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Response of the Double Concave Friction Pendulum System under Triaxial Ground Excitations
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 173 (2017) 1870 – 1877
11th International Symposium on Plasticity and Impact Mechanics, Implast 2016
Response of the double concave friction pendulum system under triaxial ground excitations Vrunda M Shaha, Dr.D.PSonib a PG Student, Civil Engineering Department, SardarVallabhbhai Patel Institute Of Technology, Vasad, 388306,India. Professor and Head, Civil Engineering Department, SardarVallabhbhai Patel Institute Of Technology, Vasad, 388306,India.
b
Abstract Modelling techniques of Double Concave Friction Pendulum (DCFP) system are presented. Seismic responses of the threedimensional single-story building isolated by DCFP with different coefficient of friction and initial time period of top and bottom sliding surfaces are investigated under triaxial ground excitations and compared with unilateral and bilateral ground excitations. The influence of vertical flexibility on horizontal response is investigated. It is observed that the triaxial ground motion has noticeable effect on response of the building relative to unilateral ground motion. The error in peak resultant absolute deck acceleration and the error in peak resultant isolator deformation, occurring by neglecting vertical ground motion component, could be significant. © 2017 2016Published The Authors. Published by Elsevier Ltd. © by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility ofthe organizing committee of Implast 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016 Keywords:Double Concave Friction Pendulum System; Vertical Component of Ground Motion; Bilateral Interaction; Seismic Isolation
1. Introduction The double concave friction pendulum system is growing interest due to its facility of providing different coefficient of friction and isolator geometry of top and bottom sliding surface. As shown in figure 1 , the displacement capacity of double concave friction pendulum system is twice compared to traditional friction pendulum system [Fenz and Constantinou, 2006][1].
Fig.1 maximum sliding displacements of Double Concave Friction Pendulum System
Jangid R S (1997)[2] presented thatthe response of structures with sliding support to bi-directional (i.e. two
1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016
doi:10.1016/j.proeng.2016.12.240
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horizontal components) harmonic ground motion is investigated. The responses of the system with bidirectional interaction are compared with 2D idealization in two orthogonal directions. And observed that the sliding base displacement may be underestimated if the effects of bi-directional interaction of frictional forces are neglected , which is crucial from the design point of view.JOSE L.et al. (1998)[7]presented that for Different modelling aspects of single-storey structures and a real four-storey building frame isolated using the FPS and subjected to earthquake ground motions in two direction are investigated. It is observed that there is 20 per cent error in mean results when neglecting vertical motion of the building. Uplift leading 3 times increase in column base shear when ignoring the vertical dynamics of the building.Soni et al. (2010)[4]presented that Modelling techniques and Seismic responses of the three-dimensional single-story building isolated by DVFPI with different μ and initial time period of top and bottom sliding surfaces are investigated under triaxial ground motions and compared with unilateral and bilateral ground excitations. It is observed that triaxial ground motion has noticeable effect on response of the building relative to unilateral ground motion. There is significant error in peak resultant base shear, occurring by neglecting vertical ground motion component. 2. Force - displacement relationship of the DCFP The force displacement relationship for the entire DCFP bearing was derived in [Fenz and Constantinou, 2006][1] as shown in equation (1). ܨൌ
ே ோభ ାோమ
ݑ
ఓேభ ௦ሺ௨ሶ భ ሻோభ ାఓேమ ௦ሺ௨ሶ మ ሻோమ ோభ ାோమ
(1)
Here, ܴଵ = ܴଵ െ ݄ଵ and ܴଶ =ܴଶ െ ݄ଶ ; ߤܰଵ ݊݃ݏሺݑሶ ଵ ሻ and ߤܰଶ ݊݃ݏሺݑሶ ଶ ሻ are the friction forces acting on top and bottom sliding surfaces respectively; the ݊݃ݏሺݑሶ ଵ ሻ and݊݃ݏሺݑሶ ଶ ሻ are incorporated to maintain the symmetry of the top and bottom sliding surfaces about the central vertical axis. The signum function have a value of +1 for positive value of sliding displacement and -1 for negative value of sliding displacement; ߤ = the coefficient of friction of top and bottom sliding surfaces; ܰଵ And ܰଶ = Vertical force acting on top and bottom surfaces respectively. Here assuming that ܰଵ and ܰଶ equal to weight of top and bottom sliding surface respectively.
Fig. 2 Mathematical model for a DCFP bearing
The two FPSs will move simultaneously and the DCFP will behave bi-linearly as the theoretical push-over curve when the sliding friction coefficients of both FPSs (ߤଵ and ߤଶ ) are same and a DCFP is subjected to a static load. The DCFP will behave tri-linearly when two friction coefficients are different (say ߤଵ <ߤଶ ). The overall behavior of DCFP can be obtained by defining two separate single concave elements connecting in series with small point mass modeled as articulated slider as shown in Figure 2. 3. Assumptions, structural system and ground motions A three dimensional single story rigid building resting on DCFP system installed between base mass and foundation block of the building shown in Figure 3. The assumptions considered for this whole system are as follow: 1. The superstructure is symmetrical with respect to two orthogonal horizontal directions. Therefore, there is no torsional coupling with lateral movement of the system and the system will have only lateral degrees of freedom. 2. The coefficient of friction of the DCFP bearing is assumed to be constant and independent of the relative velocity at the sliding interface.
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3. 4. 5. 6. 7.
There is no uplift and impact between the structure and the sliding surface of the DCFP bearing during earthquake. This express indirectly that the isolator is assumed to stay in contact with the sliding surfaces at all times during earthquake. The articulated slider of DCFP bearing is assumed to have point contact with both top and bottom sliding interfaces. No overturning and sliding occurs in the superstructure during sliding over DCFP bearing. The DCFP is isotropic it means there is same coefficient of friction and isolation period in each of the two principle directions of motion in the principle plane. Assume the superstructure is linearly elastic.
The superstructure of the system consists of a rigid deck slab and columns. The deck slab has lateral dimension b in X-direction and in Y-direction. The columns are too considered as mass less, axially inextensible. The deck slab is supported by columns, at top, at the edges of the deck slab and the columns are connected to a rigid base slab at the bottom of the columns. The centre of mass CM is located at the centre of the deck slab. The group of DCFP system is arranged betweenrigid base slab and foundation.
Fig. 3 Structural Model Elevation Table 1 Structural models taken in consideration for study Models Vertical stiffness of Earthquake components columns Model A Infinite X (Unilateral component) Y (Unilateral component) XY (Bilateral component) with interaction Model B
Infinite
XYZ (triaxial components) with interaction in XY direction
Model C
Finite
XYZ (triaxial components) with interaction in XY direction
The structural model taken in study is characterized with infinite stiffness of column and subjected to X (Unilateral component), Y (Unilateral component), XY (bilateral components) and XYZ(triaxial) components of earthquake.The responses of the structural models are obtained for six earthquake ground motions, which represent a wide range of ground motion characteristics that need to be considered in the design of seismic-isolated structures listed in Table 2. The response of structural model is obtained and the records represent near fault effects. 4. Governing equations of motion The governing equation of motion for the three dimensional single story superstructure, base mass and slider in Xdirection are shown in equation (2). The equation for Deck: ݉ௗ ݑሷ ௗ௫ ܿ௫ ݑሶ ௗ௫ ݇௫ ݑௗ௫ ൌ െ݉ௗ ݑሷ ௫ (2a) The equation for Base mass: ݉ ݑሷ ௫ െ ܿ௫ ݑሶ ௫ െ ݇௫ ݑௗ௫ ݇ଵ ݎଵ ݑ௫ െ ݇ଵ ݎଵ ݑ௦௫ ܨ௫ଵ ൌ െ݉ ݑሷ ௫
(2b)
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The equation of slider: ݉௦ ݑሷ ௦௫ െ ݇ଵ ݎଵ ݑ௫ ሾ݇ଵ ݎଵ ݇ଶ ݎଶ ሿݑ௦௫ ሺܨ௫ଶ െ ܨ௫ଵ ሻ ൌ െ݉௦ ݑሷ ௫ (2c) Wherein ݉ௗ ǡ ݉ ǡ ݉௦ = Deck mass, Base mass and mass of slider respectively; ܿ௫ = damping coefficient in Xdirection; ݇௫ = stiffness of superstructure in X-direction; ݑሷ ௗ௫ ǡ ݑሷ ௫ ݑሷ ௦௫ = X- directional component of deck acceleration, base mass acceleration and slider acceleration respectively; ݑሶ ௗ௫ ݑሶ ௫ = X- directional component of deck velocity and base mass velocity; ݑௗ௫ ǡ ݑ௫ ݑ௦௫ = X- directional component of deck displacement, base mass displacement and slider displacement respectively; ݎଵ ݎଶ = Earthquake influence coefficients; ܨ௫ଵ ܨ௫ଶ = Frictional forces in X- directional; ݑሷ ௫ = X- directional component of ground acceleration. These governing equation of motion for the three dimensional single story superstructure, base mass and slider can be obtained in Ydirection. Table 2 Details of earthquake ground motions used in study Earthquake Applied in X-direction components PGA Kobe (1995,Kobe University) 000 0.284
Applied in Y direction components PGA 090 0.304
Applied in Z-direction PGA(g) 0.372
Northridge (1994,New Hall Fire Station)
360
0.589
090
0.583
0.548
Lexington Dam (1989, Loma Prieta)
000
0.433
090
0.420
0.151
Corritilos (1989, Loma Prieta)
000
0.618
090
0.469
0.431
Sylmar (1987, Olive view Medical centre)
000
0.558
090
0.503
0.40
Coyoto Lake (1979,Gilroy Array #2)
140
0.248
050
0.186
0.162
5. Solution of equations of motion Due to nonlinear force deformation behavior of DCFP bearing and considerable difference in damping of superstructure and the isolation system, the governing equation of motion of the base isolated structure are solved in incremental form using Newmark’s step by step method assuming linear variation in acceleration over small time interval, ∆t, in this study. The friction forces are obtained by hysteretic model. The hysteretic model is a continuous model of friction forces proposed by Constantinouet al. [1990] using Bouc-wen equation [Wen, 1976]. In hysteretic model, the interaction between frictional forces can be expressed by coupling of hysteretic displacements components in two directions. A very small time step of the order of 1X10 -5 has been found suitable for step by step solution because of highly nonlinear behavior of the system. 6. Effect of vertical component on DCFP isolated building The normal reactions for top and bottom surface of isolation system produce due to effect of vertical component of earthquake. This can be expressed as ௨ሷ
ܰଵ ൌ ሺݓௗ ݓ ሻ ቂ
ͳቃ ௨ሷ
ܰଶ ൌ ሺݓௗ ݓ ݓ௦ ሻ ቂ
(3a) ͳቃ
(3b)
7. Numerical study To demonstrate the accuracy and validity of analysis methods, models and developed program, the Force deformation relationship for Model A underbilateral components and Model B under triaxial components of Northridge, 1994 (New Hall Fire Station) earthquake, respectively, for the example building isolated by the FPS obtained is shown below.
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Fig.4(a) Almazan et al. (1998)[7] and (b) developed program
Fig.5(a) Almazan et al. (1998)[7] and (b) developed program
7.1. Corrilitos (1989 Loma Prieta) earthquake results The response of three dimensional single story DCFP isolated building is obtained for criteria Ts=0.5s,T1=1.5s,T2=2.0s,μ1=0.04,μ2=0.05,λ= 0.001 and ξ = 2 %. The results obtained for corrilitos (1989 Loma Prieta) earthquake are plotted below. Table 3 shows peak response quantities for six earthquake ground motion for stiff building. Figure 6 shows time history of displacement for total isolator displacement for unilateral, bilateral and triaxial ground excitation in X and Y direction for Corrilitos (1989 Loma Prieta) earthquake. The time history of deck acceleration for unilateral, bilateral and triaxial ground excitation in X and Y direction for Corrilitos (1989 Loma Prieta) earthquake is shown in figure 7.Figure 8 shows time history of base shear for unilateral, bilateral and triaxial ground excitation in X and Y direction for Corrilitos (1989 Loma Prieta) earthquake. Figure 9 represent correlation between the isolator displacements in X and Y- directions for (a) top sliding surface, (b) bottom sliding surface and (c) total isolator deformation for Corrilitos (1989 Loma Prieta) earthquake. The graph shows path of movement of
Fig. 6 Time history of Displacement (a) in X (b) in Y for Corrilitos (1989 Loma Prieta) earthquake
Fig. 7Time history of Acceleration (a) in X (b) in Y for Corrilitos (1989 Loma Prieta) earthquake
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UNILATERAL BILATERAL TRIAXAIL
0.05
UNILATERAL BILATERAL TRIAXIAL
0.06
BASE SHEAR IN Y
BASE SHEAR IN X
0.04
0.00
-0.05
2
0.00 -0.02 -0.04 -0.06
0.098 0.102 0.11
-0.10
0.02
0.0803
-0.08
4
6
8
10
12
2
0.0813 0.086 TIME (sec) 6
4
8
10
12
TIME (sec)
Fig. 8Time history of base shear (a) in X (b) in Y for Corrilitos (1989 Loma Prieta) earthquake 2
BILATERAL TRIAXIAL
3
BILATERAL TRIAXIAL
BILTERAL TRIAXIAL
4 3
1
2
2
1
1
0
VBY
X2
X1
0
0
-1
-1 -2
-1
-3
-2
-4
-2
-5
-3
-3
-5
-4
-3
-2
-1
0
1
2
3
-2.0
-1.5
-1.0
-0.5
0.0
Y2
Y1
0.5
1.0
1.5
2.0
-6
2.5
-6
-4
-2
0
VBX
2
4
Fig. 9 correlation between the isolator displacements in X and Y- directions for (a) top sliding surface, (b) bottom sliding surface and (c) total isolator deformation for Corrilitos (1989 Loma Prieta) earthquake.
isolator. Figure 10 shows interaction curve for unilateral, bilateral and triaxial ground excitation for lexington dam (1989 Loma Prieta) earthquake. The interaction curve is square for unilateral ground motion,circular for bilateral ground motion and for triaxial ground motion the isolator move in side as well as outside of circular interaction curve. TRIAXIAL COMPONENTS
UNILATERAL COMPONENT
BILATERAL COMPONENTS 0.04
0.04
0.04
0.02
0.02
0.00
0.00
FY1 (W)
FY1
FY1
0.02
0.00
-0.02
-0.02
-0.02
-0.04
-0.04
-0.04
-0.04
-0.02
0.00
FX1
0.02
0.04
-0.04
-0.02
0.00
0.02
-0.04
0.04
-0.02
0.00
0.02
0.04
FX1 (W)
FX1
Fig.10 Interaction of friction forces in X and Y direction for (a) unilateral (b) bilateral and (c) triaxial ground motion. - TRIAXIAL COMPONENTS 1.55
1.6
1.4
N2 (W)
1.2
1.0
0.8
0.6
0.615 0
2
4
6
TIME(SEC)
8
10
12
Fig. 11 Time history of the normal reaction N2 (W) of 1994New Hall Fire Station ground motion
Figure 11 shows time history of normal reaction for 1994 New Hall Fire Station earthquake ground motion. Here normal reaction of triaxial ground motion changes continuously and reaches maximum of 55% of corresponding unitary value of model A. Figure 12 shows characteristic force displacement relationship of DCFP system under unilateral, bilateral and triaxial ground motion. For unilateral ground motion the hysteresis loop is straight, the line of loop is slightly curved for bilateral ground motion and for triaxial ground motion there is effect of spike due to variation in normal reaction of vertical component of earthquake. Figure 13 shows Mean peak resultant isolator deformation and deck acceleration error spectra under the six ground motions considered for unilateral, bilateral ground excitation relative to the triaxial ground excitation for stiff structure. The graphs represent that, When the vertical component of the ground motion is neglected, the mean error in peak resultant absolute deck acceleration could be increase in the range of 10-22% for unilateral and 2-18% for bilateral earthquake and the mean error in
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peak resultant isolator displacement could be under estimated in the range of 23-29% for unilateral and 0-2% for bilateral earthquake typically for the stiff structure. 0.2
0.4
BILATERAL (XY) COMPONENTS
UNILATERAL (X) COMPONENT 0.2
100
0
FBX (W)
0.0
FbX (W)
FBX (W)
0.0
-0.2
-0.1
-100
-200
-0.4
-300
-0.2 -0.6
-400
-400
-300
-200
-100
UBX(mm)
0
100
200
-0.3
-400
0.2
UNILATERAL (Y) COMPONENT 0.2
-300
-200
-100
Ubx (mm)
0
100
-0.6
200
BILATERAL (XY) COMPONENTS
0.0
UBY(mm)
0
50
100
150
-0.3
0.2
0.4
0.0
-0.2
-0.2
-300 -250 -200 -150 -100 -50
0.0
0.2
-0.1
-0.4
-0.2
TRIAXIAL COMPONENTS
0.4
FbY (W)
0.0
-0.2
-0.4
UX (mm)
0.1
FBY (W)
FBY(W)
TRIAXIAL COMPONENTS
200
0.1
-0.4
-0.6
-300
-200
-100
0
UBY (mm)
100
200
-300
-200
-100
0
100
200
UY (mm)
Fig.12 Characteristic force-displacement relationship of DCFP for unilateral, bilateral and triaxial ground excitation. Table 3 Peak response quantities of single storystiffbuilding under different components of considered earthquake ground motions ( Ts = 0.5 sT1 = 1.5 s, T2 =2.0 s, μ1= 0.04, μ2= 0.05, λ= 0.001and ξ = 2 %), Components
࢛ሷ ࢈࢞ ሺࢍሻ
࢛ሷ ࢈࢟ ሺࢍሻ
ࡲ࢈࢞ ሺࢃሻ
ࡲ࢈࢟ ሺࢃሻ
࢛࢈࢞ ሺሻ ࢛࢈࢟ ሺሻ N2(W)
Unilateral Model A
0.047
0.030
0.046
0.02
5.24
2.776
1.00
Bilateral Model A
0.059
0.045
0.041
0.028
15.505
6.89
1.00
Triaxial Model B
0.051
0.046
0.045
0.033
8.179
10.339
1.05
1994 Northridge (New Hall Fire Station)
Unilateral Model A Bilateral Model A Triaxial Model B
0.085 0.084 0.086
0.062 0.060 0.058
0.080 0.063 0.07
0.061 0.046 0.04
10.613 27.459 26.221
11.47 12.869 12.378
1.00 1.00 1.06
1989 Loma Prieta (Lexington Dam)
Unilateral Model A Bilateral Model A Triaxial Model B
0.063 0.059 0.06
0.064 0.056 0.057
0.048 0.056 0.058
0.045 0.054 0.056
6.85 9.598 10.405
5.86 8.92 8.961
1.00 1.00 1.012
Corritilos (1989 Loma Prieta)
Unilateral Model A Bilateral Model A Triaxial Model B
0.115 0.102 0.098
0.086 0.083 0.0813
0.114 0.078 0.072
0.074 0.071 0.069
2.87 4.07 4.93
4.53 5.16 5.58
1.00 1.00 1.05
1979 Coyoto Lake (Gilroy Array #2)
Unilateral Model A Bilateral Model A Triaxial Model B
0.011 0.02 0.016
0.032 0.03 0.03
0.01 0.018 0.01
0.032 0.029 0.029
0.61 3.87 1.86
1.55 2.79 1.89
1.00 1.00 1.013
1987 Sylmar – Olive view Medical centre
Unilateral Model A Bilateral Model A Triaxial Model B
0.097 0.1 0.097
0.06 0.075 0.067
0.076 0.097 0.098
0.042 0.046 0.047
14.33 21.361 21.493
17.47 18.418 16.919
1.00 1.00 1.05
Earthquake (Kobe
MEAN PEAK RESULTANT ABSOLUTE DECK ACCELERATION ERROR (%)
0
UNILATERAL BILATERAL
0
MEAN PEAK RESULTANT ISOLATOR DEFORMATON ERROR (%)
1995 Kobe University)
-10
-20 0.0
0.3
0.6
TS (SEC)
0.9
1.2
1.5
UNILATERAL BILATERAL -10
-20
-30 0.0
0.3
0.6
0.9
1.2
1.5
TS (SEC)
Fig. 13 Mean peak resultant isolator deformation and deck acceleration error spectra under the six ground motions considered for unilateral, bilateral ground excitation relative to the triaxial ground excitation for stiff structure.
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Vrunda M. Shah and D.P. Soni / Procedia Engineering 173 (2017) 1870 – 1877
Figure 14 shows Mean peak resultant deck acceleration error and mean peak resultant isolator deformation error for the six earthquake ground motions considered for rigid building relative to flexible building. The graphs shows that, When vertical flexibility of structure is ignored, the error in peak resultant absolute deck acceleration could be in range of 1 to 18 % and the error in peak resultant isolator deformation could be in range of 1 to 12% for rigid building relative to flexible building.
PEAK RESULTANT ABSOLUTE DECK ACCELERATION ERROOR
40
20
NWH LEN COR COY MEAN ERROR
KOBE SYL
PEAK RESULTANT ABSOLUTE ISOLATOR DEFORMATION ERROOR
50
30
KOBE SYL
10
20
10
0
-10
-20 0.03
NWH LEX COR COY MEAN ERROR
0.04
0.05
0.06
Tv
0.07
0.08
0.09
0.10
0
-10 0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Tv
Fig.14 Mean peak resultant deck acceleration error and isolator deformation error for the six earthquake ground motions considered for Model B relative to Model C.
8. Conclusion 1) There is significant difference in the results of unilateral (X or Y) components, bilateral (XY) components and triaxial components of ground excitations for structural system with DCFP bearing. 2) The hysteresis loop for force displacement relationship for triaxial ground motion shows spike effect due to variation in normal reaction of vertical component of earthquake. 3) When the vertical component of the ground motion is neglected, the mean error in peak resultant absolute deck acceleration could be increase in the range of 10-22% for unilateral and 2-18% for bilateral earthquake typically for the stiff structure. 4) When the vertical component of the ground motion is neglected, the peak resultant isolator displacement could be under estimated in the range of 23-29% for unilateral and 0-2% for bilateral earthquake typically for the stiff structure. 5) When vertical flexibility of structure is ignored, the error in peak resultant absolute deck acceleration could be in range of 1 to 18 % for rigid building relative to flexible building. 6) When vertical flexibility of structure is ignored, the error in peak resultant isolator deformation could be in range of 1 to 12% for rigid building relative to flexible building. Acknowledgements I thank the Almighty for giving me the courage and strength to complete and accomplish this research. , I would like to thank my family members, especially. I would like to take sincere thanks towards my esteemed guide, Dr. Devesh P. Soni, Head of Department of Civil Engineering, SardarVallbhbhai Institute of Technology, for his guidance. References [1] Michael C. Constantinou , ‘Behaviour of the double concave Friction Pendulum bearing’,Earthquake engineering and Structureal dynamics; 09/(2006), 35(11):1403 – 1424. [2] Jangid R S,’Response of pure-friction sliding structures to bi-directional harmonic ground motion’, Engineering Structures, 1997, Vol. 19, No. 2, pp. 97 104. [3] Jangid R S, ‘Seismic response of sliding structures to bidirectional earthquake Excitation’, Engineering an structural dynamics, 1996, vol. 25, 1301-1306. [4] Soni et al. , 'Response of the Double Variable Frequency Pendulum Isolator under Triaxial Ground Excitations', Journal of Earthquake Engineering, 2010, 14: 4, 527 — 558. [5] Jangid R S , ‘Response of sliding structures to bi-directional excitation’, journal of sound and vibration), 2001, 243(5), 929}944. [6] Jangid R S,‘Computational numerical models for seismic response of Structures isolated by sliding systems’, Structural control and health monitoring Struct. Control health monit, 2005; 12:117–137. [7] Jose l.et al. (1998),‘modelling aspects of structures isolated with the Frictional pendulum system’, earthquake engineering and structural dynamics Earthquake engng. Struct. Dyn., 1998, 27, 845ð867. [8] Montazar (2008),‘Effects of vertical ground motions on the seismic response of isolated structures with XY-Friction Pendulum system’, The 14th World Conference on Earthquake Engineering , Beijing, China, October 2008, 12-17. [9] Montazar (2008),‘Effects of vertical ground motions on the seismic response of isolated structures with XY-Friction Pendulum system’, The 14th World Conference on Earthquake Engineering , Beijing, China, October 2008, 12-17. [10] Michael C. Constantinou , ‘Friction pendulum double concave bearing’, Technical report on October 29, 2004.
ScienceDirect Procedia Engineering 173 (2017) 1870 – 1877
11th International Symposium on Plasticity and Impact Mechanics, Implast 2016
Response of the double concave friction pendulum system under triaxial ground excitations Vrunda M Shaha, Dr.D.PSonib a PG Student, Civil Engineering Department, SardarVallabhbhai Patel Institute Of Technology, Vasad, 388306,India. Professor and Head, Civil Engineering Department, SardarVallabhbhai Patel Institute Of Technology, Vasad, 388306,India.
b
Abstract Modelling techniques of Double Concave Friction Pendulum (DCFP) system are presented. Seismic responses of the threedimensional single-story building isolated by DCFP with different coefficient of friction and initial time period of top and bottom sliding surfaces are investigated under triaxial ground excitations and compared with unilateral and bilateral ground excitations. The influence of vertical flexibility on horizontal response is investigated. It is observed that the triaxial ground motion has noticeable effect on response of the building relative to unilateral ground motion. The error in peak resultant absolute deck acceleration and the error in peak resultant isolator deformation, occurring by neglecting vertical ground motion component, could be significant. © 2017 2016Published The Authors. Published by Elsevier Ltd. © by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license Peer-review under responsibility ofthe organizing committee of Implast 2016. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016 Keywords:Double Concave Friction Pendulum System; Vertical Component of Ground Motion; Bilateral Interaction; Seismic Isolation
1. Introduction The double concave friction pendulum system is growing interest due to its facility of providing different coefficient of friction and isolator geometry of top and bottom sliding surface. As shown in figure 1 , the displacement capacity of double concave friction pendulum system is twice compared to traditional friction pendulum system [Fenz and Constantinou, 2006][1].
Fig.1 maximum sliding displacements of Double Concave Friction Pendulum System
Jangid R S (1997)[2] presented thatthe response of structures with sliding support to bi-directional (i.e. two
1877-7058 © 2017 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of Implast 2016
doi:10.1016/j.proeng.2016.12.240
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horizontal components) harmonic ground motion is investigated. The responses of the system with bidirectional interaction are compared with 2D idealization in two orthogonal directions. And observed that the sliding base displacement may be underestimated if the effects of bi-directional interaction of frictional forces are neglected , which is crucial from the design point of view.JOSE L.et al. (1998)[7]presented that for Different modelling aspects of single-storey structures and a real four-storey building frame isolated using the FPS and subjected to earthquake ground motions in two direction are investigated. It is observed that there is 20 per cent error in mean results when neglecting vertical motion of the building. Uplift leading 3 times increase in column base shear when ignoring the vertical dynamics of the building.Soni et al. (2010)[4]presented that Modelling techniques and Seismic responses of the three-dimensional single-story building isolated by DVFPI with different μ and initial time period of top and bottom sliding surfaces are investigated under triaxial ground motions and compared with unilateral and bilateral ground excitations. It is observed that triaxial ground motion has noticeable effect on response of the building relative to unilateral ground motion. There is significant error in peak resultant base shear, occurring by neglecting vertical ground motion component. 2. Force - displacement relationship of the DCFP The force displacement relationship for the entire DCFP bearing was derived in [Fenz and Constantinou, 2006][1] as shown in equation (1). ܨൌ
ே ோభ ାோమ
ݑ
ఓேభ ௦ሺ௨ሶ భ ሻோభ ାఓேమ ௦ሺ௨ሶ మ ሻோమ ோభ ାோమ
(1)
Here, ܴଵ = ܴଵ െ ݄ଵ and ܴଶ =ܴଶ െ ݄ଶ ; ߤܰଵ ݊݃ݏሺݑሶ ଵ ሻ and ߤܰଶ ݊݃ݏሺݑሶ ଶ ሻ are the friction forces acting on top and bottom sliding surfaces respectively; the ݊݃ݏሺݑሶ ଵ ሻ and݊݃ݏሺݑሶ ଶ ሻ are incorporated to maintain the symmetry of the top and bottom sliding surfaces about the central vertical axis. The signum function have a value of +1 for positive value of sliding displacement and -1 for negative value of sliding displacement; ߤ = the coefficient of friction of top and bottom sliding surfaces; ܰଵ And ܰଶ = Vertical force acting on top and bottom surfaces respectively. Here assuming that ܰଵ and ܰଶ equal to weight of top and bottom sliding surface respectively.
Fig. 2 Mathematical model for a DCFP bearing
The two FPSs will move simultaneously and the DCFP will behave bi-linearly as the theoretical push-over curve when the sliding friction coefficients of both FPSs (ߤଵ and ߤଶ ) are same and a DCFP is subjected to a static load. The DCFP will behave tri-linearly when two friction coefficients are different (say ߤଵ <ߤଶ ). The overall behavior of DCFP can be obtained by defining two separate single concave elements connecting in series with small point mass modeled as articulated slider as shown in Figure 2. 3. Assumptions, structural system and ground motions A three dimensional single story rigid building resting on DCFP system installed between base mass and foundation block of the building shown in Figure 3. The assumptions considered for this whole system are as follow: 1. The superstructure is symmetrical with respect to two orthogonal horizontal directions. Therefore, there is no torsional coupling with lateral movement of the system and the system will have only lateral degrees of freedom. 2. The coefficient of friction of the DCFP bearing is assumed to be constant and independent of the relative velocity at the sliding interface.
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3. 4. 5. 6. 7.
There is no uplift and impact between the structure and the sliding surface of the DCFP bearing during earthquake. This express indirectly that the isolator is assumed to stay in contact with the sliding surfaces at all times during earthquake. The articulated slider of DCFP bearing is assumed to have point contact with both top and bottom sliding interfaces. No overturning and sliding occurs in the superstructure during sliding over DCFP bearing. The DCFP is isotropic it means there is same coefficient of friction and isolation period in each of the two principle directions of motion in the principle plane. Assume the superstructure is linearly elastic.
The superstructure of the system consists of a rigid deck slab and columns. The deck slab has lateral dimension b in X-direction and in Y-direction. The columns are too considered as mass less, axially inextensible. The deck slab is supported by columns, at top, at the edges of the deck slab and the columns are connected to a rigid base slab at the bottom of the columns. The centre of mass CM is located at the centre of the deck slab. The group of DCFP system is arranged betweenrigid base slab and foundation.
Fig. 3 Structural Model Elevation Table 1 Structural models taken in consideration for study Models Vertical stiffness of Earthquake components columns Model A Infinite X (Unilateral component) Y (Unilateral component) XY (Bilateral component) with interaction Model B
Infinite
XYZ (triaxial components) with interaction in XY direction
Model C
Finite
XYZ (triaxial components) with interaction in XY direction
The structural model taken in study is characterized with infinite stiffness of column and subjected to X (Unilateral component), Y (Unilateral component), XY (bilateral components) and XYZ(triaxial) components of earthquake.The responses of the structural models are obtained for six earthquake ground motions, which represent a wide range of ground motion characteristics that need to be considered in the design of seismic-isolated structures listed in Table 2. The response of structural model is obtained and the records represent near fault effects. 4. Governing equations of motion The governing equation of motion for the three dimensional single story superstructure, base mass and slider in Xdirection are shown in equation (2). The equation for Deck: ݉ௗ ݑሷ ௗ௫ ܿ௫ ݑሶ ௗ௫ ݇௫ ݑௗ௫ ൌ െ݉ௗ ݑሷ ௫ (2a) The equation for Base mass: ݉ ݑሷ ௫ െ ܿ௫ ݑሶ ௫ െ ݇௫ ݑௗ௫ ݇ଵ ݎଵ ݑ௫ െ ݇ଵ ݎଵ ݑ௦௫ ܨ௫ଵ ൌ െ݉ ݑሷ ௫
(2b)
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The equation of slider: ݉௦ ݑሷ ௦௫ െ ݇ଵ ݎଵ ݑ௫ ሾ݇ଵ ݎଵ ݇ଶ ݎଶ ሿݑ௦௫ ሺܨ௫ଶ െ ܨ௫ଵ ሻ ൌ െ݉௦ ݑሷ ௫ (2c) Wherein ݉ௗ ǡ ݉ ǡ ݉௦ = Deck mass, Base mass and mass of slider respectively; ܿ௫ = damping coefficient in Xdirection; ݇௫ = stiffness of superstructure in X-direction; ݑሷ ௗ௫ ǡ ݑሷ ௫ ݑሷ ௦௫ = X- directional component of deck acceleration, base mass acceleration and slider acceleration respectively; ݑሶ ௗ௫ ݑሶ ௫ = X- directional component of deck velocity and base mass velocity; ݑௗ௫ ǡ ݑ௫ ݑ௦௫ = X- directional component of deck displacement, base mass displacement and slider displacement respectively; ݎଵ ݎଶ = Earthquake influence coefficients; ܨ௫ଵ ܨ௫ଶ = Frictional forces in X- directional; ݑሷ ௫ = X- directional component of ground acceleration. These governing equation of motion for the three dimensional single story superstructure, base mass and slider can be obtained in Ydirection. Table 2 Details of earthquake ground motions used in study Earthquake Applied in X-direction components PGA Kobe (1995,Kobe University) 000 0.284
Applied in Y direction components PGA 090 0.304
Applied in Z-direction PGA(g) 0.372
Northridge (1994,New Hall Fire Station)
360
0.589
090
0.583
0.548
Lexington Dam (1989, Loma Prieta)
000
0.433
090
0.420
0.151
Corritilos (1989, Loma Prieta)
000
0.618
090
0.469
0.431
Sylmar (1987, Olive view Medical centre)
000
0.558
090
0.503
0.40
Coyoto Lake (1979,Gilroy Array #2)
140
0.248
050
0.186
0.162
5. Solution of equations of motion Due to nonlinear force deformation behavior of DCFP bearing and considerable difference in damping of superstructure and the isolation system, the governing equation of motion of the base isolated structure are solved in incremental form using Newmark’s step by step method assuming linear variation in acceleration over small time interval, ∆t, in this study. The friction forces are obtained by hysteretic model. The hysteretic model is a continuous model of friction forces proposed by Constantinouet al. [1990] using Bouc-wen equation [Wen, 1976]. In hysteretic model, the interaction between frictional forces can be expressed by coupling of hysteretic displacements components in two directions. A very small time step of the order of 1X10 -5 has been found suitable for step by step solution because of highly nonlinear behavior of the system. 6. Effect of vertical component on DCFP isolated building The normal reactions for top and bottom surface of isolation system produce due to effect of vertical component of earthquake. This can be expressed as ௨ሷ
ܰଵ ൌ ሺݓௗ ݓ ሻ ቂ
ͳቃ ௨ሷ
ܰଶ ൌ ሺݓௗ ݓ ݓ௦ ሻ ቂ
(3a) ͳቃ
(3b)
7. Numerical study To demonstrate the accuracy and validity of analysis methods, models and developed program, the Force deformation relationship for Model A underbilateral components and Model B under triaxial components of Northridge, 1994 (New Hall Fire Station) earthquake, respectively, for the example building isolated by the FPS obtained is shown below.
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Fig.4(a) Almazan et al. (1998)[7] and (b) developed program
Fig.5(a) Almazan et al. (1998)[7] and (b) developed program
7.1. Corrilitos (1989 Loma Prieta) earthquake results The response of three dimensional single story DCFP isolated building is obtained for criteria Ts=0.5s,T1=1.5s,T2=2.0s,μ1=0.04,μ2=0.05,λ= 0.001 and ξ = 2 %. The results obtained for corrilitos (1989 Loma Prieta) earthquake are plotted below. Table 3 shows peak response quantities for six earthquake ground motion for stiff building. Figure 6 shows time history of displacement for total isolator displacement for unilateral, bilateral and triaxial ground excitation in X and Y direction for Corrilitos (1989 Loma Prieta) earthquake. The time history of deck acceleration for unilateral, bilateral and triaxial ground excitation in X and Y direction for Corrilitos (1989 Loma Prieta) earthquake is shown in figure 7.Figure 8 shows time history of base shear for unilateral, bilateral and triaxial ground excitation in X and Y direction for Corrilitos (1989 Loma Prieta) earthquake. Figure 9 represent correlation between the isolator displacements in X and Y- directions for (a) top sliding surface, (b) bottom sliding surface and (c) total isolator deformation for Corrilitos (1989 Loma Prieta) earthquake. The graph shows path of movement of
Fig. 6 Time history of Displacement (a) in X (b) in Y for Corrilitos (1989 Loma Prieta) earthquake
Fig. 7Time history of Acceleration (a) in X (b) in Y for Corrilitos (1989 Loma Prieta) earthquake
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UNILATERAL BILATERAL TRIAXAIL
0.05
UNILATERAL BILATERAL TRIAXIAL
0.06
BASE SHEAR IN Y
BASE SHEAR IN X
0.04
0.00
-0.05
2
0.00 -0.02 -0.04 -0.06
0.098 0.102 0.11
-0.10
0.02
0.0803
-0.08
4
6
8
10
12
2
0.0813 0.086 TIME (sec) 6
4
8
10
12
TIME (sec)
Fig. 8Time history of base shear (a) in X (b) in Y for Corrilitos (1989 Loma Prieta) earthquake 2
BILATERAL TRIAXIAL
3
BILATERAL TRIAXIAL
BILTERAL TRIAXIAL
4 3
1
2
2
1
1
0
VBY
X2
X1
0
0
-1
-1 -2
-1
-3
-2
-4
-2
-5
-3
-3
-5
-4
-3
-2
-1
0
1
2
3
-2.0
-1.5
-1.0
-0.5
0.0
Y2
Y1
0.5
1.0
1.5
2.0
-6
2.5
-6
-4
-2
0
VBX
2
4
Fig. 9 correlation between the isolator displacements in X and Y- directions for (a) top sliding surface, (b) bottom sliding surface and (c) total isolator deformation for Corrilitos (1989 Loma Prieta) earthquake.
isolator. Figure 10 shows interaction curve for unilateral, bilateral and triaxial ground excitation for lexington dam (1989 Loma Prieta) earthquake. The interaction curve is square for unilateral ground motion,circular for bilateral ground motion and for triaxial ground motion the isolator move in side as well as outside of circular interaction curve. TRIAXIAL COMPONENTS
UNILATERAL COMPONENT
BILATERAL COMPONENTS 0.04
0.04
0.04
0.02
0.02
0.00
0.00
FY1 (W)
FY1
FY1
0.02
0.00
-0.02
-0.02
-0.02
-0.04
-0.04
-0.04
-0.04
-0.02
0.00
FX1
0.02
0.04
-0.04
-0.02
0.00
0.02
-0.04
0.04
-0.02
0.00
0.02
0.04
FX1 (W)
FX1
Fig.10 Interaction of friction forces in X and Y direction for (a) unilateral (b) bilateral and (c) triaxial ground motion. - TRIAXIAL COMPONENTS 1.55
1.6
1.4
N2 (W)
1.2
1.0
0.8
0.6
0.615 0
2
4
6
TIME(SEC)
8
10
12
Fig. 11 Time history of the normal reaction N2 (W) of 1994New Hall Fire Station ground motion
Figure 11 shows time history of normal reaction for 1994 New Hall Fire Station earthquake ground motion. Here normal reaction of triaxial ground motion changes continuously and reaches maximum of 55% of corresponding unitary value of model A. Figure 12 shows characteristic force displacement relationship of DCFP system under unilateral, bilateral and triaxial ground motion. For unilateral ground motion the hysteresis loop is straight, the line of loop is slightly curved for bilateral ground motion and for triaxial ground motion there is effect of spike due to variation in normal reaction of vertical component of earthquake. Figure 13 shows Mean peak resultant isolator deformation and deck acceleration error spectra under the six ground motions considered for unilateral, bilateral ground excitation relative to the triaxial ground excitation for stiff structure. The graphs represent that, When the vertical component of the ground motion is neglected, the mean error in peak resultant absolute deck acceleration could be increase in the range of 10-22% for unilateral and 2-18% for bilateral earthquake and the mean error in
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peak resultant isolator displacement could be under estimated in the range of 23-29% for unilateral and 0-2% for bilateral earthquake typically for the stiff structure. 0.2
0.4
BILATERAL (XY) COMPONENTS
UNILATERAL (X) COMPONENT 0.2
100
0
FBX (W)
0.0
FbX (W)
FBX (W)
0.0
-0.2
-0.1
-100
-200
-0.4
-300
-0.2 -0.6
-400
-400
-300
-200
-100
UBX(mm)
0
100
200
-0.3
-400
0.2
UNILATERAL (Y) COMPONENT 0.2
-300
-200
-100
Ubx (mm)
0
100
-0.6
200
BILATERAL (XY) COMPONENTS
0.0
UBY(mm)
0
50
100
150
-0.3
0.2
0.4
0.0
-0.2
-0.2
-300 -250 -200 -150 -100 -50
0.0
0.2
-0.1
-0.4
-0.2
TRIAXIAL COMPONENTS
0.4
FbY (W)
0.0
-0.2
-0.4
UX (mm)
0.1
FBY (W)
FBY(W)
TRIAXIAL COMPONENTS
200
0.1
-0.4
-0.6
-300
-200
-100
0
UBY (mm)
100
200
-300
-200
-100
0
100
200
UY (mm)
Fig.12 Characteristic force-displacement relationship of DCFP for unilateral, bilateral and triaxial ground excitation. Table 3 Peak response quantities of single storystiffbuilding under different components of considered earthquake ground motions ( Ts = 0.5 sT1 = 1.5 s, T2 =2.0 s, μ1= 0.04, μ2= 0.05, λ= 0.001and ξ = 2 %), Components
࢛ሷ ࢈࢞ ሺࢍሻ
࢛ሷ ࢈࢟ ሺࢍሻ
ࡲ࢈࢞ ሺࢃሻ
ࡲ࢈࢟ ሺࢃሻ
࢛࢈࢞ ሺሻ ࢛࢈࢟ ሺሻ N2(W)
Unilateral Model A
0.047
0.030
0.046
0.02
5.24
2.776
1.00
Bilateral Model A
0.059
0.045
0.041
0.028
15.505
6.89
1.00
Triaxial Model B
0.051
0.046
0.045
0.033
8.179
10.339
1.05
1994 Northridge (New Hall Fire Station)
Unilateral Model A Bilateral Model A Triaxial Model B
0.085 0.084 0.086
0.062 0.060 0.058
0.080 0.063 0.07
0.061 0.046 0.04
10.613 27.459 26.221
11.47 12.869 12.378
1.00 1.00 1.06
1989 Loma Prieta (Lexington Dam)
Unilateral Model A Bilateral Model A Triaxial Model B
0.063 0.059 0.06
0.064 0.056 0.057
0.048 0.056 0.058
0.045 0.054 0.056
6.85 9.598 10.405
5.86 8.92 8.961
1.00 1.00 1.012
Corritilos (1989 Loma Prieta)
Unilateral Model A Bilateral Model A Triaxial Model B
0.115 0.102 0.098
0.086 0.083 0.0813
0.114 0.078 0.072
0.074 0.071 0.069
2.87 4.07 4.93
4.53 5.16 5.58
1.00 1.00 1.05
1979 Coyoto Lake (Gilroy Array #2)
Unilateral Model A Bilateral Model A Triaxial Model B
0.011 0.02 0.016
0.032 0.03 0.03
0.01 0.018 0.01
0.032 0.029 0.029
0.61 3.87 1.86
1.55 2.79 1.89
1.00 1.00 1.013
1987 Sylmar – Olive view Medical centre
Unilateral Model A Bilateral Model A Triaxial Model B
0.097 0.1 0.097
0.06 0.075 0.067
0.076 0.097 0.098
0.042 0.046 0.047
14.33 21.361 21.493
17.47 18.418 16.919
1.00 1.00 1.05
Earthquake (Kobe
MEAN PEAK RESULTANT ABSOLUTE DECK ACCELERATION ERROR (%)
0
UNILATERAL BILATERAL
0
MEAN PEAK RESULTANT ISOLATOR DEFORMATON ERROR (%)
1995 Kobe University)
-10
-20 0.0
0.3
0.6
TS (SEC)
0.9
1.2
1.5
UNILATERAL BILATERAL -10
-20
-30 0.0
0.3
0.6
0.9
1.2
1.5
TS (SEC)
Fig. 13 Mean peak resultant isolator deformation and deck acceleration error spectra under the six ground motions considered for unilateral, bilateral ground excitation relative to the triaxial ground excitation for stiff structure.
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Figure 14 shows Mean peak resultant deck acceleration error and mean peak resultant isolator deformation error for the six earthquake ground motions considered for rigid building relative to flexible building. The graphs shows that, When vertical flexibility of structure is ignored, the error in peak resultant absolute deck acceleration could be in range of 1 to 18 % and the error in peak resultant isolator deformation could be in range of 1 to 12% for rigid building relative to flexible building.
PEAK RESULTANT ABSOLUTE DECK ACCELERATION ERROOR
40
20
NWH LEN COR COY MEAN ERROR
KOBE SYL
PEAK RESULTANT ABSOLUTE ISOLATOR DEFORMATION ERROOR
50
30
KOBE SYL
10
20
10
0
-10
-20 0.03
NWH LEX COR COY MEAN ERROR
0.04
0.05
0.06
Tv
0.07
0.08
0.09
0.10
0
-10 0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
Tv
Fig.14 Mean peak resultant deck acceleration error and isolator deformation error for the six earthquake ground motions considered for Model B relative to Model C.
8. Conclusion 1) There is significant difference in the results of unilateral (X or Y) components, bilateral (XY) components and triaxial components of ground excitations for structural system with DCFP bearing. 2) The hysteresis loop for force displacement relationship for triaxial ground motion shows spike effect due to variation in normal reaction of vertical component of earthquake. 3) When the vertical component of the ground motion is neglected, the mean error in peak resultant absolute deck acceleration could be increase in the range of 10-22% for unilateral and 2-18% for bilateral earthquake typically for the stiff structure. 4) When the vertical component of the ground motion is neglected, the peak resultant isolator displacement could be under estimated in the range of 23-29% for unilateral and 0-2% for bilateral earthquake typically for the stiff structure. 5) When vertical flexibility of structure is ignored, the error in peak resultant absolute deck acceleration could be in range of 1 to 18 % for rigid building relative to flexible building. 6) When vertical flexibility of structure is ignored, the error in peak resultant isolator deformation could be in range of 1 to 12% for rigid building relative to flexible building. Acknowledgements I thank the Almighty for giving me the courage and strength to complete and accomplish this research. , I would like to thank my family members, especially. I would like to take sincere thanks towards my esteemed guide, Dr. Devesh P. Soni, Head of Department of Civil Engineering, SardarVallbhbhai Institute of Technology, for his guidance. References [1] Michael C. Constantinou , ‘Behaviour of the double concave Friction Pendulum bearing’,Earthquake engineering and Structureal dynamics; 09/(2006), 35(11):1403 – 1424. [2] Jangid R S,’Response of pure-friction sliding structures to bi-directional harmonic ground motion’, Engineering Structures, 1997, Vol. 19, No. 2, pp. 97 104. [3] Jangid R S, ‘Seismic response of sliding structures to bidirectional earthquake Excitation’, Engineering an structural dynamics, 1996, vol. 25, 1301-1306. [4] Soni et al. , 'Response of the Double Variable Frequency Pendulum Isolator under Triaxial Ground Excitations', Journal of Earthquake Engineering, 2010, 14: 4, 527 — 558. [5] Jangid R S , ‘Response of sliding structures to bi-directional excitation’, journal of sound and vibration), 2001, 243(5), 929}944. [6] Jangid R S,‘Computational numerical models for seismic response of Structures isolated by sliding systems’, Structural control and health monitoring Struct. Control health monit, 2005; 12:117–137. [7] Jose l.et al. (1998),‘modelling aspects of structures isolated with the Frictional pendulum system’, earthquake engineering and structural dynamics Earthquake engng. Struct. Dyn., 1998, 27, 845ð867. [8] Montazar (2008),‘Effects of vertical ground motions on the seismic response of isolated structures with XY-Friction Pendulum system’, The 14th World Conference on Earthquake Engineering , Beijing, China, October 2008, 12-17. [9] Montazar (2008),‘Effects of vertical ground motions on the seismic response of isolated structures with XY-Friction Pendulum system’, The 14th World Conference on Earthquake Engineering , Beijing, China, October 2008, 12-17. [10] Michael C. Constantinou , ‘Friction pendulum double concave bearing’, Technical report on October 29, 2004.