## 6 Research on the Initial Stiffness of Connection Based on the Component Method
### The Calculation Results of Initial Stiffness  
The effective stiffness \(k_{\rm eff\,r}\) of bolt row r can be derived from  
\[k_{\rm eff,r}=\frac{1}{k_{1,r}+\frac{1}{k_{2,r}}+\frac{1}{k_{3,r}}+\frac{1}{k_ {4,r}}} \tag{13}\]  
where \(k_{\rm i,r}\) is the stiffness for the i-th component corresponding to bolt row r.  
A single equivalent spring with stiffness \(k_{\rm eq}\) replaces components related to the first bolt row and second bolt row. The following expression can be derived for \(k_{\rm eq}\).  
\[k_{\rm eq}=\frac{(\sum_{i}k_{\rm eff,i}z_{\rm i})^{2}}{\sum_{i}k_{\rm eff,i}z_ {\rm i}^{2}} \tag{14}\]  
where \(z_{\rm r}\) defines the location of bolt row r with respect to the assumed center of rotation.  
Figure 25: Calculation model of concrete stiffness in compression.  
The corresponding equivalent lever arm \(z_{\text{eq}}\) is derived from  
\[\text{z}_{\text{eq}}=\frac{\sum_{i}k_{\text{eff},\text{i}}\text{z}_{\text{i}}^{2 }}{\sum_{i}k_{\text{eff},\text{i}}\text{z}_{\text{i}}} \tag{15}\]  
The overall connection initial stiffness (\(K_{\text{i}}\)) can be calculated from Equation (16):  
\[K_{i}=\frac{M}{\theta}=\frac{\text{z}_{\text{eq}}^{2}}{\sum\limits_{i}\frac{ \text{1}}{k_{i}}}=\frac{\text{z}_{\text{eq}}^{2}}{\frac{\text{1}}{k_{\text{eq }}}+\frac{\text{1}}{k_{\text{5}}}+\frac{\text{1}}{k_{\text{6}}}}=k_{\text{eq}} \text{z}_{\text{eq}}^{2} \tag{16}\]  
Equation (16) is first applied to test specimens ST-1 to ST-6 for validating the improved component model for predicting the initial stiffness. The results from the proposed component model, the test and the corresponding FE models are summarized in Table 7. Compared with the test and the FE results, the theoretical predictions basically agree, and the error margin for individual specimens is 17%.  
To further validate the proposed component model described above, an extensive parametric analysis was carried out on the typical connection configurations. Hence, a total of 17 FE models for various tube thickness, end-plate thickness, vertical bolt distances, and bolt diameters were modeled to validate the proposed component method. Table 8 provides the comparison of the stiffnesses from the FE predictions to the ones from the component-based model for the various research cases. The component-based model calculations are within \(\pm\)17% of those of the FE models and mostly within an error margin of 10%. Although the initial stiffness error margins are very high, due to the complexity of the experiment, these values might be acceptable. Furthermore, it is also worth noting that an average value of the Theory/FE ratio is equal to 0.95 with a standard deviation equal to 0.072.  
Therefore, it can be seen that the predicted initial stiffnesses are generally in good agreement with those of the test results and the FE analysis results for typical cases, proving the reliability of the proposed method for predicting the initial stiffness of a connection.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Specimen** & **Component Model \(K_{\text{theory}}\)** & **Test \(K_{\text{test}}\)** & **Ratio \(K_{\text{theory}}\)/\(K_{\text{test}}\)** & **FE Model \(K_{\text{FE}}\)** & **Ratio \(K_{\text{theory}}\)/\(K_{\text{FE}}\)** \\ \hline ST-1 & 9.90 & 10.83 & 0.91 & 11.96 & 0.83 \\ ST-2 & 14.73 & 15.42 & 0.95 & 16.70 & 0.88 \\ ST-3 & 12.15 & 15.21 & 0.80 & 12.56 & 0.97 \\ ST-4 & 20.33 & 19.18 & 1.06 & 20.60 & 0.98 \\ ST-5 & 19.99 & 17.77 & 1.12 & 19.62 & 1.02 \\ ST-6 & 20.32 & 17.57 & 1.16 & 19.20 & 1.06 \\ \hline \hline \end{tabular} Note: Units for stiffness, \(K\): kN\(\cdot\)m/mrad.  
\end{table}
Table 7: Comparison of the component model to the test results and corresponding FE analysis results.