# Study of Lead Rubber Bearings for Vibration Reduction in High-Tech Factories
## 3 High-Tech Factory and Finite Element Model
### 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 \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Member** & **Axis** & **Label** &
\begin{tabular}{c} **Member Dimensions (mm)** \\ **d\(\times\)b\(\times\)tw\(\times\)tf** \\ \end{tabular} \\ \hline Column & ALL & C1 & \(\text{BOX}\ 900\times 900\times 35\times 35\) \\ \hline Braces & X & B1 & \(\text{RH}\ 400\times 400\times 13\times 21\) \\ \hline Braces & Y & B2 & \(\text{RH}\ 414\times 405\times 18\times 28\) \\ \hline Braces & Y & B3 & \(\text{RH}\ 428\times 407\times 20\times 35\) \\ \hline Girder & X & G1 & \(\text{RH}\ 588\times 300\times 12\times 20\) \\ \hline Girder & Y & G2 & \(\text{RH}\ 375\times 200\times 10\times 25\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The steel sections in the second and third levels of the high-tech factory, as shown in Figure 3 (A572 steel with \(\text{Fy}=345\) MPa). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Name** & **Model/Parameter** & \(\mathbf{K_{e}(kN/m)}\) & \(\mathbf{F_{y}(kN)}\) & \(\mathbf{\alpha}\) & \(\mathbf{K_{v}(kN/m)}\) \\ \hline LRB1 & EIRL-G4-1000-170 & 1.9732\(\times 10^{4}\) & 196.0850 & 0.0769 & 4.660\(\times 10^{6}\) \\ \hline LRB2 & EIRL-G4-700-130 & 1.4058\(\times 10^{4}\) & 114.8299 & 0.0769 & 3.259\(\times 10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LRB material properties (= 0.1, and = 0.9 based on Equation (1)). Figure 3: Two typical frames of the high-tech factories. 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. 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 \(0.17\times 10^{-2}\) m\({}^{2}\), \(I_{x}\) of 0.19E-4 m\({}^{4}\), and \(I_{y}\) of \(0.6\times 10^{-4}\) m\({}^{4}\). The 2-node 3D beam element is used to simulate rails supported by the 1.3 m interval springs with the stiffness of \(4.8\times 10^{5}\) 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.