**1. Introduction**

The lead rubber bearing (LRB) has the advantage of increasing a building's natural period, which is away from the seismic period range, to avoid the amplification caused by earthquakes, so it is ideal to reduce seismic loads using LRBs for high-tech factories. However, the LRB may increase environmental vibrations induced by moving vehicles and wind loads, which will damage the high-tech production. In the literature, the issue of LRBs for high-tech factories is rarely studied. However, LRBs have many references in building and bridge research and testing. Turkington et al. [1,2] demonstrated the bridge isolation design process that can be applied to all earthquakes and used the numerical simulation of LRB bridges to obtain the long-term periodic displacement and effective damping, due to LRBs, which can improve the seismic capacity of general bridges. Fujita et al. [3] conducted a base isolation test for a building and found that LRBs can effectively reduce the building response. Salic et al. [4] used LRB numerical simulation on eight-layer structures to propose that the structure increases the natural period to avoid the shortest period of earthquake damage. Kalpakidis et al. [5] proposed a theory that predicts the dependence of feature intensity and energy time to predict the behavior of LRBs to simplify the analysis. Kalpakidis and Constantinou [6] proposed the necessary conditions for reducing the LRB scaling test and the need to consider the temperature rise of the lead core. Islam et al. [7] made a multi-layer building foundation combined with finite element simulations of LRBs, suggesting that this isolation technology has the ability to survive buildings under strong earthquakes. Li et al. [8] studied the rational yield ratio of isolation system for buildings, considering the influences of total heights, yield ratios, and seismically isolated schemes, and the rational range of the yield ratio is recommended to be 2%–3%.

In a number of references, correlation studies on the e ffects of LRB parameters are used to understand the best design parameters. Warn et al. [9] studied the relationship between lateral displacement and vertical sti ffness of LRB and found that the vertical sti ffness decreases with the increase of lateral displacement. Weisman and Warn [10] conducted experiments and numerical simulations to understand the relationship between LRB critical loads and lateral displacements and found that the critical loads decrease with the increase of lateral displacements. Al-Kutti and Islam [11] proposed that LRB systems with higher characteristic strength and relatively less isolation periods behave better to reduce structural o ffset, and LRBs with lower characteristic strength and a high isolation period can control the basic shear, providing a small acceleration and low inertia. Several references investigated the biaxial interactions of LRBs, which is convenient for understanding interaction e ffects. Nagarajaiah et al. [12] considered the formula proposed by Park to simulate the biaxial interaction of LRB. Huang et al. [13] proposed a two-way simulation formula for LRBs and made some experiments to compare the uniaxial and biaxial e ffects. Abe et al. [14] conducted a biaxial test on LRBs to understand the e ffect of the torsional coupling e ffect. It is suggested that the two-axis interaction cannot be ignored. Falborski and Jankowski [15] used the experiment to verify the effectiveness of an isolation system made of polymeric bearings in reducing structural vibrations and demonstrated that the application of this bearing can significantly reduce the lateral acceleration.

Although the application of LRBs is quite mature, there is very little or probably no research that focuses on high-tech factories directly. The reason is because the equipment used to produce high-tech productions requires strict micro-vibration standards, but it is unclear whether micro-vibration will increase significantly when LRBs are installed in high-tech factories. This study thus investigates both the seismic and ambient vibrations, due to the LRB installed in the high-tech factory, using the finite element method, while the ambient vibrations are induced by the wind load and moving crane.

#### **2. Finite Element Modeling of Lead Rubber Bearings**

The LRB, as shown in Figure 1, is a single or multiple lead core built into laminated rubber to reduce structural horizontal vibration during earthquakes. Because the laminated rubber has high vertical sti ffness, low horizontal sti ffness, and high recovery and lead metal has low yield stress, combining the characteristics of the two makes the LRB a good vibration isolation device.

**Figure 1.** Illustration of the lead rubber bearing (LRB) containing a lead core and laminated rubber.

Nagarajaiah et al. [12] proposed a two-way LRB model as below:

$$\langle \mathbf{P} \rangle = \left\{ \begin{array}{c} P\_x \\ P\_y \end{array} \right\} = \alpha \frac{F\_y}{Y} \langle \mathbf{U} \rangle + (1 - \alpha) F\_y \langle \mathbf{Z} \rangle = k\_d \langle \mathbf{U} \rangle + (1 - \alpha) F\_y \langle \mathbf{Z} \rangle \tag{1}$$

$$Y\left\{\begin{array}{c}\dot{Z}\_{x}\\\dot{Z}\_{y}\end{array}\right\}=\left\{A[\mathbf{I}]-\begin{bmatrix}z\_{x}^{2}\big(\chi\operatorname{Sign}\big(\dot{\operatorname{Li}}\_{x}Z\_{x}\big)+\beta\right) & Z\_{x}Z\_{y}\big(\chi\operatorname{Sign}\big(\dot{\operatorname{Li}}\_{y}Z\_{y}\big)+\beta\Big) \\\ Z\_{x}Z\_{y}\big(\chi\operatorname{Sign}\big(\dot{\operatorname{Li}}\_{x}Z\_{x}\big)+\beta\Big) & z\_{y}^{2}\big(\chi\operatorname{Sign}\big(\dot{\operatorname{Li}}\_{y}Z\_{y}\big)+\beta\Big) \end{array}\right\}\tag{2}$$

where {*P*} = [*Px, Py*]<sup>T</sup> is the LRB force vector, {U} = [*Ux, Uy*]<sup>T</sup> is the in-plane displacement vector between the LRB two sides, α is the ratio of the final LRB stiffness over the initial LRB stiffness (α = *kd*/*ke*, *ke*= *<sup>F</sup>*y/*<sup>Y</sup>* = initial LRB stiffness), *F*y is the LRB yielding force, *Y* is the LRB deflection at the yielding force, *kd* is the yielding LRB stiffness, {Z} = [*Zx, Zy*]<sup>T</sup> is the LRB nonlinear variable, γ, β, and *A* are dimensionless parameters to control the shape of the hysteresis loop used in the two-way theory, where *A*/((γ + β)=1, and [I] is a unit vector. Equation (2) is nonlinear, and the Newton–Raphson method can be used to find {Z} using .U obtained from the finite element analysis. The details can be found in [16]. The finite element stiffness matrix of the shear force contribution is:

$$K\_{\text{shear}} = \begin{bmatrix} K\_{11} & K\_{12} \\ K\_{21} & K\_{22} \end{bmatrix} \tag{3}$$

$$\begin{aligned} \text{Where, } K\_{11} &= \frac{\partial P\_x}{\partial \Pi\_x} = a \binom{\mathbf{F}\_y}{\mathbf{Y}} + (1 - a) F\_y \frac{\partial \mathcal{Z}\_x}{\partial \Pi\_x}, K\_{12} = \frac{\partial P\_x}{\partial \Pi\_y} = (1 - a) F\_y \frac{\partial \mathcal{Z}\_x}{\partial \Pi\_y} \\\ K\_{21} &= \frac{\partial P\_y}{\partial \Pi\_x} = (1 - a) F\_y \frac{\partial \mathcal{Z}\_y}{\partial \Pi\_x}, \text{ and } K\_{22} = \frac{\partial P\_y}{\partial \Pi\_y} = a \binom{\mathbf{F}\_y}{\mathbf{Y}} + (1 - a) F\_y \frac{\partial \mathcal{Z}\_y}{\partial \Pi\_y} \end{aligned} \tag{4}$$

where ∂*Zx* ∂*Ux* , ∂*Zx* ∂*Uy* , ∂*Zy* ∂*Ux* , and ∂*Zy* ∂*Uy* , can be found in [16]. Equations (3) and (4) produce an unsymmetrical global stiffness matrix, which may cause the double requirement of computer memory and time. Thus, one can empirically set *K*21 = *K*12 = (*<sup>K</sup>*12 + *<sup>K</sup>*21)/2, and use the Newton–Raphson method to obtain the solution with small unbalance forces. The solution is still accurate, since equation (1) is used to find the LRB internal force vector without errors, but the Newton–Raphson iterations may increase when one direction is loading and the other is unloading. For the vertical direction of the LRB stiffness (*Kv*), a linear spring is used. The original LRB hysteretic curve under low speed loads can be modified as the functions of the wave frequency, wave speed, and axial load [17,18]. For simplicity, we used the original LRB model for finite element analyses.

Laboratory experiments were conducted to find the LRB characteristics at a vertical compressive force of 6300 kN and a maximum horizontal displacement of 0.149 m. The experimental results are shown in Figure 2, plotted as the dot line, where the LRB calibrated material properties are *K*e = *Kd*/α = 1.9 × 10<sup>5</sup> kN/m (initial stiffness), *Kv* = 5 × 10<sup>7</sup> kN/m, *F*y = 370 kN, α = 0.0288, β = 0.1, and γ = 0.9 based on equation (1). Finite element analysis using the proposed LRB element mentioned above was then performed to find the hysteresis curve, as shown in the black line in Figure 2. This figure indicates a good agreemen<sup>t</sup> between the finite element analysis and experimental result.

**Figure 2.** Experimental and finite element results for the LRB hysteresis curve.

#### **3. High-Tech Factory and Finite Element Model**

#### *3.1. Illustration of the Structure of the High-Tech Factory*

Before explaining the studied factory, we first briefly introduce the micro vibration standards in the high-tech industry. Gordon [19] recommended the vibration criterion (VC) for high-tech factories, where five levels include VC-E to VC-A under the velocity vibration at the floor slab from 42 dB to 66 dB with the increment of 6 dB, where the dB calculation can be found in the references [19,20]. The studied high-tech factory located in southern Taiwan is a three-story building mainly used for producing photovoltaic panels, where the first level is the VC-C reinforced concrete (RC) structure, the second level is the VC-B steel structure, and the third level is the VC-A steel structure. Intensive RC columns are used in the first level to avoid environmental vibration, while large span truss frames are used in the second and third levels to achieve greater production space. It is noted that the studied factory has no currently installed LRBs, and we use it to perform the seismic and micro vibration analyses with and without LRBs. Figure 3 shows the two typical frames in the X and Y direction. In the RC level, the column span is 6 m in the X direction and 5.2 m in the Y direction, where there are 71 and 27 column lines in the X and Y directions, respectively. As shown in Figure 3, the RC columns connected to the steel frame have a big square size of 1.5 m, and others are 0.6 m. For the two steel levels, the column span is 12 m in the X direction and 32 m in the Y direction, while the section properties are shown in Table 1. The thickness of the RC slab is 0.725, 0.55, and 0.45 m for the first to third level, respectively, and the main purpose of thick slabs is to reduce ambient vibration. For the two steel levels, the properties of steel sections are listed in Table 1, where columns are the box section and others are the H-shape section. This high-tech factory used pile foundations of 28 m length to avoid excess environmental vibration, while the reversed circulation piles with two di fferent sections were constructed, and one is the diameter of 1.5 m connected to the big columns and the others are the diameter of 0.6 m connected to other columns. The soil profile contains 10 m inorganic clays of medium plasticity (undrained shear strength *Su* = 50 kPa), 5 m sandy soil (submerge internal frictional angle of sand φ = 33◦), 10 m clay of hard plasticity (*Su* = 150 kPa), and the rest is very hard sand (φ = 37◦). We used the axial forces from columns to select appropriate LRBs, where two types of LRBs were used at the top of piles. As shown in Figure 3, the first type, named LRB1, was used to connect with the big columns, and the second type, named LRB2, was used to connect to other columns, where Table 2 shows the material properties of the two types of LRBs.

(**a**) X-direction section.

**Figure 3.** *Cont.*

**Figure 3.** Two typical frames of the high-tech factories.

**Table 1.** The steel sections in the second and third levels of the high-tech factory, as shown in Figure 3 (A572 steel with Fy = 345 MPa).


**Table 2.** LRB material properties (= 0.1, and = 0.9 based on Equation (1)).


#### *3.2. Finite Element Model*

The finite element program from reference [21] was used in the finite element analysis, where the LRB element mentioned in Section 2 has been added into this program. The three-dimensional (3D) finite element mesh is shown in Figure 4 with the total number of degrees of freedom of 1,849,662 and 695,643 elements, where the high-tech factory, warrior slabs, crane, and rail system are included. Although the finite element is complicated, the major part of the mesh is modelled using 2-node 3D beam elements, such as beams, columns, piles, and crane rails of the factory, and the end released moments of beam elements are used to model truss members. Waffle slabs are simulated using 2-node 3D beam elements with 0.75 m interval, 0.4 m width, and 0.75, 0.55, and 0.45 m depth on the first, second, and third floor slabs, respectively, where the 0.18 m rigid zone at two beam ends is set. The slabs at the truss bottom on the second and third steel stories are modeled using 4-node plate elements with a thickness of 0.15 m. The soil-structure interaction is modelled using the API p-y, t-z, and Q-z nonlinear soil spring elements [22], where one end of these elements are connected to the beam element nodes of piles, and the other nodes are applied to the time-history seismic displacements for the earthquake load. If LRBs are included, the LRB elements mentioned in Section 2 are generated between foundation beams and the top of piles, as shown in Figure 3. The Rayleigh damping was used

in the finite element analysis, where the mass damping equals 0.3/s and the stiffness damping equals 0.0003 s, which gives approximately 4% damping ratio at frequencies of 0.6 and 40 Hz, respectively.

**Figure 4.** Three-dimensional (3D) Finite element mesh containing the high-tech factory, slabs, rails, and crane (The big and small columns are connected to LRB1 and LRBs, respectively, as shown in Table 2).

The rail and crane system on the second level, as shown in Figure 4, contains two steel rails with the properties of the axial area of 0.17 × 10−<sup>2</sup> m2, *Ix* of 0.19E-4 m4, and *Iy* of 0.6 × 10−<sup>4</sup> m4. The 2-node 3D beam element is used to simulate rails supported by the 1.3 m interval springs with the stiffness of 4.8 × 10<sup>5</sup> kN/m and the damping of 10 kN-s/m between rails and slabs. Two slave nodes, labeled as node S in Figure 4, are controlled by the master node at the beam center at each support section, while a number of slave nodes W at the rail top are set for the route of moving wheel elements. Thus, the crane finite element model can be moved on the rails which are connected to the slab of the high-tech factory. The crane, as shown in Figure 4, is generated using a beam, spring-damper, and moving wheel elements [20] with the mass of five tons. Except the API soil spring, LRB, and moving wheel elements, other finite elements are linear elastic. The consistent mass method, Newmark's integration method with the average acceleration, and the Newton–Raphson method were used to solve this nonlinear problem with a time step of 0.005 s and a simulation of 20,000 time steps for wind loads and 10,000 time steps for other loads. The finite element analysis contains two stages, where the first stage is the static analysis under the dead weight load using a step, and the second stage is the time-history analysis using 10,000 or 20,000 time steps. It is noted that a comparison against sensors' measurements under both wind-induced and crane-induced vibrations was reported in [20,23] to validate the accuracy of the finite element analysis.

#### *3.3. Illustration of Seismic Loads*

The artificial earthquake generation software Simqke [24] was used to generate the time-history seismic acceleration using the spectrum from IBC 2006 [25], as shown in Figure 5. The peak ground accelerations (PGA) of 0.25, 0.28, 0.32, 0.36, and 0.40, respectively, were used for five seismic loads in the global *X* direction with *Ts* (Figure 5) of 0.6 s, where one group is shown in Figure 6. Moreover, the important parameter *Ts* representing the dominant frequency of seismic loads, as shown in Figure 5, was set to 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, and 1.4 s, respectively, with the PGA of 0.32 g for nine seismic loads in the global *X* direction. For the other two directions, 70% and 30% of that PGA in the global *Y* and *Z* directions, respectively. This three-direction seismic accelerations are applied on the ground surface. We used ten soil layers with the interval of 5 m for the SHAKE 91 [26] input data. The SHAKE 91 program is then used to generate the acceleration field in each soil layer. Finally, the integration to obtain the displacement fields, which are applied to the node of each p-y, t-z, and Q-z curve elements for the seismic simulation.

**Figure 5.** Seismic response spectrum according to IBC 2006 [25].

**Figure 6.** *Cont.*

**Figure 6.** Artificial time-history seismic acceleration for Ts = 0.6 and peak ground accelerations (PGAs) of 0.36, 0.252, and 0.108 g in the local X, Y, and Z directions.

#### **4. Parametric Study Using LRBs in High-Tech Factories**
