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An analytical model is presented for seismic analysis of triple friction pendulum bearings and validated using 81 bearing tests, each subjected to three cycles, with a duration of 12 seconds and using 250, 200 and 100 tons vertical loads. The main objective is to develop formulas for bilinear behavior using maximum, average and minimum friction coefficients to check which is the closest to the real behavior in the laboratory tests and comparatives curves plotting to observe the standard derivation. Parameters such as friction coefficients, effective stiffness, damping factor and vibration periods are analyzed to understand the structural behavior of the TPF bearings.

Recent earthquakes have shown that, even though modern codes have limited damage to structural elements, there are significant losses in the non-structural components [

The base isolation devices are of two main types: elastomer and friction based [

The frictional devices are classified into three types: simple, double and triple concave friction pendulum bearings. The scientific research continues and a new device called fifth friction pendulum has just appeared [

It is noteworthy that despite the advantages and existing applications [

This paper will focus on the triple friction pendulum TFP bearings, since isolation devices of this type will be placed in the new research center of the Universidad de las Fuerzas Armadas-ESPE. These devices combine friction with restoring forces created by the skin characteristics and geometry of the surface plates [

Double and triple frictional devices are called second and third generation devices respectively, and have some advantages over the first generation, such as: more compact, able to adapt its performance relative to demand, increased displacement capacity and lower speed in the movement, which prevents excessive variation in the friction coefficients. Another notable aspect of the second and third generation devices is the reduction of structural responses, thereby improving the performance of nonstructural components and elements [

The TFP bearings are constituted by an inner device with radius plates R_{2}, R_{3}, and by an exterior device with radius plates R_{1}, R_{4}. So that it really has two isolation devices instead of one. This allows having smaller dimensions with respect to the first and second generation and having greater displacement capacity [

In the Universidad de las Fuerzas Armadas-ESPE, six buildings are being constructed with TFP type FPT8833/12-12/8-6 bearings, as shown in

loads 250, 200 and 100 tonf (EPS 2015) (Earthquake Protection Systems, Mare Island, Vallejo, California 94592-USA).

The curvature radius of the outer and inner plates of the TFP bearings may be different as well as the heights h_{i}, for i = 1:4. Thereby, the displacement capacity d_{i}, may be different too. In this way, there could be up to 12 geometric conditions and 4 different friction coefficients μ_{i} in each of the plates. In this case the five-step model proposed by [

Now, in the case of the FPT8833/12-12/8-6 bearing the geometric conditions are reduced to 6 because the radius of curvature of the outer plates is equal. The same happens with the radius of the inner plates. In addition, this bearing has similar heights as shown in

where R_{i} is the curvature radius; h_{i} is height; R_{i}_{,eff} is effective radius of curvature;

Relative displacement occurs between plates 2 and 3.

where u is the lateral displacement of the bearing; F is the applied lateral force; w is the weight applied on the bearing. To the left of

The pillow block inside the two interior plates reaches the stops and surfaces 1 and 4 start adding displacement. Normally, it is in this regime that the bearing works under an earthquake of moderate and high intensity. The governing equations are shown below. The corresponding hysteresis curve is presented in

This regime occurs when the earthquake is extremely strong and the inner plates meet the outer stops. In these conditions, the inner pillow block begins to slide on surfaces 2 and 3. The equations are shown below. The corresponding hysteresis curve is presented in

The proposed model works for Regime II. But it can also be applied to Regime I. It differs from the model proposed by [

In

Now, it is proposed, as can be seen in

where μ_{1}, μ_{2}, are friction coefficients in the inner and outer plates respectively; μ is the equivalent friction coefficient.

The equations that define the bilineal model are:

where W is the vertical load on the bearing; q is the lateral displacement in the bearing, calculated in iterative form; ξ_{eq} is the equivalent damping factor; k_{ef} is the effective or secant stiffness; T is the bearing period; g is the acceleration due to gravity.

In

The bearings were initially not centered due to shakings during their transport, so a first manual load cycle is needed to re-center the bearing (

Finally, the curve that best fits the three loading cycles is calculated. Then, the friction coefficients are determined using the five regimes model of [

In _{1} corresponds to the use of the inner surfaces coefficient of friction

μ_{2}; f_{2}, f_{3}, to the use of the outer surfaces coefficients of friction μ_{1} μ_{4}. These coefficients are determined experimentally. Using the five regimes model, EPS (2015) calculated the effective stiffness k_{ef}, the equivalent damping factor ξ_{eq} and the vibration period T associated to a lateral displacement of 12''.

In this paper, the same parameters that EPS calculated using the five-regime model are determined for the bilineal (proposed) model. They are: the effective stiffness, the equivalent damping factor and the vibration period associated to a lateral displacement of 12''.

The database proportioned by EPS (2015) included the friction coefficients in each hysteresis cycle as well as their mean values for 81 bearings. It is important to note that 61 bearing were tested with a vertical load of 250 tonf, 10 additional bearings with a vertical load of 200 tonf and the remaining 10 bearings with a load of 100 tonf. Three types of hysteresis curves were obtained, one for mean, one for maximum and other for minimum friction coefficient values.

In

In

250 (61 tests), 200 (10 tests) and 100 tonf (10 tests) are presented. It is noted that the equivalent damping factor found with the proposed model is slightly greater than that found experimentally.

In

Two points of interest are presented here, the experimental and the proposed model variations of the effective stiffness, equivalent damping factor and the vibration period. For this purpose in Tables 1-3 mean values and standard deviation data of Figures 11-16 are presented.

Values | Model | W = 250 T. | W = 200 T. | W = 100 T. | |||
---|---|---|---|---|---|---|---|

Average | Experimental | 101.49 | 3.12 | 83.02 | 1.15 | 45.91 | 0.97 |

Proposed | 99.92 | 2.96 | 81.67 | 1.02 | 44.62 | 1.13 | |

Maximum | Experimental | 106.32 | 3.11 | 86.88 | 1.37 | 47.61 | 0.92 |

Proposed | 105.51 | 3.49 | 85.87 | 1.07 | 46.57 | 0.76 | |

Minimum | Experimental | 97.95 | 2.92 | 79.98 | 1.22 | 44.69 | 1.02 |

Proposed | 95.60 | 3.26 | 78.45 | 1.50 | 43.18 | 1.56 |

Values | Model | W = 250 T. | W = 200 T. | W = 100 T. | |||
---|---|---|---|---|---|---|---|

Average | Experimental | 0.25 | 0.0103 | 0.2634 | 0.0043 | 0.29 | 0.0062 |

Proposed | 0.27 | 0.0109 | 0.2864 | 0.0045 | 0.31 | 0.0089 | |

Maximum | Experimental | 0.27 | 0.0089 | 0.2788 | 0.0036 | 0.30 | 0.0042 |

Proposed | 0.29 | 0.0116 | 0.3037 | 0.0044 | 0.33 | 0.0051 | |

Minimum | Experimental | 0.24 | 0.0099 | 0.2527 | 0.0053 | 0.28 | 0.0093 |

Proposed | 0.26 | 0.0135 | 0.2718 | 0.0073 | 0.30 | 0.0133 |

Values | Model | W = 250 T. | W = 200 T. | W = 100 T. | |||
---|---|---|---|---|---|---|---|

Average | Experimental | 3.12 | 0.0443 | 3.08 | 0.0222 | 2.93 | 0.0297 |

Proposed | 3.17 | 0.0471 | 3.14 | 0.0195 | 3.00 | 0.0384 | |

Máximos | Experimental | 3.17 | 0.0499 | 3.14 | 0.0220 | 2.97 | 0.0349 |

Proposed | 3.08 | 0.0509 | 3.06 | 0.0191 | 2.94 | 0.0239 | |

Mínimos | Experimental | 3.04 | 0.0441 | 3.01 | 0.0226 | 2.88 | 0.0465 |

Proposed | 3.24 | 0.0559 | 3.20 | 0.0306 | 3.05 | 0.0554 |

It is noted that the proposed bilinear model is consistent and provides an estimate of the response of the structure, which could be compatible with the comments provided by McVitty and Constantinou.

The proposed model is validated with experimental data provided by EPS, based on the TFP bearings used in the New Research Center at Universidad de las Fuerzas Armadas-ESPE.

Effective stiffness, damping and vibration periods using the proposed model with maximum, minimum and average friction coefficients values show that the bilinear analytical model is compatible with the experimental results.

Delgado, I., Aguiar, R. and Caiza, P. (2017) Bilinear Model Proposal for Seismic Analysis Using Triple Friction Pendulum (TFP) Bearings. Open Journal of Civil Engineering, 7, 14- 31. https://doi.org/10.4236/ojce.2017.71002