# Behaviour of Steel Tubular Knee Joint in Aluminium Frames with Tension-Tie Element
## 2 Experimental Program
### Experimental Results
#### 2.5.2 Moment-Rotation Relationship  
To characterise the flexural behaviour of the tested steel knee joints, bending moments versus rotational deformations curves, i.e., the M\({}_{\rm j}\)-\(\Phi_{\rm j}\) curves, were obtained, using primary test results and common rules to obtain equilibrium on the undeformed element, Equations (1)-(5). Having obtained force at the piston (F\({}_{\rm P}\)) and force in the tie element (F\({}_{\rm T}\)) from the test, the horizontal reaction at the beam end (R\({}_{\rm H,P}\)) can be calculated using known geometry (Figure 8) and consequently, first order bending moment at the beam to column intersection M\({}_{\rm j}\) can be expressed as follows:  
\[\rm M_{\rm j}\ =\ 739\ R_{H,P}-1323\ F_{P}+\left(185\ \cos\left(14.17^{ \circ}\right)\ -\ (89+248)\ \sin\left(14.17^{\circ}\right)\right)\ F_{T} \tag{1}\]  
\[\rm M_{\rm j}\ =\ 487\ F_{P}\ -\ 271\ F_{T} \tag{2}\]  
Note that dimensions in Equations (1) and (2) are in mm, therefore, the results of bending moment will be expressed in kNmm if forces are entered in kN. Generally, rotation at the knee joint \(\rm\phi_{\rm j}\) can be calculated as subtraction of the beam rotation (\(\rm\phi_{\rm b}\)) and the column rotation (\(\rm\phi_{\rm c}\)):  
\[\rm\phi_{\rm j}\ =\ \rm\phi_{\rm c}\ -\ \rm\phi_{\rm b} \tag{3}\]  
Assuming that the column and the beam remain undeformed due to external loads and all the deformation is concentrated in the beam-column intersection (Figure 8), rotation  
Figure 8: Loading and boundary conditions of the steel knee specimen (dimensions in mm). of the beam (\(\phi_{\rm b}\)) and of the column (\(\phi_{\rm c}\)) can be calculated using the following expressions:  
\[\phi_{\rm b}=\tan^{-1}\left[\frac{561\ -\ 352\ +\ \tan\left(14.17^{\circ}\right) \ \delta_{1}\ -\ \delta_{4}}{375\ +\ \delta_{1}}\right]-29.17^{\circ} \tag{4}\]  
\[\phi_{\rm c}\ =\ \tan^{-1}\left[\frac{\delta_{1}}{352}\right] \tag{5}\]  
where \(\delta_{1}\) and \(\delta_{4}\) are displacements in mm measured at LVDT1 and LVDT4, respectively. The angle of \(14.17^{\circ}\) represents the deviation of the column from verticality (see Figure 8). It must be noted that for the arbitrary displacement of the piston, Equation (4) will give negative values of \(\phi_{\rm b}\), while Equation (5) will give positive values of \(\phi_{\rm c}\). Such an approach was chosen since arbitrary displacement of the piston will lead to a mutually opposite direction of rotation of the beam relative to the column. Bending moment versus rotational deformation curves are shown in Figure 9. It is visible that these curves consist of three characteristic regions: the elastic (linear) region (marked as 1), the knee-range (marked as 2), and the postcritical region (marked as 3) (Figure 9b). The bending moment that characterises the resistance of the steel joint M\({}_{\rm i,R}\) is defined at the intersection of the secants of the elastic and small postcritical region (after knee-range and before ultimate moment) (Figure 9b). The initial rotational stiffness is defined as the slope of the downward curve in the first loading phase (Figure 9b). As can be seen from Figure 9a, all the specimens showed relatively similar rotational properties. The main properties of M\({}_{\rm j}\)-\(\phi_{\rm j}\) curves are given in Table 2. The lowest bending resistance M\({}_{\rm j,R}\) was obtained for specimen S3, amounting to 16.6 kNm, while specimens S1 and S2 had slightly higher values of bending resistance M\({}_{\rm j,R}\) (17.1 and 17.6 kNm, respectively). The lowest value of elastic bending moment M\({}_{\rm j,el}\) was measured for specimen S1, amounting to 12.4 kNm. Specim S3 had a 10% (13.7/12.4) higher value of elastic moment compared to specimen S1, while specimen S2 had a 20% (14.9/12.4) greater value of elastic moment compared to specimen S1. The elastic bending moment M\({}_{\rm j,el}\) showed greater variation of values compared to bending resistance M\({}_{\rm j,R}\) (1.02 \(>\) 0.408).