## 5 Finite Element Method (FEM) based Numerical Simulations  
### Finite-Element Modeling  
The FHSB T-stub connection was simulated as C3D8R elements of ABAQUS [30; 31]. Due to the ultimate strength of the MC-35 FE model is higher than the test value, the yield strengths of the MC-35 pressure plates were set to 550 MPa, and others were set to the test values; the yield strength of the HSB was set to 940 MPa; the material density was set to 7850 kg/m\({}^{3}\); the Young' modulus was set to \(2.06\times 10^{6}\) MPa and the Poisson's ratio was set to 0.3. The ideal elastic-plastic constitutive model was adopted in the FE model. Figure 11 shows the constitutive relation of steel.  
As shown in Figure 12, surface-to-surface contact was employed to describe the contact between the following surface pairs: bolt head and connection plate, nut and connection plate, bolt shank and hole wall of connection plate, as well as the connection plate and connection plate. The above contacts were considered as hard contacts in the normal direction, and isotropic Coulomb friction contacts in the tangential direction [32]. For the tangential behavior, a penalty friction of 0.2 was applied [33; 34]. The friction coefficient was defined as 0.588 (measured value) between connecting plates. Washers were not taken into account [9; 35]. The structural mesh generation technique was used for the FE model, and the mesh generation size was 0.008. Mesh generation of MC-45 FE model is shown as in Figure 13.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{\(\lambda\)} & \multirow{3}{*}{**Specimen**} & \multicolumn{2}{c}{**Initial Clamping**} & \multirow{3}{*}{**Initial**} & \multirow{3}{*}{**Compressive**} & **Ultimate** & **Improved** \\  & & & **Force P/kN** & & & **Load** & **Strength** & **Coefficient** \\ \cline{3-8}  & \multicolumn{1}{c}{**ID**} & **Left Bolt** & **Right Bolt** & \(\mathbf{F_{I}/kN}\) & \(\mathbf{F_{C}/kN}\) & \(\mathbf{F_{U}/kN}\) & \(\mathbf{\
u}\) \\ \hline \multirow{6}{*}{\(1.00\)} & NMC-45 & 166.5 & 167.2 & - & - & 587.3 & - \\  & MC-45-01 & 164.5 & 165.8 & 1350.8 & 952.4 & 1583.2 & 5.46 \\  & MC-45-02 & 169.2 & 168.8 & 1353.7 & 803.4 & 1535.9 & 5.35 \\  & MC-45-03 & 167.9 & 168.9 & 1432.6 & 902.3 & 1573.4 & 5.68 \\  & MC-45-04 & 164.0 & 163.4 & 1310.8 & 1002.3 & 1503.4 & 5.35 \\  & MC-45-05 & 163.7 & 163.7 & 1462.9 & 839.4 & 1509.2 & 5.97 \\  & MC-45-06 & 169.2 & 169.7 & 1395.9 & 776.3 & 1719.6 & 5.50 \\ \hline \multirow{6}{*}{\(1.73\)} & NMC-60 & 159.5 & 155.8 & - & - & 331.3 & - \\  & MC-60-01 & 167.4 & 168.0 & 437.7 & 293.6 & 828.3 & 2.14 \\  & MC-60-02 & 166.3 & 167.9 & 428.0 & 437.1 & 816.7 & 2.10 \\  & MC-60-03 & 162.8 & 161.8 & 450.5 & 337.1 & 769.5 & 2.27 \\  & MC-60-04 & 168.2 & 167.3 & 413.3 & 481.7 & 763.2 & 2.02 \\  & MC-60-05 & 174.0 & 173.7 & 453.8 & 461.8 & 758.0 & 2.14 \\  & MC-60-06 & 175.7 & 174.6 & 432.6 & 297.1 & 795.9 & 2.02 \\ \hline \hline \end{tabular}
\end{table}
Table 4: _Cont._  
Figure 11: Constitutive model of steel.  
Figure 14 shows the boundary conditions application of the clamping force of the FE model. All degrees of freedom on the edge of the bottom pressure plate are limited. The model was loaded with a 15 mm displacement along the Y direction of the pressure plate. During the modeling, the initial clamping force of bolt was simulated by the bolt load imposed on the specific section of the HSB. Next, the initial clamping force of bolt was applied by the \"bolt load\" command at the 3/4 section of the shank.