# Finite Element Analysis of Proposed Self-Locking Joint for Modular Steel Structures
## 3 Finite Element (FE) Model
### Elements Type and Meshing  
The nonlinear FE model was discretized by applying a suitable mesh element. All subassemblies (columns, beams, connection, and bolts) were modeled with second order isoparametric brick elements (hexahedral) due to the high accuracy that this element could provide. The gradient matrix [B] of these elements is provided in ABAQUS 6.14 [28] library and it is expressed in equation (1). The whole model can be meshed with first-order 8-node with reduced integration elements to save computing time, but due to Hour-glassing the result may be inaccurate. To reduce computing time, the model was assigned using two different nodes of similar element type: brick element C3D20R (20-node brick element with reduced integration) and C3D8R (8-node brick element with reduced integration). A second-order brick element C3D20R was used for the idealized highly sensitive region such as column-beam joint, connections, and bolts where large deformations expected to occur, and C3D8R brick elements were used in the remaining portion of the model. To capture better bending results of main members (columns and beams), the elements were refined in the thickness direction of the members, i.e., at least two elements used in the thickness direction. Same as the element type, the size of elements also varied according to the region in the model. The selection of mesh density was based on the mesh sensitivity study, as summarized in Table 2.  
Mesh sensitivity was conducted to select an appropriate mesh, which can produce acceptable simulation results with less computing time. It can be observed from Table 2 that by changing size of elements the ultimate moment capacity of the joint is also changed, this is because assigning smaller size mesh also increased the number of nodes per unit area or per unit volume, which can more precisely capture the deformation and stresses. An optimum mesh size of 15 mm is selected for the highly sensitive region, i.e., the region where the plastic hinges were expected to occur in the beams and the joint panel (area of interest) and 25 mm for the remaining parts. The number of elements near holes and at corners of the beam and column were kept denser. In total, 29,100 hexahedral elements were used in the model. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Part** & \begin{tabular}{c} **Section b \(\times\) h** \\ **(mm)** \\ \end{tabular} & \begin{tabular}{c} **Thickness** \\ **(mm)** \\ \end{tabular} & \
\begin{tabular}{c} **Length** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **Plastic Moment** \\ **Mp (kNm)** \\ \end{tabular} \\ \hline Column & 160 \(\times\) 160 & 10 & 1500 & 129.6 \\ Floor beam (Fbeam) & 160 \(\times\) 160 & 8 & 1800 & 107.1 \\ Ceiling beam (Cbeam) & 160 \(\times\) 160 & 6.3 & 1800 & 86.68 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of sections for proposed connection. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Element Type** & \multicolumn{3}{c}{**Size of Elements (mm)**} & \\ \hline
**Area of** & \begin{tabular}{c} **Area of** \\ **Interest** \\ **(Joint Panel)** \\ \end{tabular} & \begin{tabular}{c} **Area of** \\ **Interest** \\ **(Joint Panel)** \\ \end{tabular} & \begin{tabular}{c} **Total Number** \\ **Other Parts** \\ \end{tabular} & \
\begin{tabular}{c} **Total Number** \\ **of Elements** \\ **Time (Minutes)** \\ \end{tabular} &
\begin{tabular}{c} **Total** \\ **Moment** \\ **(kNm)** \\ \end{tabular} \\ \hline C3D20R & C3D8R & 15 & 25 & 29,100 & 120 & 289.3 \\ C3D20R & C3D20R & 15 & 15 & 53,361 & 1440 & 291.6 \\ C3D20R & C3D8R & 30 & 30 & 11,339 & 45 & 295.5 \\ C3D20R & C3D8I & 15 & 25 & 29,100 & 150 & 291.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mesh sensitivity study on a 3D nonlinear model in ABAQUS.