# A Review of the T-Stub Components for the Analysis of Bolted Moment Joints
## 2 Analytical Model
### T-Stubs at Elevated Temperatures (Fire)
#### 2.12.3 [PERSON]'s Model  
[PERSON] proposed a model based on the observation that depending on the ratio of flexural stiffness of the end plate which acts in series with the column flange to the axial stiffness of the bolts to evaluate the thermo-elastic and plastic behaviour of flexible T-stub [25]. In this analytical model, the degradation of the components material due to thermal loads is considered. The analysis to determine the behaviour of the T-stubs starts by calculating the required force (\(F_{y}^{1}\)) and the deflection of the T-stub (\(\delta_{y}^{1}\)) when the first yield point is formed, see Equations (94)-(98) when \(Q>0\) and Equations (99)-(102) when \(Q=0\):  
\[F_{y}^{1}=\min(F_{f},F_{bl},F_{b}), \tag{94}\]  
\[F_{f}=\eta_{Ys}\bigg{(}\frac{\frac{\eta_{E_{b}}}{\eta_{W2E}}+ \frac{\delta_{W0}}{\delta_{W0}}}{(\epsilon+m)\frac{\eta_{E_{b}}}{\eta_{W2E}}+ \varepsilon(\beta-\alpha)\frac{\eta_{E_{b}}}{\delta_{W0}}+m\beta\frac{\eta_{E_ {b}}}{\delta_{W0}}}\bigg{)}\times\min(M_{ep0},M_{cp0})\,, \tag{95}\] \[K_{b0}=\frac{E_{b0}A_{s}}{l_{b}}\] \[\overline{K_{0}}=\frac{K_{0}+K_{c0}}{K_{0}K_{0}}\] \[F_{bl}=\eta_{Ys}\Bigg{(}\frac{\frac{\eta_{E_{b}}}{\eta_{W2E}}+ \beta\frac{K_{b0}}{K_{0}}}{e(\alpha-\beta)\frac{K_{b0}}{\delta_{W0}}-\frac{ \eta_{E_{b}}}{\eta_{W2E}}}\bigg{)}\times\min(M_{ep0},M_{cp0})\,, \tag{96}\]\[F_{b}=\eta_{Yb}\left(\frac{1}{\alpha}\times\frac{\eta_{Es}}{\eta_{Eb}}\times\frac{ \overline{K_{0}}}{K_{b0}}+\frac{n_{b}\beta}{\alpha}\right)B_{y0}, \tag{97}\]  
\[\delta_{y}^{1}=\frac{F_{y}^{1}}{24\eta_{Es}\overline{K_{0}}}\left(1-\frac{24 \alpha^{2}}{\frac{\eta_{Es}\overline{K_{0}}}{n_{b}\eta_{Eb}K_{b0}}+\beta} \right), \tag{98}\]  
\[F_{y}^{1}=\min(F_{f},F_{b}), \tag{99}\]  
\[F_{f}=\eta_{Ys}\times\frac{\min\bigl{(}M_{ep0},M_{cp0}\bigr{)}}{m}, \tag{100}\]  
\[F_{b}=n_{b}\eta_{Yb}B_{y0}, \tag{101}\]  
\[F_{f}=\eta_{Ys}\bigg{(}\frac{\frac{\eta_{E_{b}}}{\eta_{W2E}}+ \frac{\delta_{W0}}{\delta_{W0}}}{(\epsilon+m)\frac{\eta_{E_{b}}}{\eta_{W2E}}+ \varepsilon(\beta-\alpha)\frac{\eta_{E_{b}}}{\delta_{W0}}+m\beta\frac{\eta_{E_ {b}}}{\delta_{W0}}}\bigg{)}\times\min(M_{ep0},M_{cp0})\,, \tag{95}\]0  
where \(n_{b}\) is the number of bolts at each bolt line, \(\eta_{Es}=E_{sT}/E_{s0}\), \(\eta_{Eb}=E_{bT}/E_{b0}\) are the ratio between the Young's modulus at an elevated temperature and at ambient temperature of the steel and bolt, respectively, \(\beta=0.5l^{2}-2/3l^{3}\), \(\alpha=0.125l-1/6l^{3}\), \(l=e/L=e/(2(m+e))\). The terms \(e\) and \(m\) are shown in \[F_{f}=\eta_{Ys}\bigg{(}\frac{\frac{\eta_{E_{b}}}{\eta_{W2E}}+ \frac{\delta_{W0}}{\delta_{W0}}}{(\epsilon+m)\frac{\eta_{E_{b}}}{\eta_{W2E}}+ \varepsilon(\beta-\alpha)\frac{\eta_{E_{b}}}{\delta_{W0}}+m\beta\frac{\eta_{E_ {b}}}{\delta_{W0}}}\bigg{)}\times\min(M_{ep0},M_{cp0})\,, \tag{95}\]1 In Equation (95) \(K_{b0}\) is the stiffness of the bolt at ambient temperature and \(A_{s}\) and \(l_{b}\) are the effective area and length of bolt; \(\overline{K_{0}}\) is the equivalent stiffness of the endplate (\(K_{c0}\)) and column (\(K_{c0}\)) at ambient conditions. \(\eta_{Ys}=f_{y\pi}/f_{y0}\), \(\eta_{Yb}=f_{ybT}/f_{y0}\), \(M_{cp0}=f_{y00}\times S_{e}\), \(M_{cp0}=f_{y00}\times S_{e}\). The terms \(f_{y\pi}\), \(f_{ybT}\) and \(f_{y0}\), \(f_{y00}\) are the yielding stress of the steel and bolt at an elevated temperature and the yielding stress of the steel and bolt at ambient temperature respectively. The terms \(f_{y00}\), \(f_{y00}\) and \(S_{e}\), \(S_{e}\) are the yielding stress at ambient temperature of the endplate and column flange and the plastic section modulus of the endplate and column flange, respectively. After the first plastic hinge has appeared a second plastic hinge is formed and three possibilities could be developed for the analysis: Case 1, Case 2 and Case 3. 1. Case 1: when the first plastic hinge occurs at the fillet of the end plate or column flange, the bending moment is equal to \(M_{cpT}\) (\(\eta_{Ys}\times M_{cp0}\)) or equal to \(M_{cpT}\) (\(\eta_{Ys}\times M_{cp0}\)).