# Structural Design and Optimization of Separated Air-Rib Tents Based on Response Surface Methodology
## 2 Numerical Calculation Model of Separate Air-Ribbed Tent  
A numerical model of the separated air-rib tent is established by ABAQUS 6.14 (SIMULIA Company, Providence, RI, USA). The separated air-rib tent consists of separated arch ribs and a tarpaulin, which is shown in Figure 2. The structural dimension parameters are derived from the best-selling product of a tent supplier. The width of the tent is 4 m, and the length is 18 m. For the ribs, the cross-section diameter of a rib is 0.3 m, and the centerline radius of a rib is 4 m. The distance between two ribs is 2 m, and ten ribs are arranged side by side. The tarpaulin and air ribs are made of two new braided materials, and the property parameters of the two materials are shown in Table 1. The inflated air ribs are mainly used to support the tent, and the working pressure of the ribs is 0.2 MPa [9]. The wind ropes are added to strengthen the stability of the separated air-rib tent. The cross-section diameter of the wind ropes is 8.9 mm, and the elastic modulus is 2.82 GPa. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Part**} & **Thickness** & **Density** & **Elastic** & **Poisson’s** & **Fracture** \\  & **(mm)** & **(g/cm\({}^{3}\))** & **Modulus** & **(GPa)** & **Ratio** & **Strength** \\ \hline Tarpaulin & 0.27 & 1.000 & 0.940 & 0.35 & 178.33 \\ Air rib & 2.60 & 1.069 & 0.470 & 0.32 & 119.47 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The property parameters of two kinds of tent materials. Figure 2: The finite element model of the separated air-rib tent. (**a**) The tarpaulin; (**b**) the air rib; (**c**) the cross section of an air rib. The tarpaulin and the air rib are set as membrane elements (M3D4R) [27], and the wind ropes are set as truss elements (T3D2) [28]. At the beginning of the simulation, the geometric is coarse meshed. To improve the calculation accuracy, the connection position between the tarpaulin and the air rib is divided into finely grid cells. The stress and the displacement change little with further refinement of the grid cells [29]. The grid cells are shown in Figure 3. The model has a total of 148,734 grid cells and 149,017 nodes. The tarpaulin has 69,002 grid cells and 69,025 nodes, and the air ribs have 75,360 grid cells and 75,600 nodes. In the model, the edge of the tarpaulin and the air ribs are fixed to the ground to restrict the XYZ three direction axial movement and rotating movement. The tarpaulin is tangent to the air rib, and the contact area is narrow to be simplified into line contact. So, the node coordinates on the contact line of the tarpaulin and the air ribs are consistent [30]. One end of a wind rope is fixed to ground, and the other end is fixed to the tarpaulin by binding constraint, and then the initial prestress is added to the wind ropes [31]. According to the extreme condition of the Load Code for the Design of Building Structures (GB50009-2012), standard dead load, wind load, and snow load are applied to the tent. The standard dead load is the gravity, the value is 9.8 m/s\({}^{2}\), and load coefficient is 1.3. The coefficients of wind load and snow load are both 1.5. Based on the force eight wind, the essential wind pressure is set to be 0.3 kN/m\({}^{2}\), and the combined load coefficient is 0.6. The coefficient is shown in Figure 4, and wind pressure applied on the tent is calculated by the following Formula (1):  
\[w_{k}=\beta_{z}u_{s}u_{z}w_{0} \tag{1}\]  
Figure 3: The finite element model of the separated air-rib tent. (**a**) The tarpaulin grid; (**b**) the air rib grid. In the formula, \(w_{k}\) is the standard value of wind load (\(\rm{kN}/\rm{m}^{2}\)), \(\beta_{z}\) is the wind vibration coefficient at height z, \(\mu_{s}\) is the wind load figure coefficient, \(\mu_{z}\) is the coefficient of variation of wind pressure height, taken as 1.0; \(w_{0}\) is the basic wind pressure (\(\rm{kN}/\rm{m}^{2}\)). The pressure of the return period of 100 years in North China is defined as average basic snow, namely 0.5 \(\rm{kN}/\rm{m}^{2}\).