# A Review of the T-Stub Components for the Analysis of Bolted Moment Joints
## 2 Analytical Model
### Component Approach (EC3) T-Stub in Tension  
The characterisation of the behaviour of a joint can be studied by the analytical approach, which is called the component method. The component method is a practical method initially used in EC 1992 that enables predicting the response of any steel or composite joint based on the knowledge of the mechanical and geometrical properties of the material [5]. As mentioned before, in the component method, a joint is considered a set of individual basic components. For this review, the component T-stub of the end-plate connection is considered, located in the tension zone, see Figure 2. The T-stub resistance is predicted by the analytical model found in the Eurocode 3 part 1-8 [1]. This prediction is made in nominal values, without the uses of safety factors. According to this standard, the resistance of the T-stub could be determined by three different failures modes:  
1. Mode 1: the flange of the T-stub reaches the yielding stress before the bolts get into the plastic zone. This type of behaviour could be classified as a pinned joint, see Equation (1) or Equation (2) and Figure 5a. 2. Mode 2: this mode is characterised by the yielding of the flange of the T-stub and the failure of the bolts (bolt fracture). This type of behaviour could be classified as a semi rigid joint, see Equation (3) and Figure 5b. 3. Mode 3: this mode is characterised by the failure of the bolt without the flange reach the yielding point. This type of behaviour usually happens when flanges are quite thick, and this type of behaviour could be classified as a rigid joint, see Equation (5) and Figure 5c: \[F_{Rd,1}=\frac{4\times M_{pl,1,\ Rd}}{m},\] (1) \[F_{Rd,1}=\frac{(8n+2e_{w})M_{pl,1,\ Rd}}{2 mn-e_{w}(m+n)},\] (2) \[F_{Rd,2}=\frac{2M_{pl_{2,\ Rd}}+n\sum B_{t,Rd}}{m+n},\] (3) \[M_{pl,i,\ Rd}=\frac{1/4\sum beff_{i}\times tf^{2}\times f_{y}}{\gamma_{M0}},\] (4) \[F_{Rd,3}=\sum B_{t,Rd},\] (5) \[B_{t,Rd}=k_{2}f_{ub}A_{s}/\ \gamma_{M2},\] (6)  
where \(F_{Rd,i}\) is the T-stub design resistance for each failure mode, \(M_{pl,i,\ Rd}\) is the plastic flexural resistance of the T-stub based on the \(beff\) (see Equation (7) and Figure 6) and \(tf\), which are the effective width and the flange thickness. \(e_{w}\), \(m\) and \(n\) are a quarter of the diameter washer, the bolt distance to the plastic hinge form near to the web and \(n\) is the minimum distance to the edge, respectively, \(n=e\) (see Figure 4) but \(n\leq 1.25m\). \(B_{t,Rd}\) is the bolt tension resistance, \(k_{2}\) is the factor that takes a value of 0.68 for countersunk bolts and 0.9 for other cases, \(A_{s}\) is the tensile area of the bolt. \(f_{y}\) and \(f_{u}\) are the yield and the ultimate stresses. Finally, \(\gamma_{M0}\) and \(\gamma_{M2}\) are the partial factors for design and take the values of 1 and 1.25, respectively. \[beff=\min(beff_{1},beff_{2},beff_{3},beff_{4},beff_{5})=\left\{ \begin{array}{c} beff_{1}=2\pi m\\ beff_{2}=4m+1.25e\\ beff_{3}=\pi m+0.5p\\ beff_{4}=2m+0.625e+0.5p\\ beff_{5}=b\end{array} \right., \tag{7}\]Figure 5: T-stub Failure modes: (**a**) Mode 1; (**b**) Mode 2; (**c**) Mode 3. Figure 6: Cont.