# Virtual Scene Construction for Seismic Damage of Building Ceilings and Furniture
## 3 Technical Method
### Simulation of the Movement of Damaged Nonstructural Components  
#### 3.3.1 Suspended Ceilings  
To simulate the subsequent movement of damaged ceilings, four steps need to be completed: (a) determine the simulation scenarios, including the number and positions of falling tiles; (b) create a physical model of the falling tiles for the simulations; (c) validate the physics model and determine its key coefficients; and (d) implement the simulations using a physics engine. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component** & **Typical PG ID** &
\begin{tabular}{c} **The Required** \\ **EDP** \\ \end{tabular} & **DS** & **Fragility Curves** \\ \hline \multirow{4}{*}{Suspended ceiling} & \multirow{4}{*}{C3032.001a} & \multirow{4}{*}{PFA} & DS1: 5\% of ceiling & \multirow{4}{*}{Figure 2} \\  & & & & grid and tile \\  & & & & \\  & & & & grid and tile \\  & & & & DS3: 50\% of ceiling \\  & & & & grid and tile \\  & & & & damage \\ \hline \multirow{4}{*}{Movable furniture} & \multirow{4}{*}{E2022.001} & \multirow{4}{*}{PFA} & DS1: Wall units & \multirow{4}{*}{Reference [22]} \\  & & & & need to be adjusted \\ \cline{1-1}  & & & & and straightened. \\ \cline{1-1}  & & & & \\ \cline{1-1}  & & & & \\ \cline{1-1}  & & & & \\ \cline{1-1}  & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Typical fragility data for suspended ceilings and movable furniture  
Figure 2: Fragility curve for the suspended ceiling. 1. Simulation scenarios  
The number and positions of falling tiles are critical for determining the simulation scenario for suspended ceilings. The area ratio of falling tiles can be determined for each DS according to FEMA P-58. Taking a typical suspended ceiling as an example (the corresponding PG ID is C3032.001a), the area ratios of falling tiles are listed for three DSs in Table 1. Owing to the complex interactions between ceiling tiles during an earthquake, the positions of falling tiles are random [36; 37]. Therefore, the number of falling ceiling tiles can be determined by the corresponding ratio from the DS, while their positions can be assigned randomly in the simulation scenario. 2. Physical model of falling tiles  
When a suspended ceiling suffers seismic damage, the tiles will separate from the skeleton of the ceiling and fall to the floor. As shown in Figure 3, the typical falling process for a separated tile can be divided into four stages. In the first stage, the tile has a horizontal projectile motion with an initial velocity. The trace of the tile in the \(x\)-direction can be calculated with Equation (1)  
\[x=v_{0}t \tag{1}\]  
where \(v_{0}\) is the initial velocity, which is equal to the floor velocity when the tile separates from the ceiling and can be obtained from the seismic response results of the THA. Note that the air drag force (denoted as \(f_{y}\)) in the vertical direction cannot be ignored because the area of the vertical face of a tile generally is much larger than that of the other faces, which can be calculated using Equation (2)  
\[f_{y}=\frac{1}{2}c\rho sv_{y}^{2} \tag{2}\]  
where \(c\) and \(\rho\) are the drag coefficient and the density of the air (i.e., 1.29 kg/m\({}^{3}\)), respectively; \(s\) and \(v_{y}\) are the vertical area and velocity of the tile, respectively. Defining \(k=c\rho s/2\), the movement equation for the falling tile in the \(y\)-direction can be given as  
\[mg-k\left(\frac{dy}{dt}\right)^{2}=m\frac{d^{2}y}{dt^{2}} \tag{3}\]  
After solving Equation (3), the trace of the tile in the \(y\)-direction can be calculated with Equation (4)  
\[y=\frac{m\ln\Bigl{(}\cosh\Bigl{(}t\sqrt{gk/m}\Bigr{)}\Bigr{)}}{k} \tag{4}\]  
Figure 3: Falling process for a ceiling tile in the physical model. where \(m\) and \(g\) are the mass of the tile and the acceleration of gravity, respectively. In the second stage, the tile collides with the floor.