The FPBearingPTV command creates a single Friction Pendulum bearing element, which is capable of accounting for the changes in the coefficient of friction at the sliding surface with instantaneous values of the sliding velocity, axial pressure and temperature at the sliding surface. The constitutive modelling is similar to the existing singleFPBearing element, otherwise. The FPBearingPTV element has been verified and validated in accordance with the ASME guidelines, details of which are presented in Chapter 4 of Kumar et al. (2015a).
element FPBearingPTV $eleTag $iNode $jNode $MuRef $IsPressureDependent $pRef $isTemperatureDependent $Diffusivity $Conductivity $IsVelocityDependent $rateParameter $ReffectiveFP $Radius_Contact $kInitial $theMaterialA $theMaterialB $theMaterialC $theMaterialD $x1 $x2 $x3 $y1 $y2 $y3 $shearDist $doRayleigh $mass $iter $tol $unit
|$eleTag||unique element object tag|
|$iNode $jNode||End nodes|
|$MuRef||Reference coefficient of friction|
|$IsPressureDependent||1.0 if the coefficient of friction is a function of instantaneous axial pressure|
|$pRef||Reference axial pressure (the bearing pressure under static loads)|
|$IsTemperatureDependent||1.0 if the coefficient of friction is a function of instantaneous
temperature at the sliding surface
|$Diffusivity||Thermal diffusivity of steel|
|$Conductivity||Thermal conductivity of steel|
|$IsVelocityDependent||1.0 if the coefficient of friction is a function of instantaneous velocity at the sliding surface|
|$rateParameter||The exponent that determines the shape of the coefficient of friction vs. sliding velocity curve|
|$ReffectiveFP||Effective radius of curvature of the sliding surface of the FPbearing|
|$Radius_Contact||Radius of contact area at the sliding surface|
|$kInitial||Lateral stiffness of the sliding bearing before sliding begins|
|$theMaterialA||Tag for the uniaxial material in the axial direction|
|$theMaterialB||Tag for the uniaxial material in the torsional direction|
|$theMaterialC||Tag for the uniaxial material for rocking about local Y axis|
|$theMaterialD||Tag for the uniaxial material for rocking about local Z axis|
|$x1 $x2 $x3||Vector components to define local X axis|
|$y1 $y2 $y3||Vector components to define local Y axis|
|$shearDist||Shear distance from iNode as a fraction of the length of the element|
|$doRayleigh||To include Rayleigh damping from the bearing|
|$iter||Maximum number of iterations to satisfy the equilibrium of element|
|$tol||Convergence tolerance to satisfy the equilibrium of the element|
|$unit||Tag to identify the unit from the list below.|
|1: N, m, s, C|
|2: kN, m, s, C|
|3: N, mm, s, C|
|4: kN, mm, s, C|
|5: lb, in, s, C|
|6: kip, in, s, C|
|7: lb, ft, s, C|
|8: kip, ft, s, C|
NOTE: Updating the coefficient of friction during analysis
The coefficient of friction at the sliding surface of a sliding bearing changes continuously with instantaneous values of sliding velocity, temperature at the sliding surface and axial pressure. The following definition of the coefficient of friction is implemented in the element.
kp = 0.70^((p-p0)/50) (2)
kt = 0.79(0.70^(T/50)+0.40) (3)
where kv,kp and kt and are the factors to account for the effects of sliding velocity, axial pressure and temperature at the sliding surface, respectively, v,p and T are velocity of sliding, axial pressure and temperature at the sliding surface, respectively, controls the shape of the kv vs. v curve, and p0 is the reference axial pressure.
The reference coefficient of friction, Uref , is defined as the coefficient of friction at a reference axial pressure on the bearing p0, measured at 20˚C at a high velocity of sliding (e.g., 1000 mm/s). The coefficient of friction, adjusted for the effects of sliding velocity, axial pressure
and temperature, U(p,T,v), is obtained as follows.
U(p,T,v) = Uref(kpktkv)(4)
where all parameters were defined previously. More details on this definition of the coefficient of friction are presented in Kumar et al. (2015a, 2015b).
The global and local forces, displacements, velocities and accelerations can be output through node and element recorders. In addition, temperature, three friction factors ( in sequence), and adjusted coefficient of friction can be output using the element recorder with tags Temperature, FrictionFactors, MuAdjusted, respectively. Examples are given below.
recorder Element -file Results/Temperature.out -time -ele 1 Temperature;
recorder Element -file Results/Mu.out -time -ele 1 MuAdjusted;
recorder Element -file Results/MuFactors.out -time -ele 1 MuFactors;
EXAMPLE (All numbers are in SI units (kg, m, C, S)):
set iNode 1;
set jNode 2;
set R 2.2352 ;
set Mu_Ref 0.06 ;
set p_Ref 50000000 ;
set kp_Factor 1 ;
set kT_Factor 1 ;
set kv_Factor 1 ;
set DF 4.44e-6;
set TK 18.0;
set a 100.0;
set Radius 0.2;
set pi [expr 22.0/7.0];
set Mass_Slider [expr $p_Ref*1.0*$pi*$Radius*$Radius/9.81];
set kInit [expr $Mass_Slider*$accelGravity*$Mu_Ref/$uy];
element FPBearingPTV 1 $iNode $jNode $Mu_Ref $kp_Factor $p_Ref $kT_Factor $DF
$TK $kv_Factor $a $R $Radius $kInit 1 2 3 4 0.0 0.0 1.0 1.0 0.0 0.0 0.0 0 0.0 100 1.0E-8 1 ;
FPWithUpdate.tcl files models a single concave sliding bearing with the mass concentrated on the slider. Download the example file and the ground motions.
Kumar, M., Whittaker, A. S., and Constantinou (2015a). “Seismic isolation of nuclear power plants using sliding bearings,” Report MCEER-15- 0006, University at Buffalo, State University of New York, Buffalo, NY.
Kumar, M., Whittaker, A. S., and Constantinou, M. C. (2015b). "Characterizing friction in sliding isolation bearings," Earthquake Engineering & Structural Dynamics, Vol. 44, No. 9, pp. 1409-1425.