Keywords

1 Introduction

For the seismic isolation bridges, seismic isolation bearings are usually set in the bridge, and the natural vibration period of the structure is extended and the damping is increased to consume seismic energy and reduce the seismic response of the structure, while it also satisfies the functional requirements in normal servicing condition. Therefore, the rationality and reliability of the seismic isolation device are the key factors to meet the seismic requirements of the bridge. In order to timely and accurately identify the seismic risk sources and risk factors, and assess the seismic safety of seismic isolation bridges, the service station, function and performance level of seismic isolation devices are the main investigation and evaluation contents [1,2,3,4,5]. At present, the existing inspection standards on technical condition of bridges lack of the contents on isolation devices, in which the apparent state information is the main contents and the relevant quantitative indicators is lack [6,7,8].

The FPB is as the common typical seismic bearing in seismic isolation highway bridges in China. It uses the pendulum principle to realize the seismic absorption function by discharging seismic energy at the sliding interface friction, and realizes the seismic isolation function by extending the movement period of the beam through spherical swing. Researchers have carried out experimental and theoretical studies on the mechanical properties of different types of FPB [9,10,11], and the results show that the geometric configuration and friction characteristics of the sliding interface are the main factors affecting the function of seismic reduction and isolation, and are the key parameters for the seismic design and the design mechanical index of FPB. For the friction characteristics of the sliding interfaces, the sensitive parameters include surface pressure, sliding velocity, temperature, loading time, cumulative motion, roughness of the stainless steel surface, lubrication, wear, etc., and theoretical calculation models of the friction characteristics under the influence of different factors is proposed [12,13,14,15,16,17]. However, the ideal assumption method is usually used to determine the effective stiffness and friction parameters, which can result in great uncertainty of performance analysis results of the service seismic isolation devices and bridges.

In order to analyze the mechanical performance level under the effect of interface sliding friction state, it is further to improve the seismic safety of the serviced seismic isolation bridges, the FPB is as the research subject. Starting with theory calculation models of mechanical performance and interface friction characteristics, the testing and finite element numerical analysis methods are applied to analyze the mechanical performance of FPB under influence of different sensitive factors. It is to identify the correlation between service status and interface friction characteristics, mechanical performance, which is further to timely and accurately identify the seismic risk sources, risk factors and assess the service status.

2 Theoretical Calculation Model of FPB and Friction Characters of Sliding Interface

The load-displacement hysteresis curve for FPB is shown in Fig. 1.

Fig. 1.
figure 1

The load-displacement hysteresis curve

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The equivalent stiffness Keff of the bearing is calculated as follows:

$$ K_{eff} = \left( {\frac{1}{R} + \frac{\mu }{D}} \right)W $$
(1)

where: D is the design displacement (mm); R is the equivalent radius(mm); μ is the coefficient of dynamic friction (recommended value 0.05); W is the vertical load (kN).

The equivalent self-oscillation period Te of the bearing is calculated as follows:

$$ T_e = 2\pi \sqrt {{\frac{W}{{K_{eff} g}}}} $$
(2)

The equivalent damping ratio ξe of the bearing is calculated as follows:

$$ \xi_e = \frac{E_D }{{2\pi K_{eff} D^2 }} = \frac{4\mu WD}{{2\pi K_{eff} D^2 }} $$
(3)

The isolation period T of the bearing is calculated as follows

$$ T = 2\pi \sqrt {\frac{R}{g}} $$
(4)

Based on the above theoretical mechanical calculation model of FPB, and considering the time-varying nature of influencing factors of FPB, it can be seen that the main factors affecting its seismic performance are the sliding interface friction coefficient.

For the sensitive parameters of sliding interface friction characteristics and their theoretical calculation models, Mokha et al. [12], and Constantinou et al. [13] proposed different theoretical formulas for sliding friction coefficient with sliding speed and compressive stresses. In order to determine the sliding interface state under earthquake load, based on the classical Coulomb theory, researchers gave the discriminant conditions for the state of the sliding interface under the simple harmonic of acceleration time history, but the applicability of the theory needs to be further explored and analyzed.

During normal operational service, the sliding speed of the isolation sliding bearing is low, but it can produce certain wear under cyclic sliding displacement. A simple wear model is used to predict the wear of the bearing [14] as follows:

$$ h_s = K_s \cdot P \cdot d $$
(5)

Where: hs is the depth of wear; Ks is the wear coefficient; P is the bearing pressure; and d is the sliding distance. The wear coefficient of the sliding surface of FPB is approximately 5 × 10−10MPa−1.

The sliding interface shows the wear phenomenon with the increase of cumulative distance, and then the correlation between the friction coefficient and the cumulative distance under a constant pressure is proposed [14]:

$$ \left\{ \begin{gathered} \mu = A + B\ln (D + C) \hfill \\ \mu = A + B\ln (D_s + C) \hfill \\ \end{gathered} \right. $$
(6)

Where: A, B, and C are constants; D is the cumulative movement distance; Ds is the critical point of cumulative movement distance.

3 Finite Element Analysis of Mechanical Properties Change Rule of FPB

3.1 Three-Dimensional Solid Finite Element Model of Bearing

According to the geometrical configuration parameters (Fig. 2 and Table 1) of FPB3000-ZX-e150-0 adopted for the test, Abaqus finite element software is used to establish the corresponding three-dimensional solid finite element model as shown in Fig. 3.

Fig. 2.
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Geometrical configuration of FPB

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Fig. 3.
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Split diagram of FPB numerical model

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Table 1. Geometric configuration specific parameters of FPB3000-ZX-e150-0
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In the finite element numerical model of the FPB, the property parameters of the materials used are shown in Table 2. Stainless steel seating plate adopts the double line stress-strain intrinsic model. Polytetrafluoroethylene (PTFE) adopts the ideal elastic-plastic model, and its stress-strain relationship curve is shown in Fig. 4. Face-to-face contact is used between upper seating plate and PTFE plate, middle seating plate and PTFE plate, with penalty function in the tangential direction and “hard” contact in the normal direction. The friction contaction element is applied to simulate the changing of friction characteristics. For middle seating plate and PTFE plate, the join connection unit is used.

Table 2. Material properties of FPB
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$$ \left\{ \begin{aligned} & \sigma = E\varepsilon \;\;\;\;\;\;\;\;\;\;\;\;\;|\varepsilon | \le \varepsilon_S \\ & \sigma = \sigma_S {\text{sgn}} (\varepsilon )\;\;\;\;|\varepsilon | > \varepsilon_S \\ \end{aligned} \right. $$
(7)
Fig. 4.
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Stress-strain relationship curve of PTFE

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Fig. 5.
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Horizontal displacement loading curve

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According to the loading mode of quasi-static test, constant vertical load is applied on the top of the bearing, and horizontal unidirectional reciprocating displacement \({\text{d}} = {\text{dx}}\sin (2\pi {\text{ft}})\) is applied. The horizontal displacement curve of FPB is shown in Fig. 5. The compressive stress is 2MPa, 4MPa, and 6MPa respectively.

3.2 Test Calibration of Three-Dimensional Solid Finite Element Numerical Model of Bearing

Test Scheme

The Double Spherical Seismic Isolation Bearing (DSSIB) is adopted in this test, and the model number FPB3000-ZX-e150-0. The design parameters of the test bearing are shown in Table 3.

Table 3. Design parameters of the test bearing
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Using the proposed Pseudo-static experimental to simulate the cycling laoding and cumulative movement displacement, the schematic loading diagram and the field loading test diagram of DSSIB are shown in Fig. 6. Under the constant vertical load, the displacement loading system was adopted in the horizontal one-way, and the loading rules follow \({\text{d}} = {\text{dx}}\sin (2\pi {\text{ft}})\), with each level of horizontal displacement cyclic loading of 3 revolutions. The vertical compressive stresses are 2 MPa, 4 MPa and 6MPa, respectively, and the corresponding vertical loads were 884 kN, 1768 kN and 2652 kN, respectively, corresponding to test conditions P-1, P-2 and P-3, respectively. Throughout the process of the proposed static test of DSSIB, the horizontal displacements and the horizontal forces of the top plate under the bearing are measured and extracted.

Test and Simulation Results Comparison of the Mechanical Properties

The P1, P2, and P3 (corresponding to the vertical compressive stress 2 MPa, 4 MPa, and 6 MPa, respectively), the comparison between the test and simulation results of the fore -displacement hysteric curve is shown in Fig. 6. It shows that the overall shape of the hysteresis curve, residual displacement, lateral stiffness are basically the same, especially the horizontal displacement is less than 120 mm, and they all show the shape of “parallelogram”, which is consistent with the theoretical hysteresis curve model (Fig. 1). To further validate the reliability of the simulation results under significant horizontal displacements, the calculated results of the equivalent period and equivalent damping ratio of the bearing under the horizontal displacement amplitude of 120 mm are compared, as shown in Table 4. As shown in Table 4, the error between the test and the simulated value of the equivalent period is less than 5%, and the maximum error of the equivalent damping ratio is 5.2%. It can be seen that the evolution law of the mechanical properties of FPB can be analyzed reliably by the three-dimensional solid finite element model of FPB.

Fig. 6.
figure 6

Comparison between hysteresis curve test results and simulation results

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Table 4. Comparison of experimental results and simulation results of equivalent period and equivalent damping ratio
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3.3 Finite Element Analysis of the Sensitive Parameters of the Mechanical Properties of FPB

Taking into account the correlation of the sensitive parameters of its sliding interface state, the change law of the mechanical properties of FPB under the influence of factors such as compressive stress, sliding speed, cumulative moving distance and inconsistent sliding interface state is deeply analyzed, which provides data support for the performance evaluation of FPB in the whole life.

3.4 Compressive Stress

Based on the existing correlation between the friction coefficient and the compressive stress [15], the analysis cases of the friction coefficient of the sliding interface under different compressive stresses are shown in Table 5.

Table 5. Analysis cases of compressive stress parameters
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According to the calculation theory of basic mechanical properties of the FPB, combined with the force - displacement hysteresis curve of the bearing (Fig. 7a), the comparison results of mechanical properties of the bearing under different compressive stresses are shown in Table 6. As can be seen from Table 6, with the increase of compressive stress, the equivalent horizontal stiffness, total energy consumption and equivalent period gradually increase, in which the equivalent horizontal stiffness increases exponentially. However, the damping ratio decreases with the increase of compressive stress, and tends to be stable gradually.

Fig. 7.
figure 7

Force-displacement curves companions of the bearing

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Table 6. Comparison results of the mechanical properties of the bearing under different compressive stresses
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  1. (2)

    Sliding speed

According to the correlation relationship between the friction coefficient and the sliding speed [12, 13], the friction coefficient of the sliding interface of the bearing under different sliding speeds is determined, and the corresponding analysis conditions are shown in Table 7.

Table 7. Analysis conditions of sliding speed parameters
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According to the calculation theory of the basic mechanical properties of FPB, combined with the force - displacement hysteresis curve of the bearing (Fig. 7b), the calculation results of the mechanical properties of the bearing under different sliding speeds are shown in Table 8. As shown in Table 8, with the increase of sliding speed, the maximum variation ranges of equivalent stiffness, total energy consumption, equivalent damping ratio and equivalent period are 20.3%, 33.0%, 11.1% and 8.9%, respectively.

Table 8. Calculation results of the mechanical properties under different sliding speeds
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  1. (3)

    Cumulative movement distance

The existing research results show that the friction coefficient gradually increases with the increase of cumulative movement distance, until it tends to be stable [14]. Assuming that the critical point of the cumulative movement distance is 1500 m, the friction coefficient correspond to 0.062 and 0.034 under the conditions of vertical compressive stress 2 MPa and 6 Mpa respectively, and the corresponding analyzed conditions are shown in Table 9.

Table 9. Analysis conditions of cumulative movement distance parameters
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Combined with the force-displacement hysteresis curve of the bearing (Fig. 8c), the calculation results of the mechanical properties of the bearing under different cumulative movement distances are shown in Table 10. As shown in Table 10, when the cumulative movement distance reaches the critical point, compared with p-2, the equivalent stiffness, the total energy dissipation value, the equivalent damping ratio increases by 26.2%, 49.2%, 18.7% respectively, and the equivalent period decreases by 11.2%; compared with p-6, the equivalent stiffness, the total energy dissipation value, the equivalent damping ratio increases by 23.7%, 39.8%, 13.5%respectively, and the equivalent period decreases by 10.2%.It can be seen that the cumulative movement distance can affect its mechanical properties, especially in the case of smaller compressive stress.

Table 10. Calculation results of the mechanical properties of the bearing under different cumulative movement distances
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  1. (4)

    Non-consistent sliding interface state

DSSIB contains upper and lower sliding interfaces, two sliding interfaces, the upper and lower sliding interfaces are in different states. Therefore, the corresponding analysis conditions are formulated as shown in Table 11.

Table 11. Analysis conditions of different sliding interface condition parameters
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Combined with the force-displacement hysteresis curve of the bearing (Fig. 8d), the calculation results of the mechanical properties of the bearing under the non-consistent interface condition are shown in Table 12. Compared with vp-1, with the increase of μdown, the equivalent stiffness, the total energy dissipation, the equivalent damping ratio maximum increases by13.8%, 37.7%, 11.7% respectively; but the equivalent period maximum decreases by6.1%. With the increase of μup, the equivalent stiffness, the total energy dissipation, the equivalent damping ratio maximum increases by103.2%, 247.7%, 58.1% respectively, and the equivalent period maximum decreases by 29.7%. This shows that the inconsistent sliding interface conditions can significantly affect the performance of FRP, especially for μup.

Table 12. Calculated results of the mechanical properties of the bearing under the condition of non-uniform sliding interface
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4 Conclusion

Starting with the interface friction characteristics and mechanical performance of FPB, the testing and finite element numerical analysis methods are applied to analyze the mechanical performance of FPB under influence of different sensitive factors. The conclusions are as follows:

  1. (1)

    According to the geometric configuration and analysis essentials of the FPB, the corresponding three-dimensional solid refined finite element numerical model of the support is established. Based on the theoretical calculation model of the mechanical properties of the friction pendulum support, the maximum deviation of the simulation results and the test results of the horizontal force-displacement hysteresis curve, equivalent period and equivalent damping ratio of the friction pendulum support is 5.2%, which verifies the rationality and reliability of the three-dimensional solid refined finite element numerical model.

  2. (2)

    Furthermore, the finite element numerical simulation method and the theoretical calculation formula of the interface friction coefficient are used to analyze the evolution of key mechanical parameters such as the equivalent stiffness, total energy dissipation, equivalent period and equivalent damping ratio of the FPB. The analysis results show that compressive stress is the most sensitive factor affecting the mechanical properties of FPB, with the increase of compressive stress, the equivalent horizontal stiffness increases exponentially, but the bearing damping ratio decreases. If the compressive stress is constant, the cumulative sliding distance and sliding velocity can obviously affect the performance of the FPB, but the variation range of the equivalent damping ratio is less than 10%. In addition, when the friction characteristics of the upper and lower sliding interfaces of the FPB are not consistent, the condition of the up sliding interface has a more significant impact on the mechanical properties.

In summary, the interface friction state can significantly affect the mechanical properties of FPB. According to the mechanical properties evolution law of the FPB under different interface friction states, it provided the data and theoretical basis for further qualitative evaluation of the service performance level of the serviced FPB, and lays a foundation for the evaluation of the seismic performance of the in-serviced seismic isolation bridge. In addition, in order to better clarify the applicability of the theoretical calculation model of friction coefficient, the verification and applicability of the theoretical model will be further studied in combination with the model test method.