinou (2008d, e) in order to model behavior of the Triple FP bearings in all five regimes of operations. The series model, although unable to provide information on the motion of the internal components, is an exact representation of the triple FP bearing which indeed behaves as a series arrangement of single FP elements. However, the series model requires the introduction of artificial masses at the connecting joints of the FP elements and the use of a large number of degrees of freedom per bearing. These requirements typically have the following effects: 1) An increased computational effort for analysis. 2) The analysis produces results that are sensitive to the selection of values for secondary parameters used in the model, such as link element masses, effective stiffness and the elastic stiffness. 3) The required number of Ritz vector modes for use in fast nonlinear analysis is increased. The parallel model is a much simpler model capable of describing the behavior of the special * * * * case Triple FP bearing, for which Reff 1  Reff 4 , Reff 2  Reff 3 , d1  d 4 , d 2  d 4 , 1   4 ,  2   3 and the bearing does not enter the final regime of operation (stiffening). The parallel model was originally described in Sarlis et al (2009), where it was implemented in computer program 3DBASIS (Nagarajaiah et al, 1989 and Tsopelas et al, 1994 and 2005). This report describes the implementation of the parallel model in program SAP2000 (and, through its similarity, to program ETABS). The parallel model consists of a parallel arrangement of FP elements instead of a series arrangement of such elements. It has the following limitations: 1) Is incapable of exactly capturing the motion of the internal components of the triple FP bearing. However, the motion may be derived on the basis of the theory described in Fenz and Constantinou (2008a, 2008b), except for very small displacements. 2) The above limitation results in inability to exactly describe the velocity dependence of the friction coefficient. This limitation is bypassed by using the theory in Fenz and Constantinou (2008a, 2008b) to partition the total bearing motion to the two main sliding surfaces. Accordingly, the model describes well the velocity dependence of the friction coefficient except for the case of very small velocities. 3) The model cannot describe the behavior of the bearing in regime V as shown in Figure 2 (stiffening range). While the parallel model has been extended to describe the more general configuration of the triple FP where different friction coefficients are used at each sliding interface, this report specifically focus on the special case isolator due to its simplicity and wide practical use. The extended, so-called bundle model, will be described in another report.

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3.2 Description of Parallel Model The parallel model consists of two FP elements arranged in parallel and connected to two shared nodes. The first element (called FP1) represents a flat slider. The second element (called FP2) is a single Friction Pendulum element. The two elements share the same joints (nodes) and they overlap each other. The assembly is shown in Figure 4. The top joint is connected to the structure above, while the bottom joint is either fixed on the ground or rigidly connected to the structure below. Due to their arrangement, and provided the two elements are assigned equal axial stiffness, the elements equally share the total load on the isolator. The properties of the FP elements are selected as schematically depicted in Figure 5 to represent the global behavior of the Triple FP isolator (details of the parameters are provided in the sequel).

Figure 4 Parallel Model of Triple FP in SAP2000  

 

F

F

 

 1W

K1

 2W

K2 u

u

Y1 0 

*

 2W

W R2

FP1

FP2

Figure 5 Force Displacement Loops of Elements FP1 and FP2 of the Parallel Model The use of overlapping elements results in zero rotational stiffness of the element, which correctly represents the actual behavior. Note that earlier descriptions of the parallel element (Sarlis et al, 2009) placed the two elements of the model at a distance between them that created an artificial rotational stiffness. The rotational stiffness may affect the accuracy of the results of analysis. 8

3.3 Parameters of Parallel Model 3.3.1 Limit on Displacement The parallel model cannot capture the stiffening behavior of the triple FP bearing and therefore should not be used when the calculated displacement from response history analysis exceeds the limit (see Table 1):

udr 4  2  1  2  Reff 2  d1* 

(8)

3.3.2 Friction Coefficient The friction coefficients of elements FP1 and FP2 (see Figure 5) are calculated so the sum of the forces in the assembly of FP1 and FP2 elements equals the actual forces shown in Figure 3:

 2W   1W  K 2 Y 1

(9) (10)

1W   1W   2 W In the above equations, 1 is the friction coefficient of the FP1 element

 2 is the friction coefficient of the FP2 element W is the axial load in the FP1 and FP2 elements, which equals to W/2. K 1 is the elastic stiffness of the FP1 element. K 2 is the elastic stiffness of the FP2 element R 2 is the radius of curvature of the FP2 element. Y 1 is the yield displacement of the FP1 element. This is a small quantity, typically assumed to be about 0.01inch or larger (although see example of calculation later in this document). i is the friction coefficient of i-th surface of the triple FP isolator, where i=1,2,3 or 4. W is the axial load carried by the bearing. Assuming that the yield displacement of FP1 in SAP2000 is a small number and using W  W / 2

 1  2 2

(11)

 2  21   2 

(12)

The friction coefficients are defined at fast velocities (related to parameter FAST in SAP2000). For slow velocities, the friction coefficient (related to parameter SLOW in SAP2000) may be assumed to be half of the fast velocity value. 9

3.3.3 Radii of Curvature The FP1 is a flat slider (in SAP2000 this condition is represented as one with radius equal to zero). The post-elastic stiffness of element FP2 is equal to the post-elastic stiffness of the bearing, so that: (13)

W W W   2 Reff 1 R 2 2 R 2 Therefore,

3.3.4

(14)

R 2  Reff 1

Elastic Stiffness

The elastic stiffness of the bearing when motion in regime II starts is equal to the elastic stiffness of the FP2 element of the parallel model. K2 

(15)

W 2 Reff 2

The elastic stiffness of element FP1 can be calculated as follows

K1 

 2W 2Y 1

3.3.5

(17)

W



2 Reff 2

Force Intercept at Zero Displacement

The force intercept at zero displacement normalized by the axial load (or coefficient of friction at zero displacement), is given by the following expressions (see Figures 3 and 5): *

2  2 

 2W K 2 R2

 2  1  2 

Reff 1  Reff 2 Reff 1

*

 1   1  2 2 3.3.6

(18) (19)

Rate Parameter

The rate parameter specified for coefficients of friction  1 and  2 should be half the values of the rate parameter for the coefficients of friction 2 and 1 , respectively of surfaces 2 (or 3) and 1 (or 4) of the bearing. Note that this is due to fact that the parallel model is limited to cases in 10

which 2  3 and 1  4 for which the sliding displacements and sliding velocities on surfaces 1 and 4 are equal to half of the total displacement and velocity (this is only an approximation for small displacements). 3.3.7

Effective Stiffness

The effective stiffness should be specified to be low in order to minimize damping leakage in the isolation system. In fast nonlinear analysis (FNA), SAP2000 constructs a global damping matrix using the specified effective stiffness of the link elements in the nonlinear isolation system. Accordingly, artificial viscous damping elements are introduced in the isolation system. Specification of a small effective stiffness ensures that the artificial damping is low to affect the calculated response and prevents magnification of damping in nonlinear dynamic analysis as the element develops an instantaneous effective stiffness that is less than the initial specified value. It is advised that an appropriate value for the effective stiffness to use is the value of the post elastic stiffness (or less) of each element, although some trial and error investigation may be needed to confirm that that damping leakage is minimal. Note that the effective stiffness is also used to calculate Ritz vector modes and, therefore, the values specified have some additional effect in FNA. As it will be discussed later in this report, the results of FNA can be exact if an appropriate number of Ritz vector modes are used regardless of the specified value of effective stiffness. 3.3.8

Vertical Stiffness

It is important that the FP1 and FP2 elements in the parallel model have the same vertical stiffness so that they equally share the axial load on the bearing. Otherwise, the model is invalid. For the triple FP bearing, it is appropriate to calculate the vertical stiffness as the axial stiffness of a column having the height of the bearing, diameter of the inner slider and modulus of elasticity equal to about one half the modulus of elasticity of steel in order to account for the some limited flexibility in the bearing. The calculated vertical stiffness is then equally divided to the FP1 and FP2 elements. 3.3.9

Shear Deformation Location

The shear deformation location should be specified at the mid-point of the two nodes of the FP1 and FP2 elements. This parameter has significance only in the calculation of the moment transferred by the bearing to the structure above and below (including P- moments, if such effects are included). The use of the mid-point for the shear deformation location is strictly valid for the special case isolator considered herein. (When friction on the two main sliding surfaces is different, the shear deformation location should be selected on the basis of the partition of the isolator displacement-theory in Fenz and Constantinou, 2008a and 2008c. It will be closer to the surface of the largest friction).

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3.3.10 Element Height The element height should be the same as the total height of the isolator including the thickness of the top and bottom concave plates in order to correctly calculate the moment acting at the bottom of the bearing concave plates (including P- moments, if such effects are included). 3.3.11 Rotational Stiffness (degrees of freedom R2, R3) The rotational stiffness of elements FP1 and FP2 should be zero. Given that the two elements are at the same location, the combined rotational (or bending) stiffness of the model is zero. This is very close to the actual behavior of the isolator, which has minor resistance to rotation (the resistance results from friction tractions on the inner spherical sliding surfaces). 3.3.12 Torsional Stiffness (degree of freedom R1) The torsional stiffness of elements FP1 and FP2 should be zero. The isolator has insignificant rotational resistance. 3.3.13 Link Element Mass For each of the FP1 and FP2 elements, we recommend the use of a value equal to half of the total mass of the isolator. Use of larger values for the link element mass may speed up execution of analysis although results may be slightly affected. 3.3.14 Link Element Rotational Mass Moment of Inertia A rotational mass moment of inertia must be assigned for the FP1 and FP2 elements for proper execution of the program. We recommend the approximate calculation of a mass moment of inertia for the bearing using equation (20) below and then dividing the moment equally to the two elements. Use of larger values for the link element mass moment of inertia may speed up execution of analysis although results may be slightly affected. I FP 

misolator  ( DR 2  h 2 ) 12

(20)

In this equation, misolator is the mass of the bearing, DR is the diameter of the rigid slider of the bearing and h is the bearing height. 3.3.16 Addition of Gap Element to Parallel Model It would appear as straight forward exercise to add gap elements in parallel to the Triple FP isolator parallel model to capture the stiffening behavior of the isolator in regime V. However, the behavior produced is not exactly correct. This is due to the fact that a gap element in SAP2000 engages and disengages at specified displacements (the gap displacement) while this is 12

not true for the Triple FP bearing. Use of a gap element will produce correct stiffening behavior but the unloading branch of the hysteresis loop will be incorrect. Note that the same problem exists when other approximations are used to model the Triple FP model (e.g., the bilinear hysteretic model) together with gap elements. Only the series model (Fenz and Constantinou, 2008d) of the Triple FP bearing is capable of correctly modeling this behavior. However, a gap element added in parallel to the parallel element of the Triple FP bearing produces correct results for the very special case of 1  2  3  4 . A summary of element properties for the parallel model is presented in Table 2. Table 2 FP1 and FP2 Element Properties in Parallel Model of Triple FP Bearing

Supported Weight W Elastic Stiffness K 1 , K 2

 2W

FP1

FP2

W

W

2

W

2 W

2 Reff 2

2Reff 2



2Y 1

Friction Coefficient  1 ,  2

2 2

Normalized Force Intercept at * * Zero Displacement  1 ,  2

2 2

a1

Rate Parameter a

21   2 

2(1   2 )(

Reff 1  Reff 2 ) Reff 1

a1

2

2

Radius of Curvature R1 , R 2

0(flat)

Reff 1

Element Height

h h 2

h h 2

0 0

0 0

misolator / 2

misolator / 2

Shear Deformation Location Rotational Stiffness (R2,R3) Torsional Stiffness (R1) Link Element Mass1 Rotational Mass Moment of Inertia1

I FP 



2

misolator DR  h 2 24



I FP 



2

misolator DR  h 2



24

1 Minimum value. A larger value may be specified. DR=diameter of rigid slider, h=bearing height, Y 1 =yield displacement equal to about 0.01inch or larger

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3.4 3.4.1

Global Analysis Parameters in SAP2000 Damping Specification in Fast Nonlinear Analysis (FNA)

The specification of structural damping in the analysis of seismically isolated structures (with nonlinear isolator elements) may appear to be of secondary importance. However, if improperly specified, damping may have significant effects. When modeling an isolated structure, the inherent damping of the structure exclusive of the isolation system is assigned based on the type structure (height, materials, construction method) and expected level of deformation. For a building, the damping ratio is only approximately known for the superstructure when fixed at the isolator locations. For example, the data presented in Satake et al (2003) may be used to select the values of damping ratio for the superstructure of isolated buildings. It should be noted that when data on damping for conventional structures are used in the modeling of seismically isolated structures, each mode of vibration of the fixed-based structure corresponds to two modes of vibration of the corresponding seismically isolated structure. For example, both the first and second modes, in say the longitudinal direction, of a seismically isolated exhibit a shape for the superstructure part that is basically identical to that of the first mode of the non-isolated structure. Accordingly, when a value of damping ratio is selected for the first mode of the non-isolated structure, that value should be used for the first six modes of the corresponding three-dimensional isolated structure (first and second modes in each of the two principal directions and torsion). Note that this is applicable when diaphragm constraints are used but it may be more complex otherwise (see discussion on Purely Isolated Modes below). Program SAP2000 constructs a global damping matrix which involves the degrees of freedom of the isolation system. The calculation of the global damping matrix makes use of the specified effective stiffness of each isolator element and of the mass and moment of inertia of the part of the structure between the superstructure and the isolator supports (e.g., basemat above isolators). This leads to the introduction of artificial viscous elements in the isolation system. We term this phenomenon “damping leakage in the isolation system”. This may significantly affect the calculation of isolator displacements by artificially damping its response as if viscous dampers were present in the isolation system. For example, consider that the isolation system has bilinear hysteretic behavior with yield displacement of 1inch. The damping is specified as 5% of critical in each mode of vibration and the effective stiffness of each isolator is specified as the elastic stiffness (that is, the 5% damping ratio is anchored on the elastic stiffness). Say that upon yielding of the isolator in dynamic analysis the displacement reaches 20inch amplitude. The effective damping is then equal to 0.0520/1=0.224! The magnification of effective damping by the square root of the ductility ratio is a well understood effect (for example see Chapter 18 of ASCE-7, Minimum Design Loads for Buildings and Other Structures, 2010). In Fast Nonlinear Analysis (FNA), the stiffness matrix remains always the same and is based on the specified effective stiffness of the isolator elements. Nonlinear forces from the isolator elements are treated as loads on the right hand side of the equations of motion. This allows the user to have control on the amount of damping that is leaked into the isolation system by

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specifying a small effective stiffness of the isolators. When a small effective stiffness is specified, one can distinguish the following Ritz vectors modes: 

Purely Isolated: An example of purely isolated Ritz vector mode is shown in Figure 6. In this mode, the superstructure responds nearly as a rigid block. The number of purely isolated modes depends on the extent of discretization of the structure. When a diaphragm constraint is specified for all floor levels and all masses are specified as point masses, the number of such modes is reduced to one for two-dimensional analysis and to three for three-dimensional analysis.

Figure 6 Example of Purely Isolated Ritz Vector Mode 

Mixed Modes: These modes are characterized by the mixed response of the superstructure and the isolation system. When a diaphragm constraint is specified for all floor levels and all masses are specified as point masses, the number of such modes is reduced to a number equal to the number of floors for two-dimensional analysis and to three times the number of floors for three-dimensional analysis. Figure 7 shows examples of mixed modes.

Figure 7 Examples of Mixed Ritz Vector Modes 

Local or vertical Modes: These are high frequency modes that mostly involve local response of members or the vertical response of the structure on the isolators. These 15

modes typically create numerical difficulties and negatively affect the computational effort. However, they are important in the calculation of quantities such as accelerations, member internal forces and isolator uplift. Figure 8 shows an example of local mode.

Figure 8 Example of Local Ritz Vector Mode The distinction between modes becomes complicated when vertical modes are of relatively low frequency so that they appear in between the mixed modes or when combined horizontal and vertical responses exist in a mode. In such cases, the modes should be classified as mixed rather than local. Damping may be specified in SAP2000 as follows: 

Constant Damping for All Modes

Specifying constant damping for all modes together with small effective stiffness is the simplest approach for reducing, but not eliminating, damping leakage. Note that in this approach the specified damping ratio of each mode is the same. 

Constant Damping with Override

This method is similar to the previous one but combined with zero damping override for the purely isolated modes in order to minimize damping leakage in the isolation system. This approach significantly affects the computational speed of the analysis so that, practically, may be used when the number of isolators is small. However, the user may want to use override in order to specify different (typically larger) damping ratio values for the higher modes. 

Rayleigh Damping

Rayleigh damping is based on the specification of damping ratio values at two selected frequencies. A simple approach is to select the two frequencies such that they bound the frequencies of the purely isolated and mixed modes as illustrated in Figure 9.

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Damping Ratio

Purely Isolated + Mixed Modes

Local or vertical Modes

ζ Selected Frequencies

ωi

ωj Frequency

Figure 9 Selection of Frequencies for Rayleigh Damping Assignment Two important considerations related to Rayleigh damping specifications are: 1) The two specified frequencies should be the first isolated mode frequency and the last mixed mode frequency. Accordingly, frequencies  i and  j are significantly different (the difference is even larger when low effective element stiffness is specified). Therefore, modes that lie in between these two frequencies will have significantly lower damping than the specified value, as illustrated in Figure 9. This typically leads in overestimation of the superstructure response (accelerations, drifts and isolator uplift). 2) Rayleigh damping provides the advantage of having higher damping for the higher modes. This is both correct (e.g., see Satake et al, 2003) and also computationally efficient as it reduces numerical difficulties and increases the speed of computation. 

Rayleigh Damping with Override

This method is similar to the Rayleigh Damping approach but allows for more control. It is the most effective way of specifying damping in a seismically isolated structure when the number of isolators is large and the speed of analysis is a concern. The procedure requires identification of the purely isolated modes, mixed modes and local modes. This is relatively easy when low values for the isolator element effective stiffness are used. The Rayleigh coefficients are calculated using the lowest and highest frequency of the mixed modes. Modes that have a frequency lower than the lowest mixed mode (these are the purely isolated modes) should be assigned zero (or very low) damping using the damping override option in SAP2000. Figure 10 illustrates the assigned damping ratio based on this approach.

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Damping Ratio

Purely Isolated Modes

Mixed Modes

Local or vertical Modes

ζ

ωi

ωj Frequency

Figure 10 Damping Ratio in Rayleigh Damping with Override An advantage of the Rayleigh Damping with Override approach is that frequencies  i and  j are closely spaced so that the damping ratio in the modes with frequencies between the two selected frequencies are not substantially less than the specified value of damping ratio. The analysis speed with this approach is similar to the Rayleigh Damping case but damping leakage is better controlled due to the damping override. 

Interpolated Damping

In this method, a different damping ratio may be specified for various frequency ranges. Therefore the user can implement zero damping or nearly so for the frequency range of the purely isolated modes, a specified damping ratio for the mixed modes, and a higher damping ratio for local modes. For the local modes, either a high constant damping ratio can be specified or a linearly varying damping ratio may be specified up to a maximum value of damping ratio. This is illustrated in Figure 11. In terms of minimizing damping leakage and accuracy of superstructure response, this damping specification approach can produce results similar to the constant damping with override and Rayleigh damping with override but with increased speed of analysis.

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Damping Ratio

Purely Isolated Modes

Local or vertical Modes

ζ

ωi

ωj Frequency

Figure 11 Damping Ratio in Interpolated Damping 3.4.2

Ritz Vector Modes

Fast nonlinear analysis requires the specification of the number of Ritz vectors modes for use in the analysis. A sufficient number of Ritz vector modes are needed to accurately capture the response of the analyzed structure. When the number of isolators is large, Ritz vector modes associated with high frequencies may be included which often creates numerical difficulties. The use of Rayleigh damping with override for the damping specification may be most useful in such cases. An important consideration in the selection of the number of Ritz vector modes has to do with the development of gravity load on the isolators (which is very important for sliding isolators). In FNA and prior to start of the seismic analysis, the gravity load is applied through analysis in which the structure is subjected to vertical ground acceleration that slowly increases with time. If the number of Ritz vector modes is not sufficiently large to fully develop the gravity load effects, the analysis with sliding isolators is equivalent to one in which the structure is subjected a vertical constant downward acceleration. This results in over-prediction of isolator displacements. Figure 12 provides an example from analysis of a bridge structure (bridge is the example in Constantinou et al, 2007). The figure shows the sum of axial forces on the bridge supports (which equal to the weight of the bridge) as function of time and the number of Ritz vector modes considered. The history of axial load evolves as the vertical acceleration is slowly imposed to develop the gravity load prior to starting the dynamic response history analysis. The weight is nearly equal to 6000kip. It requires at least 33 Ritz vector modes to develop correctly the load. The typically utilized minimum number of Ritz vector modes equal to 3 times the number of isolators plus rigid masses would have resulted in this case a requirement for use of 30 vectors, which should be sufficient for this example. Note that when the number of Ritz vector modes is insufficient for the development of the isolator gravity load, member forces are also underestimated which affects the adequacy assessment of the analyzed structure. It should also be noted that the required number of Ritz vector modes may depend on the form of the ramp 19

function used to develop the gravity load. That is, the cut-off time of 10 second in Figure 12 may affect the required number of Ritz vector modes. Moreover, it is important to first determine the number of Ritz vector modes needed to correctly develop the load and then conduct analyses with that number and with larger number in order to observe the sensitivity of the calculated response on the number of modes used. It is possible to correctly develop the load with a small number of Ritz vector modes but require many more modes to obtain correct results in dynamic analysis.

Total Bridge Weight (kips)

6000

4000

FNA-129 MODES FNA-5 MODES FNA-15 MODES

2000

FNA-30 MODES FNA- 50 MODES FNA-33 MODES

0 0

10

20

Time (sec)

Figure 12 History of Axial Load on Isolators as Function of Number of Ritz Vectors 3.4.3

Direct Integration

Program SAP2000 has the option of Direct Integration for dynamic analysis. In this approach, the effect of any nonlinear elements is accounted for by the use of instantaneous stiffness and damping matrices. This prevents the program user from controlling damping leakage in the isolation system. Actually, artificial damping in the isolation system may substantially fluctuate depending on the instantaneous stiffness of the nonlinear elements. We do not recommend use of the direct integration method unless damping is specified as zero or nearly so.

20

4.

ISOLATOR P-Δ MOMENT ANALYSIS IN SAP2000

4.1 Introduction Isolator P- effects can have important contributions to structural drift and acceleration. Figure 13 illustrates the transfer of forces at the sliding interfaces of a Double or a Triple FP bearing (a similar situation exists in elastomeric bearings). The moment P1will cause rotation of the column on top of the deformed isolator that, in turn, will result in increased first story drift.

Figure 13 Transfer of Force in Double or Triple FP Bearings P- effects can be explicitly accounted for in dynamic analysis in SAP2000 by use of the direct integration method of analysis. There is no explicit way of accounting for these effects in FNA analysis. 4.2 Approximate P-Δ Analysis when using Fast Nonlinear Analysis In FNA analysis, P-Δ effects cannot be explicitly analyzed. However, it is possible to approximately account for these effects by performing two analyses as described below: 1) First a dynamic response history analysis is performed, ignoring P-Δ effects, so that histories of the bearing axial loads and the displacements for each isolator are calculated. The P- moment at the top of each isolator is calculated (for the Triple FP isolator or any elastomeric isolator, the moment is equal to one half of the axial load times the displacement-note that the component of moment resulting from the shear force times the 21

height is accounted for directly by the program and need not be added to the axial force component). 2) Dynamic response history analysis is performed with the calculated P- moment applied as time history function at the top and bottom joints of each isolator and ensuring that the direction of the moments is such that it causes a shear force in the same direction as the isolator displacement. Note that the P- moment at the bottom joints is not needed if the isolators are directly supported on the ground. The procedure described above is one of a number of options for approximately accounting for isolator P- effects. Engineers have already tried other approximate ways for accounting for these effects such as first performing the response history analysis and then performing static analysis to only calculate the P- moment effects. Such a procedure can provide information on the effects on the structural drift but not on structural accelerations.

22

5. VERIFICATION OF ACCURACY OF PARALLEL MODEL 5.1 Description of Analyzed Two-Dimensional Structure and Verification Study A four story isolated structure is used herein to demonstrate aspects of modeling and analysis procedures described above. It is two-dimensional slice of an actual building. It is a moment frame with all connections being rigid. The structure is supported by six isolators. It is assumed that each isolator carries a load of 900kip, for a total weight on the isolators equal to 5400kip. In the analysis model, the mass is lumped at the joints as shown in Figure 14 with each joint assigned the same mass. Table 3 presents the periods of vibration of the first four modes of the superstructure when assumed fixed to the ground. Table 3 Periods of Vibration of Superstructure when Fixed at the Base Mode 1 2 3 4

Period (sec) 0.776 0.253 0.138 0.093

Figure 14 Schematic of Four Story Planar Structure (dimensions in inch) In modeling the isolated structure, structural damping of 2% of critical for all modes is assumed (constant damping model). The effective stiffness of each isolator is specified as equal to the post-elastic stiffness. Analysis is performed using the TCU-065-E component of the 1999 ChiChi, Taiwan earthquake. 23

The isolator properties used in the analysis are presented in Table 4. Table 4 Properties of Triple FP Isolators

Reff1  Reff4 (inch)

82.5

Reff2  Reff3 (inch)

7.5

d1*  d 4* (inch)

17.8

d 2*  d 3* (inch)

0.94

1   4

0.108

 2  3

0.030

a1  a 2  a 3  a 4 (sec/in)

2.54

All friction values are for high speed conditions=fmax

In order to verify the accuracy of the parallel model, analysis is performed using the parallel model described herein and the series model described in Fenz and Constantinou, 2008d. It is presumed that the series model provides an exact description of behavior of the Triple FP isolator. The series model as implemented in program SAP2000 is depicted in Figure 15.

Figure 15 Assembly of Friction Pendulum Link Elements, Gap Elements and Rigid Beam Elements Used in the Series Model of Triple FP Bearing in SAP2000 Table 5 presents the parameters of the series model of Triple FP bearings in SAP2000.

24

Table 5 Parameters of the Series Model of Triple FP Bearing in SAP2000 Friction Radii of Gap Elastic Rate Element Coefficient Curvature Displacement Stiffness Parameter FP1

 2  3

Reff 1  Reff 2  Reff 3

FP2

1

Reff 2  Reff 1  Reff 2

FP3

4

Reff 1  Reff 2 Reff 1 Reff 4  Reff 3

Reff 3  Reff 4  Reff 3

Reff 4

1  a2  a3  2 2

K

2W

d1

K

1W 2Y

Reff 1  Reff 2

d4

K

4W

Reff 4

NA

2Y

2Y

Reff 1

Reff 4  Reff 3

a1 a4

Yield displacement Y should be about equal to 0.01inch

Table 6 presents numerical values of the properties implemented in SAP2000 for the series and parallel models of each bearing. Results of analysis obtained with the two models are compared in Figures 16 to 20. Analysis was performed using the FNA method and utilizing 53 Ritz vector modes for the parallel model and 77 Ritz vector modes for the series model. These correspond to the maximum Ritz vector modes calculated by SAP2000. Evidently, the two models produce identical results for all practical purposes. Some small differences in the results are attributed to small differences in structural damping modeling and in small differences in modeling velocity dependence of the friction coefficient in the two models.

25

Table 6 Values of Properties of the Series and Parallel Models of Bearings in SAP2000 Series Model

Parallel Model

FP1

FP2

FP3

FP1

FP2

Element Height (inch)

8

2

8

16

16

Shear Deformation Location (in)-(distance from top joint of FP element)

8

2

0

8

8

Element Mass (kip-s2/in)

0.001

0.001

0.001

0.0025

0.0025

Supported Weight (kip)

900

900

900

450

450

Vertical Stiffness1 (kip/in)

213530

213530

213530

35588.4

35588.4

Elastic Stiffness1 (kip/in)

1350

4860

4860

807.9

54.55

Yield Displacement1 (inch)

0.01

0.01

0.01

0.0155

-

Effective Stiffness1 (kip/in)

60

12

12

0

5.45

Friction Coefficient SLOW

0.015

0.054

0.054

0.030

0.0709

Friction Coefficient FAST

0.030

0.108

0.108

0.060

0.1418

Radius (inch)

15

75

75

0

82.5

Rate Parameter1 (in/sec)

1.27

2.794

2.794

1.27

1.27

Rotational/Torsional Stiffness (R1,R2,R3)

0

Fixed

Fixed

0

0

Rotational Moment of Inertia (kip-in-sec2)

1.0

0

0

0.5

0.5

1.

See Appendix A for details of calculation

26

Figure 16 Comparison of Isolator Displacements

Figure 17 Comparison of Base Shear-Isolator Displacement Loops

27

Figure 18 Comparison of First Story Drift Ratio

Figure 19 Comparison of Roof Acceleration

28

Figure 20 Comparison of Moments at Top of Isolators 5.2 Description of Analyzed Three-Dimensional Structure and Verification Study A three-dimensional structure is generated using three frames identical to the two-dimensional frame of Figure 14, placed at distance of 384inch, center to center. All connections are assumed rigid. Figure 21 illustrates the three-dimensional model. The TCU-065-E and TCU-065-N components of the 1999 Chi-Chi, Taiwan earthquake are used as seismic excitation along the principal building directions. Structural damping is again specified as 2% of critical in each mode (constant damping model) with the effective stiffness of each of the18 isolators specified equal to the post-elastic stiffness. Each isolator is modeled using the data in Tables 4, 5 and 6. Analysis was performed using the FNA method and utilizing 196 Ritz vector modes for the parallel model and 200 Ritz vector modes for the series model. The results are shown in Figures 22 to 24. Evidently, the two models produce identical results for all practical purposes.

Figure 21 Schematic of Four Story Three-Dimensional Structure 29

Figure 22 Comparison of Isolator displacement Orbits and Histories 30

Figure 23 Comparison of Isolator Force-Displacement Loops in Orthogonal Directions

31

Figure 23 Comparison of First Story Drift Ratio

Figure 24 Comparison of Roof Accelerations 32

5.3 Effect of Ritz Vector Modes The number of Ritz vector modes affects the accuracy of analysis through (a) its effect on the development of the axial load on the isolators and the development of actions in the structural members, and (b) the incomplete description of the structural response when an insufficient number of modes are used. The two-dimensional, 4-story structure example of Section 5.1 with constant 2% damping (and effective isolator stiffness specified equal to the post-elastic isolator stiffness) was analyzed using the parallel Triple FP model in fast nonlinear analysis with varying number of Ritz vector modes. Selected peak response results are presented in Table 7. Table 7 Effect of Number of Ritz Vector Modes on Peak Response Results for TwoDimensional Structure No. of Ritz Vector Modes 2 5 10 20 30 40 50

Isolator Base Displacement Shear/Weight (inch) 0.234 0.235 0.236 0.235 0.235 0.235 0.235

22.18 22.22 22.38 22.31 22.29 22.29 22.29

First Story Drift Ratio (%)

Roof Acceleration (g)

CPU Time (sec)

0.18 0.33 0.34 0.32 0.31 0.31 0.31

0.24 0.34 0.43 1.55 0.43 0.42 0.42

14 17 84 158 188 229 257

Note that stable response prediction in this example is obtained when the number of modes exceeds 30. Also, note that the prediction of acceleration response is very sensitive to the number of modes. It is apparent that some experimentation is needed to determine a sufficient number of modes for use in the analysis. 5.4 Effect of Gap Element As described in Section 3.3.16, the addition of a gap element in parallel with the parallel model for the Triple FP bearing does not correctly capture the behavior of the isolator. To demonstrate the effects of such modeling, the two-dimensional structure of Section 5.1 was analyzed with the TCU-065-E component of the 1999 Chi-Chi earthquake scaled up by a factor equal to 1.4 so that the bearing enters the stiffening regime V. Each isolator was modeled using the parallel model described in Section 5.1 and with a gap element added. The gap opening was calculated using equation (8) and gap element stiffness after closing was selected in accordance with equation (21) below. Note that the gap element stiffness is such that the sum of the post elastic stiffness of the parallel model and the gap element stiffness equals to the stiffness of the FP bearing when sliding occurs simultaneously on surfaces 2 and 3.

33

K gap 

W 2 Reff 2



W

(21)

2 Reff 1

Moreover, and just for the purpose of demonstrating the gap element effects, the displacement capacities of surfaces 2 and 3 have been considered unlimited so that the parallel model can be used. Figure 25 compares the isolation system force-displacement loops for the case of the series model (exact) to the approximate parallel model with added gap element. The error introduced by the incorrect use of the gap element in the parallel model is seen in the enlarged figure. Nevertheless, the calculated displacement of the isolation system is practically unaffected.

Figure 25 Comparison of Results Obtained with the Exact Series Model and the Approximate Parallel Model with Gap Element 34

The enlarged version of Figure 25 demonstrates the error introduced by the gap element in the parallel model. The gap element engages and disengages at the same displacement as specified in the gap element parameters. The stiffening behavior of the Triple FP bearing, however, starts and ends at different displacements, which depend on the geometric and frictional properties of the bearing. Moreover, the fluctuating loop seen in Figure 25 for the series model is a result of numerical difficulty in the integration process.

35

6. STUDY OF DAMPING SPECIFICATION EFFECTS The two-dimensional model of Figure 14 is used to demonstrate the damping specification effects discussed in Section 3.4.1. 6.1 Demonstration of Damping Leakage Figure 26 compares the isolator displacement in the two-dimensional example of Section 5.1 when modeling damping using the constant damping model with 2% damping ratio and when specifying zero damping. All other parameters in the analysis model are identical for the two analyses. Note the effective stiffness of each isolator has been specified as equal to the postelastic stiffness. The difference in the peak isolator displacement observed in Figure 26 is primarily attributed to leakage of damping in the isolation system when the 2% constant damping specification is used. While the difference in peak isolator displacement is small in this case, it is noticeably larger when 5% damping is specified.

Figure 26 Comparison of Isolator Force-Displacement in Analysis with Different Damping Specification 6.2 Effect of Effective Stiffness Value on Damping Leakage The effect of the specified value of effective stiffness of each isolator used in analysis is demonstrated by analyzing the two-dimensional example of Section 5.1 using the constant damping model with various values of the effective isolator stiffness. Results are compared in Figure 27 when the damping ratio is specified as 2% in each mode and in Figure 28 when the damping ratio is specified as 5% in each mode. The effective stiffness is specified as either equal to the post-elastic stiffness (case KEFF=KPE) or equal to the elastic stiffness (case KEFF=KEL). 36

Figure 27 Comparison of Isolator Force-Displacement Loops in Case of 2% Constant Damping and with Different Values of Effective Stiffness (equal to the post-elastic stiffness or equal to the elastic stiffness) The results in Figure 27 demonstrate a significant effect when a large value of stiffness is used. As discussed earlier, the observed substantial difference in peak displacement is the result of magnification of effective damping that leaked into the isolation system and is magnified as inelastic action occurs in the isolation system.

Figure 28 Comparison of Isolator Force-Displacement Loops in Case of Zero Damping and 5% Damping and with Effective Stiffness Equal to the Elastic Stiffness The results of Figure 28 show that an incorrect specification of damping (although seemingly appropriate) may lead to totally erroneous results. Incorrect specification of the effective stiffness results in so much damping leakage in the isolation system that the predicted peak isolator displacement is reduced to less than half of the actual value. 37

6.3 Use of Constant Damping with Override Figure 29 compares isolator force-displacement loops in the example of Section 5.1 when damping is specified either as zero or as 2% constant damping with a 0% damping override for the first mode and with effective stiffness equal to the post-elastic stiffness. The results of the two models of damping are identical for all practical purposes. Note that constant damping with override, as described in Section 3.4.1, is an appropriate damping specification for seismically isolated structures. In this example, non-zero damping was specified but prevented from leaking into the isolation system.

Figure 29 Comparison of Isolator Force-Displacement Loops in Case of Zero Damping and 2% Constant Damping with Override and with Effective Stiffness Equal to the Post-Elastic Stiffness 6.4 Use of Rayleigh Damping Table 8 presents information on the calculated modes of the 4-story isolated structure. The identification was based on observation of the shapes as discussed in Section 3.4.1. Table 8 Properties of Modes of Four Story Isolated Structure Mode No. Mode Identification Period (sec) 1 Purely Isolated 4.167 2 0.462 3 0.216 Mixed Modes 4 0.131 5 0.091 6 0.0885 Local or vertical 0.0884 7 Damping was modeled either as constant in all modes with value equal to 2% or as Rayleigh damping with values of 2% at frequencies of 0.24 and 11.1Hz (which correspond to the first and 38

sixth modes as shown in Table 8). The effective stiffness of the isolators was specified equal to the post-elastic stiffness for both analyses. Results for the two cases are compared in Figure 30. Evidently the two methods of specifying damping produce practically identical results.

Figure 30 Comparison of Isolator Force-Displacement Loops in Case of 2% Constant Damping and 2% Rayleigh Damping and with Effective Stiffness Equal to the Post-Elastic Stiffness Nevertheless, the use of 2% Rayleigh damping as described above still had some damping leakage in the isolation system. This is demonstrated in Figure 31 where the zero damping case is compared to the 2% Rayleigh damping case.

Figure 31 Comparison of Isolator Force-Displacement Loops in Case of Zero Damping and 2% Rayleigh Damping and with Effective Stiffness Equal to the Post-Elastic Stiffness Use of Rayleigh damping as described in Section 3.4.1 may significantly increase the speed of the analysis. Figure 32 compares the CPU time required for the analysis of the two cases of 2% 39

constant damping and 2% Rayleigh damping described above as function of the number of Ritz vector modes used in the analysis.

Figure 32 Comparison of CPU Time for Analysis Using the Constant Damping and Rayleigh Damping Methods Small CPU time is achieved in the Rayleigh damping specification method because the method assigns large damping ratio values to very high frequency modes which effectively removes their effects and reduces numerical difficulties associated with high frequency response. Despite the differences in computational speed, the two methods produce practically identical results as demonstrated in Figure 30. 6.5 Use of Rayleigh Damping with Override Figure 33 compares results of analysis for the two-dimensional structure of Section 5.1 when damping is specified as (a) 2% using the Rayleigh method for frequencies f i  2.16 Hz and f j  11.1Hz (bounding frequencies of mixed modes) with 0% damping override for the first mode (purely isolated mode) and (b) as zero. For both cases the effective stiffness was specified equal to the post-elastic stiffness. Note that the case of zero damping is used herein to provide the most accurate results for the isolation system displacement as it does not have damping leakage in the isolation system. However, the specification of zero damping affects the computation of accelerations and bearing uplift displacements, which are then typically overestimated.

40

Figure 33 Comparison of Isolator Force-Displacement Loops in Case of Zero Damping and 2% Rayleigh Damping at f i  2.16 Hz , f j  11.1Hz with 0% Override for the First Mode

6.6 Use of Interpolated Damping In this example, damping was specified in three ranges as follows: (a) 0% for frequencies less than 2.13Hz, (b) 2% for frequencies in the range of 2.13 to 11.1 Hz and (c) linearly varying from 2% at frequency of 11.1Hz to a value of 40% at frequency of 1515Hz. The latter frequency corresponds to the last mode included in the analysis. Results are compared in Figure 34. The two methods of damping specification produce practically identical results. 6.7 Comparison of Results for all Analyzed Cases Table 9 compares peak response quantities in the analysis of the two-dimensional structure of Section 5.1 using the five methods of damping specification studied. Figure 35 presents a comparison of damping ratio values as function of frequency for the five cases. On the basis of these results we conclude that Rayleigh damping with override and interpolated damping are most appropriate for use in the analysis of seismically isolated structures.

41

Figure 34 Comparison of Isolator Force-Displacement Loops in Case of Zero Damping and Interpolated Damping

0.05

Rayleigh Rayleigh with Override Interpolated Constant Damping Constant Damping with Override

Damping Ratio

0.04

0.03

0.02

0.01

0.00 0

10

20

30

40

50

60

70

80

90

100

110

Frequency (Hz) Figure 35 Damping Ratio as Function of Frequency for Studied Cases

42

120

Table 9 Comparison of Peak Response Results in Studied Cases of Damping Specification 2% 2% Constant 2% 2% Rayleigh Interpolated Zero Damping Rayleigh Constant with damping Damping Damping with Damping Override Override Base Shear/ 0.235 0.242 0.235 0.241 0.241 0.242 Weight Isolator 22.3 23.4 22.2 23.3 23.3 23.4 Displacement (inch) First Story 0.31 0.32 0.32 0.32 0.32 0.34 Drift Ratio (%) Roof 0.42 0.42 0.48 0.43 0.42 0.51 Acceleration (g)

43

7. DIRECT INTEGRATION METHOD OF ANALYSIS The Direct Integration (DI) method of analysis is presumed to provide the most accurate results and accordingly often used to verify results of fast nonlinear analysis. Yet it is known, and will be demonstrated herein too, that the direct integration and the fast nonlinear method of analysis do not, in general, produce the same results. We will show herein that actually the fast nonlinear analysis method allows the user for better control of the analysis procedures and most often produces results of higher fidelity than direct integration. In the direct integration method the damping matrix is continuously updated during the integration process, whereas in the fast nonlinear analysis method the damping matrix remains constant. It is apparent that the two analysis methods are comparable in terms of results only when damping is zero. Figures 36 and 37 compare results of analysis of the two-dimensional structure of Section 5.1 when damping is specified as zero and analysis is performed (a) with direct integration using a time step equal to 1/10th of the excitation time step (integration time step equal to 0.0005 sec ), and (b) with fast nonlinear analysis using 53 Ritz vector modes. The two methods produce identical results.

Figure 36 Comparison of Isolator Force-displacement Loops Obtained with Direct Integration and with Fast Nonlinear Analysis in Zero-Damped Structure

44

Figure 37 Comparison of First Story Drift Histories Obtained with Direct Integration and with Fast Nonlinear Analysis in Zero-Damped Structure Figure 38 compares the isolator force-displacement loops of the same structure with non-zero damping when analyzed with direct integration (integration time step equal to 0.0005 sec ) and fast nonlinear analysis (53 Ritz vector modes). Damping was specified as 2% using the Rayleigh damping method with limit frequencies equal to 0.24 and 11.1Hz (first and sixth modes). Direct integration in this case produces a lesser isolator displacement. This is likely due to more leakage of damping in the isolation system in the direct integration method rather than errors in the fast nonlinear analysis. On the basis of these examples, the authors of this report believe that 1) The fast nonlinear analysis method, when properly implemented, is sufficiently accurate. 2) Comparison of fast nonlinear analysis results to direct integration analysis results for verification of the analysis model is not warranted nor recommended. 3) When verification of the analysis model is required, other programs should be used. For example, program 3D-BASIS (Nagarajaiah et al, 1989; Tsopelas et al, 1994 and 2005; Sarlis et al, 2009) may be used.

45

Figure 38 Comparison of Isolator Force-displacement Loops Obtained with Direct Integration and with Fast Nonlinear Analysis in 2%-Damped Structure

46

8. EXAMPLE OF ANALYSIS WITH ISOLATOR P-Δ EFFECTS IN SAP2000 8.1 Introduction The two-dimensional structure of Section 5.1 is used for demonstration of analysis with isolator P- effects. However, in order to magnify the P- effects for the demonstration, the sections of the girders above the isolators (see Figure 14) were changed from W36X300 to W27X217 and the earthquake was scaled up by a factor equal to 1.2. Exact P- analysis requires use of the direct integration method. Approximate P- analysis is performed using the fast nonlinear analysis. Due to the differences in modeling damping in the two methods of analysis, we utilize zero damping in order to have comparable results. 8.2 Comparison of Response with Exact and Approximate Isolator P- Effects Analysis was performed in SAP2000 with direct integration and with the P- effects first deactivated and then activated. When P-Δ effects are activated, the calculated P-Δ moment was equally distributed as moments at ends I and J of each element. A comparison of the calculated first story drift ratio history in the two cases is presented in Figure 39. In this case, isolator P- has some effect on drift. Analysis was performed in two steps for approximate isolator P- effects as described in Section 4. In this case the isolator displacements was first calculated in dynamic analysis without the P- effects, and then used to calculate P- moment histories. The calculated moment histories were added as time history functions, acting on top of each isolator. This requires the introduction of separate load patterns, separate time history and a separate point moment for each isolator. Dynamic analysis was then repeated with both the P- moment and the ground acceleration histories as input. Figure 40 compares the calculated first story drift ratio history, with P- effects accounted for either by the exact method in direct integration or by the approximate 2-step method in fast nonlinear analysis. The results are identical.

47

Figure 39 Isolator P- Effect on First Story Drift

48

Figure 40 Comparison of First Story Drift History Calculated with Exact and Approximate Isolator P- Effects

49

9. CONCLUSIONS This document presented the following: 1) 2) 3) 4) 5)

A summary description of the Triple Friction Pendulum isolator behavior. Presentation of the parallel model for modeling Triple Friction Pendulum isolators. Information on the implementation of the parallel model in SAP2000. Information on the use of direct integration and fast nonlinear analysis in SAP2000. Information on modeling of global damping in seismically isolated structures in SAP2000. 6) Information on the significance of Ritz vector modes in capturing correctly the response of seismically isolated structures in SAP2000. 7) Information on how isolator P- effects can be accounted for in fast nonlinear analysis in SAP2000. On the basis of the results presented in this report, the authors have the following conclusions: 1) Fast nonlinear analysis in SAP2000 (and by similarity in ETABS), when properly implemented, produces results of acceptable accuracy in the analysis of seismically isolated structures. 2) Direct integration should not be used as means of verification of the analysis model in fast nonlinear analysis unless structural damping is specified to be zero. 3) Important considerations in the application of the fast nonlinear analysis are the number of Ritz vector modes and the specification for damping. 4) A criterion for determining the minimum required number of Ritz vector modes is the development of the correct vertical load on the isolators in the initial application of vertical acceleration prior to staring the dynamic response history analysis. However, a larger number of modes may be needed to correctly perform dynamic analysis. 5) Damping may be specified in a variety of ways of which the constant damping with override and the Rayleigh damping with override result in control of the damping leakage problem in the isolation system. The Rayleigh damping with override approach typically results in least computational time. 6) Regardless of the damping specification method used, the effective stiffness of the isolators needs to specified low for preventing or reducing damping leakage in the isolation system. We recommend use of the post-elastic isolator stiffness value (or a value less than that) for the effective stiffness together with low damping ratio values.

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10. REFERENCES 1. Computers and Structures Inc. (2007), “SAP2000: INTEGRATED FINITE ELEMENT ANALYSIS AND DESIGN OF STRUCTURES”, Version 11.0.8, Berkeley, CA. 2. Constantinou, M.C., Whittaker, A.S., Fenz, D.M. and Apostolakis, G. (2007b), “Seismic Isolation of Bridges”, University at Buffalo, Report to Caltrans for contract 65A0174, June. 3. Fenz, D.M. and Constantinou, M.C. (2008a),”MECHANICAL BEHAVIOR OF MULTISPHERICAL SLIDING BEARINGS”, Report No. MCEER-08-0007, Multidisciplinary Center for Earthquake Engineering Research, Buffalo, NY. 4. Fenz, D.M. and Constantinou, M.C. (2008b), “SPHERICAL SLIDING ISOLATION BEARINGS WITH ADAPTIVE BEHAVIOR: THEORY,” Earthquake Engineering and Structural Dynamics, Vol. 37, No. 2, 163-183. 5. Fenz, D.M. and Constantinou, M.C. (2008c), “SPHERICAL SLIDING ISOLATION BEARINGS WITH ADAPTIVE BEHAVIOR: EXPERIMENTAL VERIFICATION”, Earthquake Engineering and Structural Dynamics, Vol. 37, No. 2, 185-205. 6. Fenz, D.M. and Constantinou, M.C., (2008d), “MODELING TRIPLE FRICTION PENDULUM BEARINGS FOR RESPONSE-HISTORY ANALYSIS”, Earthquake Spectra, Vol. 24, No. 4, 1011-1028. 7. Fenz, D.M. and Constantinou, M.C. (2008e),”DEVELOPMENT, IMPLEMENTATION AND VERIFICATION OF DYNAMIC ANALYSIS MODELS FOR MULTISPHERICAL SLIDING BEARINGS”, Report No. MCEER-08-0018, Multidisciplinary Center for Earthquake Engineering Research, Buffalo, NY. 8. Morgan, T. A. (2007), “THE USE OF INNOVATIVE BASE ISOLATION SYSTEMS TO ACHIEVE COMPLEX SEISMIC PERFORMANCE OBJECTIVES”, Ph.D. Dissertation, Department of Civil and Environmental Engineering, University of California, Berkeley. 9. Nagarajaiah, S., Reinhorn, A.M., and Constantinou, M.C. (1989), “NONLINEAR DYNAMIC ANALYSIS OF THREE DIMENSIONAL BASE ISOLATED STRUCTURES (3D-BASIS)” Report NCEER-89-0019, National Center for Earthquake Engineering Research, State University of New York, Buffalo, NY. 10. Satake, N., Suda, K., Arakawa, T., Sasaki, A and Tamura, Y. (2003), “DAMPING EVALUATION USING FULL-SCALE DATA OF BUILDINGS IN JAPAN”, ASCE, J. Structural Engineering, 129 (4). 11. Sarlis, A.A., Tsopelas, P.C., Constantinou, M.C. and Reinhorn, A.M., “3D-BASIS-ME-MB: Computer Program for Nonlinear Dynamic Analysis of Seismically Isolated Structures, Element for Triple Pendulum Isolator and Verification Examples”(2009), supplement to MCEER Report 05-009, document distributed to the engineering community together with executable version of program and example files, University at Buffalo. 51

12. Tsopelas, P.C., Constantinou, M.C., and Reinhorn, A.M. (1994), “3D-BASIS-ME: COMPUTER PROGRAM FOR NONLINEAR DYNAMIC ANALYSIS OF SEISMICALLY ISOLATED SINGLE AND MULTIPLE STRUCTURES AND LIQUID STORAGE TANKS”, Report NCEER-94-0010, National Center for Earthquake Engineering Research, State University of New York, Buffalo, NY. 13. Tsopelas, P.C., Roussis, P.C., Constantinou, M.C., Buchanan, R. and Reinhorn, A.M. (2005), “3D-BASIS-ME-MB: COMPUTER PROGRAM FOR NONLINEAR DYNAMIC ANALYSIS OF SEISMICALLY ISOLATED STRUCTURES,” Report MCEER-05-0009, Multidisciplinary Center for Earthquake Engineering Research, State University of New York, Buffalo, NY.

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APPENDIX A CALCULATION OF MODEL PARAMETERS SERIES MODEL Vertical Stiffness The bearing vertical stiffness is approximately calculated as AE/h, where A is the area of the center slider (herein a circle of 10inch diameter), E is a representative modulus (typically assumed about half of that of steel to account for flexibilities in the bearing assembly, herein 14500ksi) and h is the height (herein 16inch). The vertical stiffness is thus equal to x52x14500/16=71176.7kip/in. The three elements of the series model each have vertical stiffness K so that the combined stiffness equals 7176.7kip/in. That is, (1/K+1/K+1/K)-1=71176.7kip/in or K=213530kip/in. Elastic Stiffness (see Table 5) Element FP1 Elastic Stiffness: K1  Element FP2 Elastic Stiffness: K2  Element FP3 Elastic Stiffness: K3 

2W 2Y

1W 2Y

4W 2Y



0.03x900  1350kip / in 2 x0.01



0.108 x900  4860kip / in 2 x0.01



0.108 x900  4860kip / in 2 x 0.01

Effective Stiffness (see Table 5) Element FP1 Effective Stiffness: K1 

W W 900    60kip / in Reff 1 Reff 2  Reff 3 7.5  7.5

Element FP2 Effective Stiffness: K2 

W W 900    12kip / in Reff 2 Reff 1  Reff 2 82.5  7.5

Element FP3 Effective Stiffness: K3 

W W 900    12kip / in Reff 3 Reff 4  Reff 3 82.5  7.5

Rate Parameter (see Table 5) Element FP1 Rate Parameter: Element FP2 Rate Parameter:

1  a2  a3  (2.54  2.54)   1.27 sec/ inch 2 2 4

Reff 1 Reff 1  Reff 2

a1 

82.5 x 2.54  2.794 sec/ inch 82.5  7.5 53

Element FP3 Rate Parameter:

Reff 4 Reff 4  Reff 3

a1 

82.5 x 2.54  2.794sec/ inch 82.5  7.5

PARALLEL MODEL Vertical Stiffness The vertical stiffness of 71176.7kip/in is equally divided to elements FP1 and FP2 of the parallel model. That is, each element is assigned vertical stiffness of 35588.4kip/in. Elastic Stiffness (see Table 2) Element FP1 Elastic Stiffness:

K1 

 2W 2Y 1



W 2 Reff 2



0.03  900 2  0.0155



900 2  7.5

 807.9kip / in

Note that the yield displacement Y 1 was assumed to be equal to 0.0155inch so that the elastic stiffness of the parallel model is the same as the elastic stiffness of the series model (see demonstration below). In general, the value of the yield displacement Y 1 is arbitrarily assumed to be about or larger than 0.01inch. Element FP2 Elastic Stiffness: K 2 

W 900   60kip / in 2 Reff 2 2 x 7.5

The value to be specified in SAP2000 is not 60kip/in but a value equal to 60kip/in minus the post-elastic stiffness of the element (SAP2000 adds the specified value of elastic stiffness to the post-elastic stiffness and uses that as the elastic stiffness):

K2 

W  Reff 1  Reff 2  W W    54.55kip / in 2 Reff 2 2 Reff 1 2 Reff 2 Reff 1

Note that the elastic stiffness of the parallel model of the bearing is equal to the elastic stiffness of the series model as a result of the choice of the yield displacement Y 1 . Specifically, the two 1

 1 1 1    models satisfy the condition K1  K 2    . In this equation, K1, K2 and K3 are the K K K 2 3   1 elastic stiffness of the three elements of the series model. 1

1

 1 1 1  1 1   1          867.9kip / in  1350 4860 4860   K1 K 2 K3  K1  K 2  807.9+60=867.9kip/in 54

Note that SAP2000 also performs the operation mentioned above (SAP2000 adds the specified value of elastic stiffness to the post-elastic stiffness and uses that as the elastic stiffness) for the elastic stiffness in each of the three elements of the series model. However, since the values of elastic stiffness of each of the elements in the series model are very large, the effect of the modification is insignificant and neglected. For example the specified value of elastic stiffness for element FP3 of the series model should be specified in SAP2000 as K3 

4W 2Y



W 0.108 x 900 900    4860  12  4848kip / in . Reff 4  Reff 3 2 x 0.01 82.5  7.5

The difference between the value of 4848 and 4860kip/in is insignificant, and neglected. Effective Stiffness The effective stiffness is specified to be the post-elastic stiffness of each element. For the FP1 W 900 element is zero (flat slider). For the FP2 element it is equal to   5.45kip / in . 2 Reff 1 2 x82.5 Note that these values are consistent with the values specified for the series model. For the parallel model, the total effective stiffness is 0+5.45=5.45kip/in. For the series model, the total 1

1

 1 1 1   1 1 1  effective stiffness is          5.45kip / in .  60 12 12   K1 K2 K3  Rate Parameter For each element, the parameter is specified as 2.54/2=1.27sec/in. This is based on the assumption that the velocity is equally partitioned between sliding surfaces (true for surfaces 1 and 4).

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