Theoretical study of the double concave friction pendulum system under variable vertical loading

Abstract

The double concave friction pendulum system has been recognized as an efficient device for decreasing the seismic response of a structure during an earthquake excitation. Previous studies have focused mainly on the properties of the double concave friction pendulum system under constant vertical loading, and the width of the hysteretic loop changed by the vertical ground motion is less considered. In view of this, a theoretical study of the double concave friction pendulum system under variable vertical loading is conducted in this article. Meanwhile, the properties of the hysteretic loops of the double concave friction pendulum system with different friction coefficients between the articulated slider with the upper and lower sliding surfaces are investigated. The results show that the hysteretic loops of the double concave friction pendulum system will be affected by the variation of the vertical loading and the difference of the friction coefficients between the articulated slider with the upper and lower sliding surfaces.

Keywords

double concave friction pendulum finite element method hysteretic loop theoretical study variable vertical loading

Introduction

Dynamic response of structure can be reduced with the installation of structure devices. The friction pendulum system (FPS), proposed by Zayas et al. (1987), is one type of base isolation technology, and the effectiveness for isolating the seismic transmitted energy has been certified by comprehensive experimental and numerical studies (Almazan et al., 2015; Tsai, 1995, 1997; Wang et al., 2015). And, it becomes a widely used device for seismic protection due to its remarkable features, such as the better control of the fundamental period of the system, the large deformation capacity by changing the geometric forms, and the stability of physical properties compared to the elastomeric bearings (Christopoulos and Filiatrault, 2006; Zayas et al., 1990). Based on the consistency of FPS, a new device called double concave friction pendulum (DCFP) is invented to supply a substantially larger displacement compared to a traditional FPS bearing of identical plan dimensions. Meanwhile, for same maximum design deformation, the DCFP or multi-concave friction pendulum can achieve the larger isolation period. Moreover, there is the capability to use sliding surfaces with varying radii of curvature and coefficients of friction, offering the designer greater flexibility to optimize performance (Fenz and Constantinou, 2008, 2010). The dynamic response of an isolated structure in Japan wherein the DCFP bearings are employed is presented by Hyakuda et al. (2001). A piecewise exact solution for predicting seismic responses of a structure which utilized DCFP is proposed by Tsai et al. (2003a, 2003b, 2004, 2005), and some experimental tests are conducted to verify the accuracy of the proposed method. Comparisons between the experimental and the analytical results reveal that the proposed method can predict well the nonlinear behavior of sliding isolators during earthquakes. Kim and Yun (2007) studied a DCFP system with trilinear behavior. The effects of DCFP systems with various friction values and restoring properties on a bridge are investigated under various earthquake excitations. Earthquake response analyses are carried out in the time domain on a simple bridge model with a DCFP system having various properties. Investigations are carried out on the reduction of the base-shear force on the bridge pier due to the DCFP. The advantages of a DCFP with trilinear behavior are explored. Two isolated structures with FPS and DCFP have been analyzed at three different hazard levels under 60 records by Malekzadeh and Taghikhany (2010). The peak roof acceleration, peak inter-story drift, and peak isolation displacement are considered as the response quantities of interest. Results show that DCFP acts as an adaptive isolation system, since stiffness and damping vary in proportion to the level of input ground motion, and can control peak floor acceleration and inter-story drift together. Khoshnoudian and Hemmati (2011) investigated the seismic response of three-dimensional base-isolated structures using DCFP systems considering trilinear and bilinear behaviors. Advantages and disadvantages of trilinear behavior over bilinear one are scrutinized by studying the effect of main parameters such as isolation period, amplitude of the ground motion, and friction coefficient of the surfaces on the peak responses under seven earthquakes. Results demonstrated that trilinear DCFP bearings, in comparison with bilinear bearings, can decrease the base shear up to about 48%. Mshah and Psoni (2017) studied the seismic responses of the three-dimensional single-story building isolated by DCFP with different coefficient of friction and initial time period of top and bottom sliding surfaces under triaxial ground excitations, and it is observed that the triaxial ground motion has noticeable effect on response of the building relative to unilateral ground motion. Mohammadi and Nouri (2017) analyzed single degree-of-freedom structures, which are connected to FPS and DCFP isolators, under time history implementing the records of near-fault and far-fault earthquakes. The obtained results indicated that in comparison with DCFP isolator, FPS isolator transfers less base shear on its structure in near-fault earthquake, whereas in comparison with FPS isolator, DCFP isolator transfers less displacements on its structure in near-fault earthquake. Nestovito and Occhiuzzi (2016) proposed a new adaptive algorithm, in conjunction with a smart-passive system and DCFP devices, and analysis shows that the approach proposed results in a more robust seismic behavior of the structure. A summary of the experimental dynamic response of curved surface sliders as resulting from more than 1000 dynamic tests on about 60 different types of devices performed at the laboratory of the EUCENTRE foundation was presented by Barone et al. (2017). The rate of change of the dynamic friction coefficient as a function of sliding velocity and contact pressure is assessed and the transition phase between dynamic and static friction, with the consequent possible evidence of static/stick-slip phenomena, has been carefully considered. Ponzo et al. (2017) investigated the restoring capability of DCFP. The accumulation of residual displacements is also investigated by means of nonlinear dynamic analysis. An experimental-based mathematical formulation describing the friction behavior of curved surface sliding bearings has been presented by Gandelli et al. (2018). The novel features introduced by the formulation are the contribution of the static friction during the sticking phase and the degradation of the coefficient of friction induced by the heat generated during the sliding motion.

In all of the aforementioned studies, the vertical loading of the DCFP is usually assumed to be a constant value. Properties of the DCFP under variable vertical loading are seldom considered. Therefore, a theoretical study of the DCFP system under variable vertical loading is conducted in this article. Meanwhile, the properties of the hysteretic loops of the DCFP with different friction coefficients between the articulated slider with the upper and lower sliding surfaces are investigated.

Outline of the theoretical mechanics of the DCFP

As shown in Figure 1, the DCFP contains two sliding surfaces. u₁ and u₂ stand for the friction coefficient, respectively. Thus, the displacement capacity of DCFP system is twice compared to traditional FPS. When the friction coefficients between the articulated slider with the upper and lower sliding surface are the same, if tangential force (T) acting on the surface of the articulated slider is greater than its friction force, the articulated slider will slide along the upper and lower sliding surfaces simultaneously. Therefore, the motion state of the DCFP can be decomposed as Figure 2. The equilibrium of the concave surfaces under horizontal and vertical loading can be expressed as follows

m (g - a) \cos θ_{1} + F_{1} \sin θ_{1} - N_{1} = 0

(1)

m (g - a) \sin θ_{1} - F_{1} \cos θ_{1} + T_{1} = 0

(2)

m (g - a) \cos θ_{2} + F_{2} \sin θ_{2} - N_{2} = 0

(3)

m (g - a) \sin θ_{2} - F_{2} \cos θ_{2} + T_{2} = 0

(4)

where m is the mass of the superstructure; g is the acceleration of gravity; a is the acceleration of the superstructure along vertical direction; and F₁, F₂ are the horizontal forces acting on the sliding surfaces. T₁, T₂ are the tangential forces generated on the upper and lower sliding surfaces; N₁, N₂ are the contact forces normal to the upper and lower sliding surfaces, respectively.

Figure 1.

Deformation of the double concave friction pendulum.

Figure 2.

Forces acting on the sliding surfaces: (a) lower sliding surface and (b) upper sliding surface.

Through equations (1) to (4), the horizontal forces acting on the sliding surfaces can be obtained as

F_{1} = k_{1} x_{1} + \frac{T_{1}}{\cos θ_{1}}

(5)

F_{2} = k_{2} x_{2} + \frac{T_{2}}{\cos θ_{2}}

(6)

where $k_{1} = m (g - a) / R_{1} \cos θ_{1}$ , $x_{1} = R_{1} \sin θ_{1}$ , $k_{2} = m (g - a) / R_{2} \cos θ_{2}$ , and $x_{2} = R_{2} \sin θ_{2}$ .

Because the total displacement of the DCFP is equal to x₁ plus x₂, the relationship between the horizontal force F and x shown in Figure 1 can be expressed as

F = \frac{k_{1} k_{2}}{k_{1} + k_{2}} x + \frac{1}{k_{1} + k_{2}} (\frac{T_{1} k_{2}}{\cos θ_{1}} + \frac{T_{2} k_{1}}{\cos θ_{2}})

(7)

The angle generated during the motion of the DCFP can be ignored due to a large value of the radius of the friction pendulum. Therefore, the stiffness of the DCFP can be obtained as follows

k_{e} = \frac{k_{1} k_{2}}{k_{1} + k_{2}} = \frac{m (g - a)}{R_{1} \cos θ_{1} + R_{2} \cos θ_{2}} \to \frac{m (g - a)}{R_{1} + R_{2}}

(8)

Equation (7) is just suitable for the case that $μ_{1} = μ_{2}$ . If $μ_{1} \neq μ_{2}$ , the articulated slider first slips along the sliding surface with small friction coefficient $(μ_{1})$ , and it keeps static status along the sliding surface with large friction coefficient $(μ_{2})$ before the deformation x₁ reaches to a critical value D*. When $x_{1} > D^{*}$ , the articulated slider will slip along both the upper and lower sliding surfaces. The value D* can be obtained through the following process

\begin{matrix} \frac{m (g - a) \sin θ_{1}}{\cos θ_{1}} + μ_{1} m (g - a) = μ_{2} m (g - a) \\ \to \frac{D^{*}}{\sqrt{R_{1}^{2} - {(D^{*})}^{2}}} = μ_{2} - μ_{1} \\ \to D^{*} = \frac{(μ_{2} - μ_{1}) R_{1}}{\sqrt{1 + {(μ_{2} - μ_{1})}^{2}}} \end{matrix}

(9)

It can be obtained through equation (9) that the critical value D* is related to the friction coefficients $μ_{1}$ , $μ_{2}$ and the radius R₁. Therefore, the stiffness of the DCFP with different friction coefficients can be expressed as follows

k_{e} = {\begin{matrix} \frac{m (g - a)}{R_{1}} x \leq D^{*} \\ \frac{m (g - a)}{R_{1} + R_{2}} x > D^{*} \end{matrix}

(10)

According to equations (8) to (10), the relationship between the horizontal force F and horizontal displacement x can be depicted as shown in Figure 3. In Figure 3, a_i, a_j stands for the vertical acceleration of the mass of the superstructure at time i and j, respectively. Therefore, it can be known that the width of the hysteretic loop of DCFP will be influenced by the variation of the vertical loading.

Figure 3.

Force–displacement relationship for DCFP: (a) $μ_{1} = μ_{2}$ and (b) $μ_{1} \neq μ_{2}$ .

Numerical study

Hysteretic property of the DCFP with $μ_{1} = μ_{2}$

In order to investigate the hysteretic property of the DCFP influenced by the variation of the vertical loading, a DCFP as shown in Table 1 is employed as a numerical example. Vertical loading cases are depicted in Table 2. The horizontal displacement x is assumed to be $180 \sin π t (unit : mm)$ . Finite element analysis (FEA) is conducted to verify the accuracy of the employed mechanical model depicted in section “Outline of the theoretical mechanics of the DCFP.” The three-dimensional finite element model shown in Figure 4 is developed in ABAQUS for the study. The materials of the bearings are Q345 steel (the elastic modulus is 210 GPa, Poisson’s ratio is 0.3, and the yield strength is 345 MPa) and the fraction material Poly tetra fluoroethylene (PTFE) (the elastic modulus is 2.6 GPa, Poisson’s ratio is 0.4, and tensile strength is 70 MPa). Hexahedron is adopted in the finite element grid, in which the steel part of the bearing uses C3D8R and the PTFE uses C3D8. The bearing’s FEA load condition restricts the degrees of freedom (DOFs) of rotation of the upper rotary plate as well as all DOFs of the bearings under sliding plate. It also applies the horizontal reciprocating displacement load on the upper rotary plate, records the horizontal force with the bearing moving, and obtains the hysteresis loops.

Table 1.

Parameters of the DCFP.

R ₁: 3.15 m

R ₂: 3.15 m

µ ₁: 0.09

µ ₂: 0.09

DCFP: double concave friction pendulum.

Table 2.

Vertical loading cases (P: kN).

Case 1: 500	Case 5: $500 + 200 \sin (2 π t)$
Case 2: $500 + 100 \sin π t$	Case 6: $500 + 200 \sin (3 π t)$
Case 3: $500 + 200 \sin π t$	Case 7: $500 + 200 \sin (π t + 0.5 π)$
Case 4: $500 + 300 \sin π t$	Case 8: $500 + 200 \sin (π t + 0.75 π)$

Figure 4.

Schematic diagram of every part of the DCFP.

The hysteretic loops of the DCFP are shown in Figure 5. It can be observed that the theoretical simulations are identical with the numerical analysis results. Thus, the accuracy of the theoretical simulation process depicted in section “Outline of the theoretical mechanics of the DCFP” is confirmed. The bilinear behavior of the DCFP is shown through Figure 5(a) under a constant vertical loading of 500 kN. Comparing the hysteretic loop of Case 1 with that of Cases 2–4, it can be known that the width of the hysteretic loops changes because the varying vertical loading is correctly represented in the mathematical model. Comparing the hysteretic loops of Case 3 with that of Cases 5–8, it can be obtained that the property of the hysteretic loops verifies with the changes of the frequency and the phase angle of the vertical loading. Therefore, consideration of the properties of the vertical loading is necessary for an accurate determination of the hysteretic loops of the DCFP during the design process.

Figure 5.

Hysteretic loops of DCFP $(μ_{1} = μ_{2})$ : (a) Case 1, (b) Case 2, (c) Case 3, (d) Case 4, (e) Case 5, (f) Case 6, (g) Case 7, and (h) Case 8.

Hysteretic property of the DCFP with $μ_{1} \neq μ_{2}$

In order to investigate the hysteretic property of the DCFP with different values of $μ_{1}$ and $μ_{2}$ , the following types (as shown in Table 3) of the DCFPs are used. The hysteretic loops of the DCFP are shown in Figure 6. Comparing Figure 6(a) with Figure 5(a), it can be observed that the shape of the hysteretic loop of the DCFP with $μ_{1} \neq μ_{2}$ no longer follows simple bilinear property. In other words, it obeys the trilinear property to some extent. Through Figure 6(b) to (d), it can be obtained that the width of the hysteretic loops changes because the varying vertical loading is correctly represented in the mathematical model. And the width of the hysteretic loops at both ends is just related to the small friction coefficient $μ_{1}$ and the value of the vertical load is also known through Figure 6. Through Figure 6, it can also be known that, if $μ_{1} \neq μ_{2}$ , the articulated slider first slips along the sliding surface with small friction coefficient $(μ_{1})$ , and it keeps static status along the sliding surface with large friction coefficient $(μ_{2})$ before the deformation x₁ reaches to a critical value D*. When $x_{1} > D^{*}$ , the articulated slider will slip along both the upper and lower sliding surfaces.

Table 3.

Types of the DCFP.

Type 1	R ₁: 3.15 m	R ₂: 3.15 m	µ ₁: 0.07	µ ₂: 0.09
Type 2	R ₁: 3.15 m	R ₂: 3.15 m	µ ₁: 0.05	µ ₂: 0.09
Type 3	R ₁: 3.15 m	R ₂: 3.15 m	µ ₁: 0.03	µ ₂: 0.09

DCFP: double concave friction pendulum.

Figure 6.

Hysteretic loops of DCFP $(μ_{1} \neq μ_{2})$ : (a and b) Type 1, (c) Type 2, and (d) Type 3.

Conclusion

Numerical investigations have been carried out on the DCFP. A mathematical process is proposed to investigate the hysteretic loops of the DCFP influenced by the variation of the vertical loading and the unequal friction coefficients $μ_{1}, μ_{2}$ . The results of this study are summarized as follows:

The correctness of the theoretic model is examined by the finite element method, and the results show that the theoretic model has a good agreement with the data calculated by the finite element method.

The hysteretic loops of the DCFP can be influenced by the variation of the vertical loading. And, the width of the hysteretic loops changes because the varying vertical loading is correctly represented in the mathematical model.

The property of the hysteretic loops of the DCFP verifies with the changes of the frequency and the phase angle of the vertical loading.

For the DCFP with $μ_{1} \neq μ_{2}$ , the relationship between the horizontal force and the horizontal displacement no longer follows simple bilinear property, and it obeys the trilinear property to some extent. The critical displacement D* controls the slip status.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial support for this research was provided by the National Natural Science Foundation of China (Grant Nos 51878314 and 51308243) and the National Key Research and Development Plan of China (2017YFC1500705).

ORCID iDs

Fangyuan Zhou

Hongping Zhu

References

Almazan

De La Llera

Inaudi

(2015) Modelling aspects of structures isolated with the frictional pendulum system. Earthquake Engineering and Structural Dynamics 27(8): 845–867.

Barone

Calvi

Pavese

(2017) Experimental dynamic response of spherical friction-based isolation devices. Journal of Earthquake Engineering. Epub ahead of print 11 October 2017. DOI: 10.1080/13632469.2017.1387201

Christopoulos

Filiatrault

(2006) Principles of passive supplemental damping and seismic isolation. Pavia: IUSS Press.

Fenz

Constantinou

(2008) Spherical sliding isolation bearings with adaptive behavior: theory. Earthquake Engineering and Structural Dynamics 37(2): 163–183.

Fenz

Constantinou

(2010) Behaviour of the double concave friction pendulum bearing. Earthquake Engineering and Structural Dynamics 35(11): 1403–1424.

Gandelli

Penati

Quaglini

et al . (2018) A novel OpenSees element for single curved surface sliding isolators. Soil Dynamics and Earthquake Engineering 119: 433–453.

Hyakuda

Saito

Matsushita

et al . (2001) The structural design and earthquake observation of a seismic isolation building using friction pendulum system. In: Proceedings of the 7th international seminar on seismic isolation, passive energy dissipation and active control of vibrations of structures, Assisi, 2–5 October 2001.

Khoshnoudian

Hemmati

(2011) Seismic response of base-isolated structures using DCFP bearings with tri-linear and bi-linear behaviors. Procedia Engineering 14: 3027–3035.

Kim

Yun

(2007) Seismic response characteristics of bridges using double concave friction pendulum bearings with tri-linear behavior. Engineering Structures 29: 3082–3093.

10.

Malekzadeh

Taghikhany

(2010) Adaptive behavior of double concave friction pendulum bearing and its advantages over friction pendulum systems. Transaction A: Civil Engineering 17(2): 81–88.

11.

Mohammadi

Nouri

(2017) Investigation on the effect of near-fault earthquake on Double Concave Friction Pendulum (DCFP) base isolated buildings. Procedia Engineering 199: 1731–1736.

12.

Nestovito

Occhiuzzi

(2016) Implementation of smart-passive dampers combined with double concave friction pendulum devices to retrofit an existing highway viaduct exploiting the seismic early warning information. Engineering Structures 120: 58–74.

13.

Ponzo

Cesare

Leccese

et al . (2017) Shake table testing on restoring capability of double concave friction pendulum seismic isolation systems. Earthquake Engineering & Structural Dynamics 46: 2337–2353.

14.

Shah

Soni

DDP

(2017) Response of the double concave friction pendulum system under triaxial ground excitations. Procedia Engineering 176: 1870–1877.

15.

Tsai

(1995) Seismic behavior of buildings with FPS isolators. In: Proceedings of the Second congress on computing in civil engineering, Atlanta, GA, 5–8 June, pp. 1203–1211. Reston, VI: ASCE.

16.

Tsai

(1997) Finite element formulations for friction pendulum seismic isolation bearings. International Journal for Numerical Methods in Engineering 40(1): 29–49.

17.

Tsai

Chiang

Chen

(2003a) Seismic behavior of MFPS isolated structure under near-fault earthquakes and strong ground motions with long predominant periods. In: Proceedings of the 2003 ASME pressure vessels and piping conference, Cleveland, OH, 20–24 July.

18.

Tsai

Chiang

Chen

(2003b) Shaking table tests of a full scale steel structure isolated with MFPS. In: Proceedings of the 2003 ASME pressure vessels and piping conference, Cleveland, OH, 20–24 July.

19.

Tsai

Chiang

Chen

(2004) Experimental study for multiple Friction Pendulum system. In: Proceedings of the 13th world conference on earthquake engineering, Vancouver, BC, Canada, 1–6 August.

20.

Tsai

Chiang

Chen

(2005) Experimental evaluation of piecewise exact solution for predicting seismic responses of spherical sliding type isolated structures. Earthquake Engineering and Structural Dynamics 34(9): 1027–1046.

21.

Wang

Chung

Liao

(2015) Seismic response analysis of bridges isolated with friction pendulum behavior. Earthquake Engineering and Structural Dynamics 27(10): 1069–1093.

22.

Zayas

Low

Mahin

(1987) The FPS earthquake resisting system experimental report. EERC Technical Report, UBC/EERC-87/01.

23.

Zayas

Low

Mahin

(1990) A simple pendulum technique for achieving seismic isolation. Earthquake Spectra 6(2): 317–333.

Theoretical study of the double concave friction pendulum system under variable vertical loading

Abstract

Keywords

Introduction

Outline of the theoretical mechanics of the DCFP

Numerical study

Hysteretic property of the DCFP with μ 1 = μ 2

Hysteretic property of the DCFP with μ 1 ≠ μ 2

Conclusion

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iDs

References

Hysteretic property of the DCFP with $μ_{1} = μ_{2}$

Hysteretic property of the DCFP with $μ_{1} \neq μ_{2}$