## 4 Design of CFS Screw Connection with Gypsum Sheathing
### Other Parameters  
Some other screw connection parameters were also considered, including \(k_{\mathrm{eT}}\), \(\Delta_{\mathrm{mT}}\) and \(\Delta_{\mathrm{uT}}\). Because the scatter is significant, fitting these parameters focused on the simplicity of operation and conservation instead of accuracy. The prediction formula of the initial screw connection stiffness at ambient and elevated temperatures is given in Equation (6). \(R_{\mathrm{kT}}\) is the reduction factor of the initial stiffness defined as the ratio of the initial stiffness at elevated temperatures to that at ambient temperature. The values of \(R_{\mathrm{kT}}\) are shown in Table 4, and they did not consider the sudden increase in initial stiffness at 100 \({}^{\circ}\)C (Tables 1 and 2). Figure 15 compares the predicted \(R_{\mathrm{kT}}\) to the test results at ambient and elevated temperatures. In most cases, the \(k_{\mathrm{eT}}\) values obtained from Equation (6) are conservative and easy for operation.  
\[k_{\mathrm{eT}}=R_{\mathrm{kT}}\cdot 1000 \tag{6}\]  
where the units of \(k_{\mathrm{eT}}\) are N/mm.  
Equation (7) is the prediction formula of \(\Delta_{\mathrm{mT}}\), which takes the larger value of of \(F_{\mathrm{mT}}/k_{\mathrm{eT}}\) and \(\alpha_{1}\cdot\Delta_{\mathrm{1T}}\). \(F_{\mathrm{mT}}\) and \(k_{\mathrm{eT}}\) can be obtained from Equations (1) and (6), respectively. \(\Delta_{\mathrm{1T}}\) is the predicted displacement corresponding to \(F_{\mathrm{mT}}\) for the screw connection with single-layer gypsum sheathing at \(T\)\({}^{\circ}\)C. Figure 16 gives the values of the predicted \(\Delta_{\mathrm{1T}}\) based on the present experiments. \(\alpha_{1}\) is the coefficient that considers the effect of double-layer gypsum sheathing on \(\Delta_{\mathrm{mT}}\), as shown in Equations (8) and (9). As shown in Table 2, the test results of \(\Delta_{\mathrm{mT}}\) for the connection series with double-layer gypsum sheathing and a loaded edge distance of 20 mm were much lower than those of the series with double-layer gypsum sheathing and a loaded edge distance of 15 or 10 mm for temperatures beyond 250 \({}^{\circ}\)C. This phenomenon seems unusual, and it was not considered in Equation (9).  
\[\Delta_{mT}=\max(F_{mT}/k_{\mathrm{eT}},\alpha_{1}\cdot\Delta_{\mathrm{1T}}) \tag{7}\]  
where the units of \(\Delta_{\mathrm{mT}}\) and \(\Delta_{\mathrm{1T}}\) are mm.  
For single-layer gypsum sheathing,  
\[\alpha_{1}\;=\;1.0 \tag{8}\]  
For double-layer gypsum sheathing,  
\[\alpha_{1}\;=\;1.3 \tag{9}\]  
In addition, the prediction formula of \(\Delta_{\mathrm{uT}}\) was also given, as shown in Equation (10). \(\alpha_{2}\) is the additional coefficient that considers the effect of double-layer gypsum sheathing on \(\Delta_{\mathrm{uT}}\) at elevated temperatures, as shown in Equations (11) and (12).  
\[\Delta_{uT}=1.5\cdot\alpha_{2}\cdot\Delta_{mT} \tag{10}\]  
where the units of \(\Delta_{\mathrm{uT}}\) are mm.  
For single-layer gypsum sheathing,  
\[\alpha_{2}\;=\;1.0 \tag{11}\]  
For double-layer gypsum sheathing,  
\[\alpha_{2}\;=\;1.0,\;T\leq 200\,^{\circ}\mathrm{C}\mathrm{\,and}\;\alpha_{2}\; =\;1.6,\;T\;>\;200\,^{\circ}\mathrm{C} \tag{12}\]Figure 16: Predicted values of \(\Delta_{\rm IT}\).  
Figure 15: Predicted initial stiffness compared to the experimental results.