## 4 Component-Based Mechanical Models for TSACWs  
### Initial Stiffness  
The initial stiffness \(S_{j,ini}\) of bolted connections is one of the most important parameters for the characterization of their (\(M\)-\(\theta\)) response. There is a large volume of published studies investigating \(S_{j,ini}\) of bolted beam-to-column connections. The first deep discussion and analysis about \(S_{j,ini}\) emerged in the 1980s with the research studies by [PERSON] et al. [27]. The authors proposed a beam model dividing the leg of the top cleat flange into two types of beam segments, flexible and rigid sections. Relevant equations of practical use have been presented to estimate \(S_{j,ini}\) for TSACWs. Later, [PERSON] and [PERSON] [29] presented a new equation that was developed according to the cantilever beam model. The EC3 standard document [13]--Annex J--implemented a component method to estimate the \(S_{j,ini}\) of TSACs. In this method, the connection behavior is simulated by a series  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Test**} & \multirow{2}{*}{**Beam**} & \multicolumn{3}{c}{**Size**} & \multirow{2}{*}{**Top Cleat**} & \multirow{2}{*}{**Web Cleat**} & \multirow{2}{*}{**Yield Stress of Angle**} \\  & & & **Bolts** & & & **(mm)** & **(N/mm2)** \\ \hline
**8S1** & H210 \(\times\) 134 \(\times\) 6.4 \(\times\)10.2 & H310 \(\times\) 254 \(\times\) 9.1 \(\times\) 16.3 & 19.1 & L152  
of different components, each one represented in the form of an elastic spring with specific stiffness and strength. The overall initial stiffness of the connection is hence notoriously estimated by assembling all the springs in a parallel series configuration, as shown in Figure 3.  
In the EC3, the input stiffness coefficients that must be considered are associated with the column shear panel zone (\(K_{1}\)), the column flange in tension (\(K_{3}\)), the column flange in compression (\(K_{2}\)), the flexural stiffness of column flange (\(K_{4}\)), the top clear flexural stiffness (\(K_{6}\)), the tensile stiffness of bolts (\(K_{10}\)), and, for non-preloaded bolts, their shear stiffness (\(K_{11}\)) and their bearing stiffness (\(K_{12}\)). In conclusion, \(S_{j,ini}\) of TSACs is given by:  
\[S_{j,\;ini}=\frac{E\;z^{2}}{\sum_{i=1}^{n}1/K_{i}} \tag{1}\]  
where \(E\) is [PERSON]'s modulus, \(z\) is the lever arm, \(K_{i}\) is the \(i\)-th component stiffness coefficient, and \(n\) is the number of joint components, as in Figure 3. Finally, \(z\) should be taken as the distance from the bolt-row in tension and mid-thickness of the leg of the seat cleat on the compression flange. While the above formulation is widely used in practice, the EC3 formulation does not include a mechanical model for TSACs. Accordingly, an extension of EC3 for TSACs is proposed in this paper. For a bolt-row in a web cleat, the stiffness illustrated in Figure 4 should be considered. The overall stiffness of basic components illustrated in Figure 4 is represented by a single equivalent stiffness coefficient \(k_{eq}\), that can be calculated as:  
\[k_{eq}=\frac{\frac{S_{j,\;ini}}{Ez}+\sum k_{eff,\;r}z_{r}}{z_{eq}} \tag{2}\]  
where \(S_{j,ini}\) is again the initial stiffness for TSACs, while \(z_{r}\) represents the distance between the center of the compression cleat and the bolt-row \(r\) of the web cleat. Moreover:  
\[k_{eff,\;r}=\frac{1}{\sum_{i}1/k_{i,\;r}} \tag{3}\]  
Figure 3: Component-based modeling of TSAC. where \(k_{i,r}\) is the stiffness coefficient representing component \(i\) relative to bolt-row \(r\), and:  
\[z_{eq}=\frac{\frac{S_{i,ini}}{E}+\sum_{r}k_{eff,r},r_{r}^{2}}{\frac{S_{i,ini}}{E }+\sum_{r}k_{eff,r},r_{r}} \tag{4}\]  
In conclusion, the initial stiffness of TSCWs can be calculated as:  
\[S_{j,\ ini}=\frac{Eze_{eq}}{k_{eq}} \tag{5}\]  
[PERSON] and [PERSON] [14] have used curve fitting software to obtain the effects of top and seat angles on the \(S_{j,ini}\) value of TSACWs.