By Michalakis Constantinou,1 Associate Member, ASCE, Anoop Mokha,2 and Andrei Reinhorn,3 Member, ASCE
ABSTRACT: A mathematical model of the factional behavior of Teflon sliding bearings for conditions of interest in base isolation is developed. The calibration of the model is based on extensive experimental data that were presented in an accompanying paper. The model is capable of accounting for: (1) Unidirectional and multidirectional motion at the Teflon-steel interface; (2) velocity and pressure dependence of the coefficient of sliding friction; and (3) breakaway (or static) friction effects. The model is characterized by four parameters. These are the minimum and maximum values of the sliding coefficient of friction, the ratio of breakaway to sliding coefficient of friction at initiation of sliding and a parameter that describes the variation of the sliding coefficient of friction with velocity. Values of these parameters are presented for sixteen combinations of type of Teflon, bearing pressure and condition (surface roughness) of mating steel surface. Applications of the model in the analysis /design of a sliding isolation system are presented and the effects of bearing pressure and breakaway friction are evaluated. Furthermore, an assessment of the implications of using Coulomb's constant friction model rather than the developed model is presented.
INTRODUCTION
Design codes require that earthquake forces be absorbed by the structural system through inelastic action that lengthens the period of the system and increases its energy dissipation capacity. Base isolation utilizes a similar concept, except that flexibility and energy absorption capacity are provided by a specially designed system that is placed between the structure and its foundation. Inelastic action, if any, may be limited within this system without necessarily imposing ductility requirements to the superstructure. If properly designed, it may provide a level of performance well beyond the normal code requirements, with potential for substantial life-cycle cost reduction (Kelly 1986; Mokha et al. 1988).
Elastomeric bearing isolation systems provide, at the present time, the simplest method of isolation. They consist of steel-reinforced natural rubber or Neoprene bearings that provide the needed flexibility. Energy dissipation capacity is provided by specially compounded rubber that exhibits high inherent damping or by the use of lead plugs or mild steel energy absorbing elements (Kelly 1986; Buckle 1986). There are about 25 buildings and 45 bridges isolated by elastomeric bearings in United States, New Zealand, and
'Assoc. Prof., Dept. of Civ. Engrg., Univ. of Buffalo, State Univ. of New York, 212 Kettler Hall, Buffalo, NY, 14260.
2Grad. Res. Asst., Dept. of Civ. Engrg., Univ. of Buffalo, State Univ. of New York, 212 Kettler Hall, Buffalo, NY.
3Assoc. Prof., Dept. of Civ. Engrg., Univ. of Buffalo, State Univ. of New York, 212 Kettler Hall, Buffalo, NY.
Japan (Kelly 1986; Buckle 1986; Constantinou 1988). The successful implementation of elastomeric bearing isolation systems has been the result of significant theoretical and experimental research, a major part-of which has been conducted in the United States.
More recently, Teflon sliding bearings have been proposed as part of aseismic isolation systems. In these systems, the entire weight of the structure is carried by Teflon-steel interfaces that provide little resistance to lateral loading by virtue of their low friction. Separate mechanisms are used for providing centering force capability and, if needed, additional energy absorption capacity. The major source of energy dissipation is the Teflon-steel interface. A major advantage of these systems is their insensitivity to the frequency content of excitation, an apparent consequence of their frictional behavior. Other advantages stem from the separation of the functions of carrying the weight and of providing horizontal stiffness at the isolation interface. The system exhibits a high degree of stability, which makes it ideal for isolating light-weight structures or structures that need to be supported by a large number of isolators.
In total, six sliding isolation systems with restoring force have been proposed, of which two have reached the stage of implementation. These are the earthquake barrier system (Caspe and Reinhorn 1986), the friction pendulum system (Zayas et al. 1987), Alexisismon (Ikonomou 1985), Taisei or TASS (Nagashima et al. 1987), resilient-friction base isolator system (R-FBI) (Mostaghel and Khodaverdian 1987), and Wabo-Fyfe earthquake protection system (Watson Bowman 1981; Constantinou et al. 1988).
A sliding isolation system has been used for the seismic protection of the Technology Research Center of Taisei Corporation in Japan (Constantinou 1988). The structure is a reinforced concrete four-story building that weighs 22 MN. The isolation system consists of eight Teflon-elastomeric bearings and eight Neoprene springs. The rigid body mode period of the structure in the sliding phase is about 5 sec (stiffness provided by Neoprene springs). As a result of the weak restoring force provided by the Neoprene springs, the transmission of force to the superstructure is primarily controlled by the frictional properties of the sliding bearings. Furthermore, the friction pendulum system has been used in the seismic isolation of a 50,000-gal emergency fire water tank in California.
Theoretical evaluations of the seismic performance of sliding isolation systems have been numerous (Mostaghel and Khodaverdian 1987; Su et al. 1987; Fan et al. 1988, among others). These studies demonstrated the feasibility of sliding isolation systems and confirmed their insensitivity to the frequency content of earthquake excitation. Invariably, these studies utilized Coulomb's law of friction.
Mokha et al. (1988) have conducted a series of tests on sheet-type Teflon sliding bearings, and obtained measurements of the breakaway (static) and sliding (kinetic) coefficients of friction for bearing pressure ranging between 1,000 and 6,500 psi (6.9 and 44.9 N/mm2) and for sliding velocity between 0.1 and 20 in./sec (0.25 and 50 cm/s). The results of these experiments are utilized herein, in the calibration of a mathematical model of friction for Teflon sliding bearings. Prime characteristic of this model is the dependency of the frictional force on the velocity of sliding. Other features of this model are the dependency of the frictional force on bearing pressure and condition of interface (type of Teflon and roughness of stainless steel).
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The developed model of friction exhibits a behavior that deviates significantly from that of Coulomb. Results are presented that show that under certain conditions the Coulomb's model may provide reliable estimates of the forces imparted to the isolated structure. However, the resulting responses with Coulomb's model contain high-frequency components that are not present in the responses with the developed model. This is a consequence of significantly more sticking that results from the independency of friction from velocity in Coulomb's model.
The developed model of friction is used in the seismic analysis of a six-story structure isolated by sliding isolation systems. This structure represents the prototype of a model that will be tested on a shake table. In these analyses, the bearing pressure at the Teflon-steel interface is parametrically varied between 1,000 and 3,000 psi (6.9 and 20.7 N/mm2) that represents the normal range of operation of sliding Teflon bearings. The results show that the design of Teflon bearings at low pressure has certain advantages over designs at higher pressure.
Furthermore, the effect of breakaway friction on the behavior of sliding isolation systems is assessed. At initiation of sliding, the structure is subjected to high frictional forces. These forces are significant for sliding systems for which the Teflon-steel interfaces operate at low velocity (e.g., R-FBI system) and in interfaces for which the variation of friction with velocity is not significant (e.g., glass-filled Teflon at very high pressure).
EXPERIMENTAL RESULTS AND MATHEMATICAL MODEL
The writers have conducted a series of tests on Teflon-steel interfaces (Mokha et al. 1988) under the following conditions:
1. Sliding interface consisted of sheet type unfilled or glass-filled Teflon at composition of 15% and 25% by weight. The Teflon was in contact with stainless steel (ASTM A240, type 304), which was commercially polished to degree No. 8. Sliding was imposed in either the direction parallel to lay or in the direction perpendicular to lay. Roughness measurements of the steel surface gave values of 0.03 |xm and 0.04 \im Ra in the two directions, respectively.
2. Bearings pressure at the interface was 1,000, 2,000, 3,000, and 6,500 psi (6.9, 13.8, 20.7, and 44.9 N/mm2).
3. Motion at the interface was either sinusoidal or of constant velocity (sawtooth cyclic displacement). Peak values of sliding velocity were between 0.1 and 20 in./sec (0.25 and 50 cm/s).
The frictional force at initiation of sliding upon division by the normal force results in the breakaway (or static) coefficient of friction, \LB. This value appears to be substantially larger than the sliding (or kinetic) value, as recorded after initiation of sliding, u^. The following observations were made on the frictional behavior of these interfaces:
1. Both the breakaway and sliding values of the friction coefficient appear to be independent of the acceleration of sliding at the interface.
2. Both values are dependent on the bearing pressure and condition of interface (type of Teflon and roughness of stainless steel).
3. The sliding value of the coefficient of friction increases with an increase
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20
18-
16-
a 0 A v SINUSOIDAL TESTS • * A T CONSTANT VEL. TESTS
+ STICK-SUP OBSERVED * TEST ON 5 " DIA. SPECIMEN
1000 psi
2000 psi
3000 psi
6500 psi
16 VELOCITY IN./SEC
24 32
FIG. 1. Variation of Sliding Coefficient of Friction with Velocity of Glass-Filled Teflon at 25% for Sliding Parallel to Lay
in the sliding velocity up to a certain value of velocity beyond which it remains constant. This value of velocity is between 4 and 8 in./sec (10 and 20 cm/s).
4. At initiation of sliding, and for low sliding velocity (0.1 in./sec in these experiments), the ratio of breakaway to sliding values of friction was between 1.5 and 4.5.
5. Interfaces that have been previously run in (even for only one cycle) exhibited, in subsequent testing, a much lower breakaway coefficient of friction. This was attributed to the transfer of thin Teflon flakes on the stainless-steel slider. As such, a sliding structure will be subjected to the effect of breakaway friction only at initial sliding and not at every instance that follows a momentary sticking.
Fig. 1 shows the sliding coefficient of friction as function of sliding velocity for glass-filled Teflon at 25%. The effect of velocity and pressure are evident in this figure. The coefficient of sliding friction at sliding velocity U may be approximated by the following equation:
V* =/max - Dfexp (-a\U\) (1)
in which /max = the coefficient of friction at large velocity of sliding (after leveling off); and Df = the difference between/max and the sliding value at very low velocity. Furthermore, a is a constant for given bearing pressure and condition of interface. Table 1 presents values of /max, Df, and a for various conditions of Teflon-steel interface and pressure that resulted in the solid-line curves in Fig. 1. It is apparent that Eq. 1 reproduces the results of experiments with good accuracy. Further results may be found in Mokha et al. (1988).
Note: UF = unfilled Teflon; 15GF = glass-filled Teflon at 15%; 25GF = glass-filled Teflon at 25%; P = sliding parallel to lay; T = sliding perpendicular to lay; and 1,000 psi = 6.9 N/mm2.
For symmetric structural systems with ground motion consisting of components in a vertical plane, the motion will be in a single direction. The frictional force, Ff, at a sliding interface, may be described by the following equation:
Ff = v„W sgn (£>) (2)
in which W = the load carried by the interface and U - the velocity at the isolation interface. Furthermore, sgn represents the signum function. The transition from sticked to sliding mode and vice versa is controlled by stick-slip conditions, as described by Mostaghel and Khodaverdian (1987) and Su et al. (1987). This model will be referred to as the modified Coulomb model.
There are certain complications that arise in the use of the modified Coulomb model. The first one arises in the use of the model for the analysis of systems with multiple support motion and in systems in which each Teflon bearing undergoes different motion than other bearings. This situation occurs in bridges in which Teflon bearings are placed on top of flexible piers. Situations like this require the use of multiple stick-slip conditions that control the transition from one mode of motion to another. Each one of these modes of motion is governed by different equations of motion. The second complication arises in the numerical solution of the equations of motion that have discontinuities (Eq. 2). Feldstein and Goodman (1973) have proved that if any algorithm of order m > 1 is applied to a differential equation with discontinuities, its order of convergence collapses to m = 1 after only one discontinuity. To avoid this disastrous effect a* procedure must be employed to locate the position of discontinuity. Such procedures have been employed
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for sliding systems by Mostaghel and Khodaverdian (1987) and Su et al. (1987). The third complication arises in the extreme difficulties encountered in extending Eq. 2 to the case of general plane motion with friction (Younis et al. 1983).
Another model of friction of Teflon-steel interfaces is presented herein. It is based on principles of the theory of viscoplasticity, and it will be referred to as the modified viscoplasticity model. This model is based on the following equation that was originally proposed by Bouc (1971) and subsequently extended and used by Wen (1976) in random vibration studies of hysteretic systems:
UF62: 1000PSI: SIN: 0.16HZ: 1 " : P
-0 .4 0
DISPLACEMENT INCHES
20 -
10 -
10 -
20 -
30 -
UF75: 1000PSI: SIN: 0.6HZ: 4" :
! f e ^ = = — -
^ v _ _
-
" 1 1
P
=^=^
1
- 1 1
DISPLACEMENT INCHES
FIG. 2. Recorded Force—Displacement Loops in Experiments No. 62 and 75
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YZ + 7|J/|Z|Z|,|~1 + $U\z\^ - AU = 0 (3)
in which U stands for the velocity; Z = a hysteretic dimensionless quantity; and B, 7, A, and T| = dimensionless constants. Furthermore, 7 represents a displacement quantity. Constantinou and Adnane (1987) have shown that when A = 1 and B + 7 = 1, the model of Eq. 3 collapses to a model of viscoplasticity that was proposed by Ozdemir (1976). In this case, 7 represents the yield displacement while T| controls the mode of transition into the inelastic range. The model exhibits a rate of dependency that reduces with increases in the value of exponent T| and/or increases in the ductility ratio (maximum value of ratio of U to 7).
The frictional force is given by
Ff = p,WZ (4)
which is essentially identical to Eq. 2. It should be noted that during sliding (yielding), Z takes values of ±1 . During sticking (elastic behavior), the absolute value of Z is less than unity. The conditions of separation and reattachment are accounted for by Eq. 3. In this respect, Z may be regarded as a continuous approximation to function sgn(C/) in Eq. 2. As such, the model is incapable of reproducing truly rigid-plastic behavior. This is not a limitation. Teflon-steel interfaces undergo some very small elastic displacement before sliding. This displacement consists primarily of elastic shear deformation of Teflon. Experimental observations (see Figs. 2 and 3) suggests a value of 7 of about 0.005-0.02 in. (0.13-0.5 mm). With such low yield displacement, the resulting ductility ratio is very large and the model exhibits an insignificant rate dependency. A value of -n = 2, with A = 1 and B + 7 = 1 (B = 0.1, 7 = 0.9) produces loops of frictional force versus sliding displacement that are in good accord with experimental results.
CO Q_
UJ
o DC O
< z o I— o DC LL.
15-
12-
9 -
6 -
3
0 -
-3 -
- 6
- 9
-12'
-15'
UF : 1000psi : T
Experiment, No.191 Model
- 0 . 6 - 0 . 4 - 0 . 2 0.0 0.2 0.4
DISPLACEMENT inches 0.6
FIG. 3. Simulation of Breakaway Friction
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15 -
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5 -
- 5 -
10 -
15 -
20 "1
( ^
ZZ^ -0 .4 0
DISPLACEMENT INCHES
-
-
r
r
I
A
J
i i i - 1 1
DISPLACEMENT INCHES
FIG. 4. Numerical Simulation of Experiments No. 62 and No. 75
Fig. 4 presents frictional force-displacement loops produced by the modified viscoplasticity model of Eqs. 1,3, and 4. These results were produced by numerically simulating experiments No. 62 and 75, the results of which are shown in Fig. 2. Both tests were conducted on 10-in. (25.4-cm) diameter specimens of unfilled Teflon at pressure of 1,000 psi (6.9 N/mm2). Motion was sinusoidal and in the direction parallel to lay. Amplitude was 1 and 4 in. (2.54 and 10.2 cm) and frequency was 0.16 and 0.6 Hz, respectively. The recorded frictional force was from two identical interfaces (Mokha et al. 1988). For the numerical simulation, this motion represented the input
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in Eq. 3 (7 = 0.005 in., A = 1, (3 = 0.1, 7 = 0.9, t\ = 2) that was solved for Z. The variation of friction with velocity was accounted for by Eq. 1 with appropriate data from Table 1. It is apparent that the proposed model reproduces the experimental results well. It is of interest to note the significant difference between the force-displacement loops in the two experiments. In experiment No. 62, the peak velocity is 1 in./sec (2.54 cm/s), which is well below the limit of about 4 in./sec (10 cm/s), beyond which friction remains constant. As such, the frictional force exhibits a continuous variation with the imposed displacement. In experiment No. 75, however, the peak velocity is 15 in./sec (38 cm/s), which is well above the constant friction limit of velocity. As such, the loop appears to be almost rectangular.
MODEL OF FRICTION OF TEFLON-STEEL INTERFACES IN GENERAL PLANE MOTION
General plane motion arises in nonsymmetric structural systems and in systems excited by multidirectional ground motion. In sliding isolation systems, asymmetric behavior is furthermore produced by differences in the frictional properties of Teflon bearings. In general, a Teflon-steel interface will undergo displacement in the two orthogonal directions with rotation in the same plane. This motion will give rise to frictional forces in the two orthogonal directions and moment. The contribution of this moment to the total torque exerted to the structure above the isolation interface is insignificant.
Let us consider general plane motion of a Teflon-steel interface with displacement components Ux and Uy and velocity components Ux and Uy in the two orthogonal directions. At any instant of time, the interface undergoes a displacement increment (displacement of the part above interface, with respect to the part below interface) in the direction specified by the angle 8, with respect to the x-axis:
- - - ( D «' The line of action of the resultant of frictional forces (exerted to the superstructure) must be the same as that of the displacement increment, and its direction must oppose the direction of the vector of incremental displacement. Furthermore, the magnitude of the resultant frictional force will depend on the magnitude of the instantaneous velocity U:
U=(U2X + U2)1/2 (6)
The frictional forces Fx and Fy (exerted to the superstructure) are given by:
Fx = \hsW cos 6, Fy = ILSW sin 6 (7)
in which |xs is described by Eq. 1 with U given by Eq. 6. Parameters/max, Df, and a depend, in general, on the direction of sliding.
The model proposed herein is based on the following system of coupled differential equations that was proposed by Park et al. (1985):
This system of equations is an extension of Eq. 3. It may be easily shown that Eqs. 8 and 9 collapse to Eq. 3 for U, = Zy = 0 and for -n = 2. Eqs. 8 and 9 are used to account for the conditions of separation and reattachment of the Teflon-steel interface in the same manner as Eq. 3. During sliding, Zx and Zy attain their maximum values and the following conditions apply:
dZx dZ„ _ i = ^ = 0 (10) dUx dUy
As such, Eqs. 8 and 9 may be expressed as
Z2X + -^- ZxZy = (11)
dUx y (3 + 7
2 dU* A
Z2y + ZxZy = (12)
y dUy y (3 + 7
where dVy dUx
— = tan 6, — = cot 6 : .. . . (13)
dUx dUy
The system of Eqs. 11-12 has the solution
Zx = cos 8, Zy = sin 6 (14) provided that A/((3 + 7) = 1. In this way, the components of frictional force in the two orthogonal directions are expressed as Fx = ixsWZx, Fy = txsWZy (15) which are now identical to Eq. 7. It should be noted that this is true only when the ratio A /O + 7) is equal to unity. As discussed earlier, this condition reduces the unidirectional version of the model to one of the theory of viscoplasticity.
The assessment of the proposed model should, of course, be based on experimental data. Such an assessment requires identification of the model parameters and validation of the model. The identification of the model parameters is based on experimental results with unidirectional motion in either the direction parallel to lay (x-direction) or the direction perpendicular to lay (v-direction). A complete validation of the model requires experimental data with bidirectional motion. In lieu of such data, the writers provide the following indirect validation of the proposed model:
1. When unidirectional motion is imposed in an arbitrary direction, the resulting frictional force is in the same direction. The model is capable of producing this behavior by virtue of Eqs. 14 and 15.
2. When multidirectional motion with out-of-phase components is imposed, the resulting force-displacement loops exhibit a marked similarity with experimental results from testing of a cantilever mild steel damper developed by Ka-jima Corporation, Japan (Yasaka et al. 1988). This device exhibits essentially elastoplastic behavior and has been recently tested with Kajima's large multidirectional testing machine (Constantinou 1988).
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0.14
- 0 . 1 0 - "3
- 0 . 1 4 I i i i i i i i i < i i i
-1.4 -1.0 -0 .6 -0.2 0.2 0.6 1.0 1.4
Ux/Uo & Uy/Uo
FIG. 5. Simulated Frictional Behavior of Teflon-Steel Interface under Bi-Direc-tional Motion of Low Velocity (Shown in Upper Left Corner)
Fig. 5 shows frictional force-displacement loops produced by the model of Eqs. 1, 8, 9, and 15 for eight-shaped type of bidirectional motion in which
Ux = U0 sin u>t, Uy = U0 sin 2cof (16)
with U„ — 1 in. (2.54 cm) and <o = 1 rad/sec (0.16 Hz). Teflon is unfilled and pressure is 1,000 psi (6.9 N/mm2). The motion in the x-direction (parallel to lay) is identical to that in experiment No. 62, of which the results are depicted in Fig. 2. The solid line in Fig. 5 represents the results of the isotropic version of the model in which |J,J (Eq. 1) is independent of the direction of sliding. In the anisotropic version of the model (dashed line), constants/max, Df, and a, in the expression for ji,, were continuously adjusted according to the direction of sliding as specified by angle 0 (Eq. 5). This adjustment was based on linear interpolation of the results of Table 1 for sliding parallel to lay (8 = 0) and for sliding perpendicular to lay (6 = ±90°).
The modification of the loop in the x-direction by the motion in the y-direction is apparent (compare with heavy solid line). At start (point 1), angle 6 is larger than 45°, resulting in lower frictional force in the x-direc-tion. As time progresses, angle 9 decreases and the frictional force increases towards the value corresponding to zero angle, which it attains at point 2. The marked similarity of the two hysteresis loops with those of Kajima's damper under eight-shaped bidirectional motion is noted.
Fig. 6 shows the computed force-displacement loops when U0 = 4 in. (10.16 cm) and co = 2.77 rad/sec (0.6 Hz). In this case, the motion in the x-direction is identical to that in experiment No. 75. In this example of highspeed motion, the anisotropic model yields results that are quite different
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0.20
0.15
U ^ 0.05
•s 0.00
- - 0 . 0 5
- 0 . 1 5
- 0 . 2 0
No. 75
- 1 . 4 - 1 . 0 - 0 . 6 - 0 . 2 0.2 0.6 1.0
Ux/Uo & Uy/Uo
FIG. 6. Simulated Frictional Behavior of Teflon-Steel Interface under Bi-Direc-tional Motion of High Velocity
than those of the isotropic model. The velocity of sliding is large enough, such that the sliding friction coefficient attains its maximum value,/max. This value is 0.119 for sliding parallel to lay as compared to 0.142 for sliding in the direction perpendicular to lay. For the type of motion of Fig. 5, maximum velocity is much lower and the coefficient of sliding friction exhibits values in the two directions that are almost equal. Further results on the bidirectional model of friction may be found in Constantinou and Mokha (1989).
APPLICATION OF RESULTS
Comparison of Proposed Model to Coulomb's Model The implications of using Coulomb's model of friction in the analysis of
structures supported by Teflon bearings are demonstrated with the following example. A rigid structure is considered supported by Teflon bearings that operate at a pressure of 1,000 psi (6.9 N/mm2). The conditions at the interface are those of unfilled Teflon with the sliding direction parallel to lay. The structure is excited by the component S00E component of the 1940 El Centra earthquake (recorded at Imperial Valley station No. 117). Fig. 7 shows the North-South time history of sliding displacement, U, normalized by the peak ground displacement, PGD. The solid line represents the exact response when accounting for the variation of the coefficient of friction with the velocity. The dashed lines show the response when using Coulomb's model with constant friction that takes either the minimum value (at zero velocity) or the maximum value (at high velocity) or an average value. For this case of "strong" excitation, the resulting sliding velocity is large enough, such that the coefficient of sliding friction attains its maximum value. As such, Coulomb's model with maximum friction predicts a displacement re-
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1.5
1.0
8 °-»H Q_
0.0
- 0 . 5 -
-1.0
1940 EL CENTRO SOOE: PGA=0.35g: PGD-=4.29in: UF : 1000PSI
0.0
• Exact Minimum -Average Maximum
2.0 4.0 6.0
TIME sec 8.0 10.0
FIG. 7. Comparison of Displacement Response of Rigid Mass as Computed by Proposed Model (Exact) and Coulomb's Model of Constant Friction
sponse that approximates the exact one but with considerably more sticking. The total acceleration response, U„ normalized by the peak ground ac
celeration, PGA, is shown in Fig. 8. While peak values are predicted quite accurately, the frequency content appears different. This is further illustrated in Fig. 9, in which the Fourier amplitude of U,/PGA is shown for the cases of exact and maximum friction. Apparently, Coulomb's model exhibits more stick-slip tendencies that result in responses with higher frequency content.
1.0
0.5 < o Q_
0.0
- 0 . 5
1940 EL CENTRO SOOE: PGA=0.35g: PGD=4.29in: UF ; 1000PSI
Exact Maximum
0.0 — I — 2.0 4.0 6.0
TIME sec 8.0 10.0
FIG. 8. Comparison of Acceleration Response of Rigid Mass as Computed by Proposed Model (Exact) and Coulomb's Model of Constant Friction
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10 15 FREQUENCY Hz
FIG. 9. Fourier Amplitude of Acceleration Responses in Fig. 8
This example and other studies by the writers with different excitations (Mokha et al. 1988) demonstrate the significance of the variation of the friction force with velocity. The use of the constant friction Coulomb model may result in a useful estimation of the peak response, if an appropriate value of the coefficient of friction is used. For strong motions, the appropriate value is the maximum value, /max.
Effect of Breakaway Friction The incorporation of breakaway friction in the presented modified vis-
coplasticity model has been accomplished by adjusting the value for ^ in Eq. 1 to a higher value for times prior to initial sliding:
M* = *(/max - Df) for \Z\ < 0.999 (17)
in which b = a constant larger than unity. (/max - Df) = the coefficient of sliding friction at essentially zero velocity of sliding (immediately after initiation of sliding). It is implied in Eq. 17 that sliding commences for values of \Z\ > 0.999. When this value is attained and thereafter, the value of the sliding coefficient of friction is given by Eq. 1. Experimental values of constant b are given in Table 1.
The validity of this simple model is illustrated in Fig. 3 in which it is compared against experimental results [test No. 191 in Mokha et al. (1988)]. The solid line represents the recorded frictional force-displacement loop from two identical unfilled Teflon-steel interfaces. Motion was sinusoidal with amplitude of 0.5 in. (1.27 cm) and frequency of 0.03 Hz in the direction perpendicular to lay. Normal force on the interface was 78.5 kips (350 kN) inducing an average pressure of 1,000 psi (6.9 N/mm2). The dashed line represents the results of numerical simulation using the modified viscoplas-ticity model with b = 3. Evidently, the model is capable of reproducing the behavior observed in laboratory tests.
The effect of breakaway friction is demonstrated by studying the response
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SINUSOIDAL : i igo=0.25g : w = 3 . 1 4 r / s Without Breakaway With Breakaway=2*Fmin
1.2 -M 0.0
0.4 -
0.3 -
0.2 -
0.1 -
0.0 -
- - - With - - With
TEFLON
/ I / I
i"Jf i
I I I
Breakaway=4*Fmin Breakaway=6*Fmin
BEARING:UF:3000psi
i i I i i I i i
0.3 0.6 0.9
TIME sec 0.3 0.6 0.9
TIME sec
FIG. 10. Effect of Breakaway Friction on the Acceleration Response of a Sinu-soidally Excited Rigid Mass
of a rigid structure supported entirely by Teflon bearings and excited at the base. This simple model has been selected in order to avoid any masking effects induced by the flexibility of the superstructure or the stiffness of a restoring force device. The base motion is assumed sinusoidal of frequency w = 3.14 rad/sec and peak acceleration, Ug0, of 0.25 g. Time histories of the total acceleration, U„ normalized by Ugo, as computed by the presented model, are shown in Fig. 10. It should be noted that U,/Ugo is proportional to the force imparted to the structure. The solid line represents the response when breakaway friction is disregarded, while the dashed lines represent the response with breakaway incorporated and for b = 2, 4, and 6. In two of the cases, the interface consists of unfilled Teflon at pressure of 1,000 and 3,000 psi (6.9 and 20.7 N/mm2), the third case consists of 25% glass-filled Teflon at 6,500-psi (44.9-N/mm2) pressure and the fourth case consists of a stack of 24 unfilled Teflon-steel interfaces at 1,000 psi (6.9 N/mm2) pressure as in the R-FBI system (Mostaghel and Khodaverdian 1987). In modeling the behavior of this system, Eq. 1 was used but with the velocity divided by the number of interfaces (24). In all cases the direction of sliding is parallel to lay.
It is clear that the effects of breakaway friction are important when constant b is larger than the ratio of the maximum-to-minimum values of the sliding coefficients of friction as experienced by the sliding interfaces. For interfaces that operate at large sliding velocity, and sliding friction attains its maximum value, this condition is expressed as
b> Jmax
(18)
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The validity of this condition is easily demonstrated in the cases of Teflon bearings in Fig. 10. For unfilled Teflon, the ratio/ma5 to (/max - Df) exceeds 4.5 at pressures of 1,000 psi-3,000 psi (6.9-20.7 N/mm2). As such, only values of b larger than 4.5 would result in significant breakaway effects. For the glass-filled Teflon interface, this ratio is 1.8 and the breakaway effects are significant even for a value b equal to 2. Similarly in the R-FBI bearing, the maximum value of sliding friction is 0.088 (attained at time of 0.9 sec when velocity is 1.8 in./sec) and the corresponding ratio is 0.088/ 0.0266 = 3.3). This results in significant breakaway effects for values of b larger than 4. As such, the peak frictional force experienced by a structure on R-FBI bearings is comparable to that experienced by structures on bearings with a single sliding interface.
The writers have observed that running in glass-filled Teflon for even one cycle eliminates breakaway friction (Mokha et al. 1988). The development of significant breakaway friction after prolonged compression without sliding is a concern that has not been addressed. The writers believe that values of constant b (in Eq. 17) of the order of 4, or even larger, should be used in lieu of contradictory experimental evidence.
Analysis of Base-Isolated Structures Incorporating Teflon Bearings and Centering Force Devices
The earthquake response of a six-story steel-moment-resisting-frame building is considered. Floor height is 12 ft (3.6 m) and columns are spaced at 16 ft (4.9 m) on center in both directions. A slice of the building of 48-ft (14.6-m) length supported by eight columns is analyzed. A computer model of the building was developed for determining the dynamic characteristics. The model has been condensed to one with six degrees of freedom, representing the lateral displacement of each floor. The dynamic characteristics of the condensed model are presented in Table 2. Typical floor weight (including fram-
TABLE 2. Characteristics of Six-Story Building (Value in Parenthesis is Experl mental Extrapolated to Prototype)
Mode
(1)
1
2
3
4
5
6
Frequency (Hz) (2)
1.07 (1.17) 3.38
(3.86) 6.02
(6.64) 8.99
(9.52) 12.01
(12.40) 14.41
(14.46)
1 (3)
0.164
0.520
-0.804
1
1
0.679
2 (4)
0.395
1
-0.863
0.104
-0 .769
-0 .919
MODE SHAPE
Floor
3 (5)
0.611
0.956
0.230
-0.996
-0 .027
1
4 (6)
0.791
0.386
1
0.240
0.805
-0 .879
5 (7)
0.923
-0 .401
0.383
0.908
-0 .946
0.580
6 (top) (8)
1
-0 .996
-0 .817
-0 .619
0.397
-0 .196
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2.0
1.5
1 1.0 H
\ 0 . 5
LU 0.0
o 4.0
< _ j 0 - 3 0 O
o i < 03 1.0
0.0
- Linear Isolation System • Frict. Isolation System,1000psi
FIG, 11. Base Displacement and Base Shear Spectra of Isolated Six-Story Struc
ture
ing, etc.) is 105 kips (465 kN). A quarter scale, artificial mass model of this building has been constructed for shake table testing. The indentified frequencies and mode shapes of the scaled structure agree well with those in Table 2.
The isolation system for this building consists of sliding unfilled Teflon bearings and rubber springs. Bearing pressure on Teflon was parametrically varied between 1,000 and 3,000 psi (6.9 and 20.7 N/mm2). The rubber springs do not carry any vertical load (e.g., TASS system) and provide a total horizontal stiffness Kb. The rigid body mode period of the isolated structure is defined by
n-2-U ' <«> in which W = the weight of the structure including the base (672 kips); and g = the gravitational constant. Furthermore, the rubber springs provide an equivalent viscous damping factor in the rigid body of vibration equal to 0.05. Damping in the superstructure is assumed to be 2% of critical damping in all modes.
Results on the peak base displacement and base shear (first-story shear) normalized by the peak ground displacement and superstructure weight, respectively are shown in Fig. 11. Excitation is the North-South SOOE component of the 1940 El Centro earthquake (40 sec) and the North-West N90W component of the 1985 Mexico City earthquake (portion between 25 and 75 sec) that was recorded at the SCT building. The results are plotted for values of period Tb between 0.2 and 4 sec. The dashed lines represent the response of the sliding system, as computed by the presented modified viscoplasticity
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model with breakaway incorporated (b = 4 in Eq. 17). The solid line represents the response of a linear isolation system with period Tb and viscous damping factor of 0 .1 . It represents a high damping rubber isolation system. In all cases, the superstructure is assumed to remain elastic. Several conclusions may be derived from the results of Fig. 11:
1. Sliding isolation systems exhibit a low sensitivity to the frequency content of excitation. For example, the use of a sliding system at pressure of 1,000 psi (6.9 N/mm2) and Tb = 4 sec will result in a base shear coefficient of about 0.16 for both El Centra and Mexico City earthquakes. In both motions, the base displacement is restricted to values below the peak ground displacement. In contrast, an elastomeric isolation system designed for Tb = 2 sec, that is appropriate for El Centra motion, will be experiencing detrimental forces for a motion of the type of Mexico City.
2. A sliding isolation system designed for a bearing pressure of 1,000 psi (6.9 N/mm2) will undergo a lower base displacement than either sliding systems designed at higher pressure or elastomeric systems. This is apparently due to higher friction in low pressure Teflon bearings. As a consequence of this higher friction, it would be expected that base shear in low-pressure sliding isolation systems is higher than in high-pressure sliding systems. Fig. 11 provides opposite evidence. Around resonance areas (values of Tb close to predominant earthquake period), low-pressure sliding isolation systems have a base shear coefficient that is lower than the corresponding coefficient in higher pressure systems. The behavior is similar to that of a viscously damped, harmonically excited single-degree-of-freedom system of which the base shear decreases with increasing damping factor in the resonance region (excitation frequency to natural frequency less than V5). It appears that low-pressure Teflon bearings are preferable to designs at higher pressure.
3. Both base displacement and shear in the sliding systems remain constant above a certain value of period Tb (about 1.5 sec for El Centra and 3 sec for Mexico City). This indicates that the restoring force provided by the rubber springs is weaker than the frictional force. Designing, in this case, for values of Tb larger than these limits provides no benefit. It may actually be worse, as more permanent displacements may result.
CONCLUSIONS
A mathematical model of friction of Teflon-steel interfaces has been presented. Fundamental novelty of the presented model is the dependency of the frictional force on velocity of sliding and on bearing pressure. The model parameters have been identified from experimental data. It should be noted that the specific findings of this paper apply to the sheet-type Teflon bearings.
The model has been used in parametric studies of the behavior of sliding isolated structures and the following conclusions have been derived:
1. The use of Coulomb's constant friction model may result in a useful estimation of the peak response, provided that an appropriate value of the coefficient of friction is used. This value depends on the resulting sliding velocity that depends on the characteristics of the structural system and excitation. Even when the coefficient of friction is correctly postulated, the resulting motion ex-
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hibits a significantly higher frequency content than the exact motion. 2. The effects of breakaway friction are important only when the ratio of
breakaway-to-minimum sliding coefficient of friction (constant b) is larger than the ratio of maximum-to-minimum sliding coefficient of friction, as experienced by the sliding interface. Experimental results on fresh Teflon-steel interfaces suggest a value of constant b of about 4. Running in of the interfaces substantially reduces this constant. However, prolonged loading may induce a substantial increase on this constant, the extent of which is unknown.
3. Isolation systems that incorporate Teflon bearings exhibit a low sensitivity to the frequency content of excitation.
4. The use of Teflon bearings at low bearing pressure results in a behavior that, under certain conditions, is superior to designs at higher pressure. This behavior occurs for values of Tb close to the predominant period in the earthquake excitation (resonance). In this case, the restoring force is stronger than the frictional force, and the system resembles a highly damped oscillator that operates at resonance.
ACKNOWLEDGMENTS
This work has been supported by the National Center for Earthquake Engineering Research (contract 87-2002A), the National Science Foundation (PYI grant No. CES-8857080), and Watson Bowman Acme Corporation. This support is gratefully acknowledged.
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