## 4 Results and Discussion  
### Hysteretic Behaviour  
The seismic performance of the SS-TTDs is readily characterised in the complete load-displacement \(Q\)-\(\delta\) hysteretic curves of Figure 15; it combines the curves obtained from the shaking table tests and from the quasi-static tests. Since the yield force and yield displacement are different under quasi-static (i.e., \(Q_{yST}=15.5\) kN, \(\delta_{yST}=1.08\) mm), and under dynamic loads (i.e., \(Q_{yDYN}=24.83\) kN, \(\delta_{yDYN}=1.73\) mm) due to strain rate effects [19; 20; 21], prior to combining the \(Q\)-\(\delta\) curves they were  
Figure 14: \(Q\)-\(\delta\) curves obtained from the quasi-static cyclic tests: (**a**) SS-TTD4, (**b**) SS-TTD5, (**c**) SS-TTD6. normalized by the corresponding yielding force and yielding displacement. From the \(Q\)-\(\delta\) curves shown in Figure 15 the following observations can be made. First, the SS-TTD exhibits a stable hysteretic behaviour. Second, the \(Q\)-\(\delta\) curves under repeated cycles of the same amplitude are almost coincident, that is, the reduction of the peak strength under repeated cycles of constant amplitude is small. Third, second order effects due to strip geometric nonlinearity appear when the imposed displacements are very large, more precisely at displacement ductility \(\mu=\delta_{max}\delta_{yST}\) above 12. This geometric nonlinearity results in a spurious increase of the restoring force. It must be noted, however, that the value of the ductility \(\mu\) (i.e., about 12) beyond which the geometric nonlinearity effects become significant is far beyond the realistic range of lateral displacements that could be allowed in actual structures equipped with dampers. More precisely, in flexible RC frames or RC waffle-flat plate systems, damage starts for interstorey drifts, _ID_, between 0.005 and 0.01 times the storey height \(H\)[17; 22]. Even if some (moderate) damage is allowed in the RC structure to reduce the cost of the dampers, the _ID_ should not exceed about 0.015\(H\), because otherwise installing the dampers would be meaningless. Assuming a realistic storey height (after scaling by the scaling factor of the test specimen 2/5) of \(H=3000\times 2/5=1200\) mm, and an angle of the SS-TTD axis with the horizontal \(\alpha=0.46\) rad, the maximum axial displacement \(\delta_{max}\) (=\(H\times ID\times\cos\alpha\)) normalized by the yield displacement \(\delta_{yDYN}\) (=1.73 mm), i.e., \(\delta_{max}/\delta_{yDYN}\), to be allowed in the SS-TTD to prevent damage in the main RC structure would range between 3.1 and 6.2, and should be smaller than 9.3 in any case. These values are below the axial displacement ductility \(\mu\) (= 12) corresponding to the onset of the second order effects. The shape of the hysteretic loop \(Q\)-\(\delta\) obtained in one cycle of imposed deformation at maximum amplitude \(\delta_{max}\) is typically characterised by the equivalent viscous ratio \(\xi\), defined as follows:  
\[\xi\ =\ \frac{1}{2\pi}\frac{W_{c}}{Q_{max}\delta_{max}}, \tag{3}\]  
where \(W_{c}\) is the energy dissipated in one cycle and \(Q_{max}\) the force attained at \(\delta_{max}\). This formula is obtained by equating the hysteretic energy dissipated in one cycle and the energy dissipated through viscous damping by means of an equivalent viscous damper in one cycle of harmonic vibration [23]. The larger \(\xi\) is, the closer the shape of the hysteretic loop to an ideal parallelogram. \(\xi\) was calculated for the hysteretic loops obtained from the SS-TTDs subjected to cycles of incremental amplitude; the results are displayed in Figure 16 against the amplitude of the cycle \(\delta_{max}\) normalized by \(\delta_{y}\). It can be seen that for practical values of \(\delta_{max}/\delta_{y}\) (between approximately 6.2 and 9.3), the equivalent viscous damping \(\xi\) remains approximately constant at 30%.