# Finite Element Analysis of Proposed Self-Locking Joint for Modular Steel Structures
## 3 Finite Element (FE) Model
### Material Modeling  
The entire collapse mechanism of the column-beam joint can either occur by yielding the joint panel (column failure) or by developing plastic hinges in connected beams, depending on design specifications [29]. In general, steel structure members are commonly made of hot-rolled steel, and properties of this type of steel may be adopted in the FE analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Contact Pair** & **Contact Type** & **Tangential Behavior** & **Normal Behavior** \\ \hline Column with Beam & Tie Constraint & - & - \\ Stiffener with column & Tie Constraint & & \\ and beam & Surface-surface & & \\ Plates & contact interaction & Penalty & Hard contact \\ Bolt shank-hole & Surface-surface & & \\ contact interaction & Penalty & Hard contact \\ Bolt head-cover plate & Tie Constraint & - & - \\ \hline \hline \end{tabular}
\end{table}
Table 3: Summary of contact interaction.  
In finite element modeling, accurate and specific material modeling is required to produce the actual behavior of the real structure. Since the model is composed of steel material, a nonlinear steel behavior represented by the bi-linear stress-strain relationship was considered as illustrated in Figure 4, where \(E_{0}\) is Young's modulus, \(E_{t}\) is tangent modulus. The plastic properties required to be defined as input in the material properties are: (a) a yield condition to specify the onset of plastic deformation, (b) a flow rule to define the plastic straining, and (c) a hardening rule to define the evolution of the yield surface with plastic straining. The yield condition was defined under the von Mises yield criterion as given in Equation (5). Flow rule defines the direction of plastic strain vector. The equivalent plastic strain (PEEQ) criterion was utilized, which can be expressed by Equation (6). For the hardening stage, isotropic hardening was assumed. The isotropic type of elasticity was selected, that is simplest form of linear elasticity is the isotropic case, and the stress-strain relationship is represented by [D] matrix as given in ABAQUS 6.14 material library [28] and expressed in Equation (7).  
\[\sigma_{v}=\sqrt{\frac{\left(\sigma_{1}-\sigma_{2}\right)^{2}+\left(\sigma_{2 }-\sigma_{3}\right)^{2}+\left(\sigma_{1}-\sigma_{3}\right)^{2}}{2}} \tag{5}\]  
where \(\sigma_{v}\) is von mises stress, \(\sigma_{1},\sigma_{2}\) and \(\sigma_{3}\) is the normal stress x,y and z component respectively.  
\[\varepsilon^{p}=\sqrt{\frac{2}{3}\varepsilon^{pl}ij\,\varepsilon^{pl}ij} \tag{6}\]  
where \(\varepsilon^{p}\) is PEEQ, \(\varepsilon^{pl}\) is plastic strain and \(i_{j}\) is strain tensor.  
\[[D]=\left\{\begin{array}{c}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ \gamma_{12}\\ \gamma_{13}\\ \gamma_{23}\end{array}\right\}=\left[\begin{array}{cccccc}\frac{1}{E}& \frac{-p}{E}&\frac{-p}{E}&0&0&0\\ \frac{-p}{E}&\frac{-p}{E}&\frac{-p}{E}&0&0&0\\ \frac{-p}{E}&\frac{-p}{E}&\frac{-p}{E}&0&0&0\\ 0&0&0&\frac{1}{C}&0&0\\ 0&0&0&0&\frac{1}{C}&0\\ 0&0&0&0&0&\frac{1}{C}\end{array}\right]\left\{\begin{array}{c}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{12}\\ \sigma_{13}\\ \sigma_{23}\end{array}\right\} \tag{7}\]  
where \(E\) is [PERSON]'s modulus, \(v\) is poisson ratio and \(G\) is shear modulus.