# Behaviour of Steel Tubular Knee Joint in Aluminium Frames with Tension-Tie Element
## 3 Numerical Analysis
### Discussion of Parametric Numerical Analysis Results  
The results of the numerical analyses are presented in Figures 15 and 16. Force-piston displacement curves for different tie element stiffnesses are shown in Figure 15. The increase in the tie element stiffness increases the maximal piston force (load) significantly. In the elastic region, the behaviour of the knee joint is relatively similar for all analysed cases. In the knee range, and especially in the postcritical region, the influence of the tie element stiffness is much more pronounced. In the case of knee joints with a softer tie element (dashed curves), differences in the force-piston displacement curves in the postcritical region are less pronounced. Figure 16 represents the moment-rotation relationship of the analysed knee joints where both moment and rotation were measured at the beam to column intersection of the knee joint using Equations (1)-(5). It can be seen from the Figure 16 that stiffness of the tie element has a negligible effect on the moment-rotation relationship of the knee joint. In the elastic range, the moment-rotation relationship of the knee joint is almost identical for all the analysed cases, therefore leading to the conclusion that the tie element does not affect the bending resistance or rotational stiffness of the knee joint. It must be noted that the tie element has a major effect on the redistribution of bending moments due to external vertical loads. Because the bending moment in the knee joint for the frame with a tie element is much lower in comparison with the portal frame without the tie element for the same level of loading, tie element is very useful for large-span aluminium portal frames. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Portal Frame Span [m].** & **Stiffness Ratio \(\mathrm{f_{red}}\) [–]** & **E’=E/\(\mathrm{f_{red}}\) [MPa]** \\ \hline
1 & 17 & 12,200 \\
2.5 & 43 & 4880 \\
5 & 86 & 2440 \\
10 & 172 & 1220 \\
15 & 258 & 815 \\
20 & 344 & 610 \\ \hline \hline \end{tabular} E—modulus of elasticity for steel taken as E = 210 GPa  
\end{table} 
Table 3: Comparison of the stiffness values of tie elements used in experimental program and the one used in practice. Slight differences in the moment-rotation curves can be observed in the knee region and post-critical region. Contrary to expectation, knee joints with softer tie element developed slightly higher values of ultimate bending moment and rotational deformation. This kind of behaviour can be explained by the fact that joints with softer tie elements have lower values of compression force, enabling them to have slightly higher ultimate moment and total rotation. Changing the stiffness of the tie element also led to different failure modes of the knee joint (Figure 17). For the knee joints with a stiffer tie element, numerical analyses showed that the ultimate knee joint failure was caused by failure of the weld between the beam section and the tensile bolt nut. In the case of knee joints with a softer tie element, tensile force in the tie element is much lower, therefore the pull-out of the tensile bolt does not occur. In that case, bending at the beam to column intersection along with web buckling was the dominant cause of joint failure. In real aluminium halls with a tension tie element, the latter failure mode is to be expected, which is favourable since that kind of failure is much more ductile compared to brittle failure where tensile bolt nut pull-out occurs. Figure 16: Joint bending moment–rotation curves for different stiffnesses of the tie element. Figure 15: Force–piston displacement curves for different stiffnesses of the tie element. Furthermore, joints with a stiffer tie element have discontinuity in force-piston displacement curves after reaching a force value of approx. 40 kN (Figure 15), which is caused by the slip of the manually prestressed bolts that connect the T-adapter to the beam of the steel knee joint.