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					                  Earthquake Engineering and Structural Dynamics




       RE-SHAPING HYSTERETIC BEHAVIOUR - SPECTRAL ANALYSIS AND
             DESIGN EQUATIONS FOR SEMI-ACTIVE STRUCTURES
              Fo
                 Journal:    Earthquake Engineering and Structural Dynamics

          Manuscript ID:     EQE-06-0043.R2
                 r
 Wiley - Manuscript type:    Research Article

  Date Submitted by the
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                             n/a
                Author:

Complete List of Authors:    Rodgers, Geoffrey; University of Canterbury, Mechanical
                             Engineering
                                     er

                             Mander, John; University of Canterbury, Dept of Civil Engineering
                             chase, geoff; University of Canterbury, Mechanical Engineering
                             Mulligan, Kerry; University of Canterbury, Mechanical Engineering
                             Deam, Bruce; University of Canterbury, Dept of Civil Engineering
                             Carr, Athol; University of Canterbury, Dept of Civil Engineering
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                             hysteretic behaviour, structural design, response spectra, semi-
              Keywords:
                             active control, reduction factors
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                    RE-SHAPING HYSTERETIC BEHAVIOUR - SPECTRAL ANALYSIS AND DESIGN
5                                         EQUATIONS FOR SEMI-ACTIVE STRUCTURES
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9                     Geoffrey W. Rodgers1, John B. Mander2, J. Geoffrey Chase1, Kerry J. Mulligan1, Bruce L.
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11                                                          Deam2 and Athol Carr2
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15             Abstract
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17             Semi-active dampers offer significant capability to reduce dynamic wind and seismic structural
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20             response. A novel resetable device with independent valve control laws that enables semi-active
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22             re-shaping of the overall structural hysteretic behaviour has been recently developed, and a one-
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24             fifth scale prototype experimentally validated. This research statistically analyses three methods
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27             of re-shaping structural hysteretic dynamics in a performance-based seismic design context.
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29             Displacement, structural force, and total base-shear response reduction factor spectra are
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               obtained for suites of ground motions from the SAC project. Results indicate that the reduction
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34             factors are suite invariant. Resisting all motion adds damping in all four quadrants and showed
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36             40-60% reductions in the structural force and displacement at the cost of a 20-60% increase in
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39             total base-shear. Resisting only motion away from equilibrium adds damping in quadrants 1 and
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41             3, and provides reductions of 20-40%, with a 20-50% increase in total base-shear. However, only
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               resisting motion towards equilibrium adds damping in quadrants 2 and 4 only, for which the
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46             structural responses and total base-shear are reduced 20-40%. The spectral analysis results are
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48             used to create empirical reduction factor equations suitable for use in performance based design
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51             methods, creating an avenue for designing these devices into structural applications. Overall, the
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53             reductions in both response and base-shear indicate the potential appeal of this semi-active
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57                 Dept. of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
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58                 Dept of Civil Engineering, University of Canterbury, Christchurch, New Zealand
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                             Earthquake Engineering and Structural Dynamics                               Page 2 of 40


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     hysteresis sculpting approach for seismic retrofit applications — largely due to the reduction of
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6    the structural force and overturning demands on the foundation system.
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11   1.0 Introduction
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14   Semi-active control is emerging as an effective method of mitigating structural damage from
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16   large environmental loads, such as wind loading and seismic excitation. It has two main benefits
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     over active and passive solutions. First, a large power/energy supply is not required for
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21   significant reduction in response. Second, semi-active systems provide the broad range of control
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23   that a tuned passive system cannot, making them better able to respond to changes in structural
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26   behaviour due to non-linearity, damage or degradation over time. Semi-active systems are also
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28   strictly dissipative and do not add energy to the structural system, guaranteeing stability with
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31   careful implementation [1]
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35   Because semi-active systems utilise the building motion to generate resistive forces, semi-active
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38   devices focus on managing these forces to dissipate energy in a controlled manner. It is well
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40   known that semi-active systems can dissipate significant energy and mitigate damage during
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     seismic events, and recent research has examined re-shaping of hysteretic behaviour [2].
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45   However, such research has not extended to investigating the relative effects of different control
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47   laws on structural response, nor has it quantified that impact statistically over suites of ground
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50   motions. Hence, there has been no examination of how to readily incorporate such novel semi-
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52   active systems into performance based design methodologies.
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               Semi-active devices are particularly suitable in situations where the device may not be required
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6              to be active for extended periods of time [3]. The potential of many classes of semi-active
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8              devices and methods, including variable stiffness and variable damping classes, to mitigate
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11             damage during seismic events is well documented [4-6]. This research investigates variable
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13             stiffness resetable devices where instead of altering the damping of the system, they alter the
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15             stiffness with the stored energy being released as the working fluid reverts to its initial pressure
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18             on resetting. This resetting gives rise to discontinuous jumps in the device stiffness. Thus, they
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20             are a stiffness based device that are used to dissipate stored restoring force energy.
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25             Semi-active resetable devices show significant promise in their ability to dissipate energy and
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27             reduce seismic structural response, but are still in their infancy. Although semi-active systems
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30             may inherently raise reliability questions, regular inspection and preventative maintenance should
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32             ensure correct operation. The small power sources required for a semi-active system make them
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               largely, if not entirely, independent of mains power and consequently unaffected by power
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37             shortages during an earthquake. Semi-active damping via resetable devices also offers the unique
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39             opportunity to sculpt or re-shape the resulting structural hysteresis loop to meet design needs,
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42             enabled by the ability to actively control the device valve and reset times. For example, given a
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44             sinusoidal response, a typical viscously damped, linear structure has the hysteresis loop
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46             definitions schematically shown in Figure 1a, where the linear force deflection response is added
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49             to the circular force-deflection response due to viscous damping to create the well-known overall
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51             hysteresis loop. Figure 1b shows the same behaviour for a simple resetable device where all
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54             stored energy is released at the peak of each sine-wave cycle and all other motion is resisted [7].
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     This form is denoted a “1-4 device” as it provides damping in all four quadrants. A stiff damper
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6    will dissipate significant energy. However, the resulting base-shear force is increased. If the
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8    control law is changed such that only motion towards the zero position (from the peak values) is
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11   resisted, the force-deflection curves that result are shown in Figure 1d. In this case, the semi-
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13   active resetable damper force actually reduces the base-shear demand by providing damping
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15   forces only in quadrants 2 and 4; this is denoted a “2-4 device”. Figure 1c shows a damper that
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18   resists motion only away from equilibrium, also increases base shear, and is denoted a “1-3
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25   Although earthquake records are random signals and vary significantly from the harmonic
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27   response shown in Figure 1, the control implementation does not change. The only feedback
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30   measurement required for implementation is displacement information to determine position and
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32   velocity, defining the current quadrant of the displacement-velocity plot, and consequently
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     defining the required valve position. The sampling rate is of importance for implementation, but
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37   a sampling rate of 1-2kHz is easily achievable with modern devices and allows a relatively quite
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39   simple practical implementation with modern electronics [8-11]. Hence, the fundamental systems
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42   are simple and the analyses presented here are quite general.
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46   Overall, all three forms of resetable device enable the opportunity to re-shape and customise the
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49   overall structural hysteretic behaviour, while also providing supplemental damping to minimise
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51   structural response. Hence, Figures 1b-d show the fundamental methods by which semi-active
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54   hysteretic damping can be re-shaped with these devices. More complex patterns and control laws
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               may also have potential. Note that these methods are enabled by the independent control of the
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6              device valves, but could be generalised to similar viscous dampers to add further oversight of the
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8              resulting shape of the hysteretical behaviour.
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13             This paper investigates the relative effect of the three different resetable control laws in Figures
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15             1b-d, with their corresponding hysteretic behaviour, on structural response due to seismic
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18             excitation. When considering the effect of the additional resetable device stiffness on the
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               spectral response analysis shows the greatest change in response, and how to relate the non-linear
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30             spectra, as they provide a suitable way to synthesize all of these factors.
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               2.0 Device Dynamics
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38             The only device control mechanism for a semi-active resetable device is the activation of the
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40             valves with a reset time of approximately 20 ms [1, 4]. The valve dynamics are more rapid than
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               the dynamic response of most structural systems; therefore they interact only minimally with the
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47             specifically, instead of altering the damping of the structure, resetable devices are fundamentally
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50             hydraulic spring elements in which the un-stretched spring length can be reset to obtain
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52             maximum energy dissipation from the structural system [3]. Energy is stored by compressing the
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               working fluid as a piston is displaced. When the piston reaches its maximum displaced position,
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     the stored energy is also at a maximum. At this point, the stored energy can be released by
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6    discharging the fluid/air to the non-working side of the device, thus resetting the un-stretched
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8    spring length, as shown in Figure 2a. This approach yields the 1-4 device behaviour of Figure 1b.
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11   Figure 2b shows a modified device design that treats each chamber independently [2]. This
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13   approach eliminates the need to rapidly dissipate energy between the two chambers, as in the
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15   coupled design of Figure 2a. The resulting independent control of the pressure and energy
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18   dissipation on each side of the piston for each portion of response motion allows greater
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     range of control laws as each valve can be operated independently.
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27   By utilising a common, well-understood fluid such as air, analyses and modelling of the device is
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30   made simpler [2]. Furthermore, by utilising air as the working fluid, the atmosphere can be used
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     the working fluid. There are several tradeoffs in the complexity of semi-active resetable devices
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37   when compared to electro-rheological or magneto-rheological smart dampers [12, 13]. The
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39   rheolgical dampers do not have valves, but their potentially large hysteretic behaviour and the lag
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42   introduced by it in slow response to inputs make control implementation more difficult.
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46   A combination of the atmospheric fluid reservoir, for a pneumatic (air-based) device, and
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49   independent valve design allows more time for pressures to equalise during the valve reset, as
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               longer period to allow the pressure to equalise, as this valve does not affect the compression in
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8              extending the valve reset time prevents the other chamber from storing energy. Thus, the newer
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11             device design in Figure 2b is more practicable than the former design in Figure 2a.
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15             Modelling the device’s force displacement characteristics uses the fundamental thermodynamic
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                                                        p1 ) Ac = (V0 + Ax )   (V0   Ax )   ]Ac              (2)
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6                                                    2 A2 P0
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                                                        V0
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     where A is the piston area. The effective stiffness of the resetable device is therefore defined:
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                                                   2 A2 P0
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26   chamber of the device in Figure 2b. Equation (4) can then be used to design a device to produce a
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     set resisting force at a given displacement, or a set added stiffness. Since Equation (4) includes
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38   peak to peak (Figure 1b) or from the zero position to a displacement peak (Figure 1c) are reset at
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41   a displacement peak where the velocity is zero. As such, the valve reset time is not an imperative
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     energy by valve reset at the equilibrium position is doing so at a non-zero velocity. This approach
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48   makes the reset times much more important, and significantly differentiates these control laws.
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53   A one-fifth scale pneumatic-based prototype device has been constructed to enable experimental
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               the 1-3 and 2-4 control laws of Figures 1c-d. The ideal model of Equations (1)-(4) has been
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6              modified using experimental results to obtain a more realistic, non-linear device model [8]. Peak
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8              stiffness value varied from 185 kN/m to 236 kN/m, which are slightly lower than the design
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11             value of 250 kN/m. Differences were attributed to air loss due to valve flexibility, and inability to
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18             levels of approximately 20kN. This force level represents an internal cylinder pressure in the
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               diaphragm valves, with a maximum operating pressure of 10bar. It is important to note that, for
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42             control laws of Figures 1c-d, respectively. The hysteretic behaviour of the experimental device is
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44             very similar in shape to the idealised model, and magnitudes match well. Note that the major
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46             difference in the tensile portion of Figure 3b is due to loss of hydraulic test system pressure, with
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49             details in Mulligan et al [2, 8]. Based on these results, and to minimise the computational power
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51             required, the simpler idealised model of Equations (1)-(4) is used in developing the response
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6    The frequency of experimental testing is important for practical implementation of resetable
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8    devices, and should be done at frequencies similar to those expected in earthquake induced
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11   structural vibrations. Therefore, Figure 4 shows the experimental results of a prototype device
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13   using the 1-3 control law at 1Hz and 2Hz for amplitudes of 16.5 and 10mm respectively. Larger
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15   amplitudes cannot be tested at higher frequencies due to limitations in the testing equipment,
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18   however the 1-2Hz frequencies in Figure 4 for relatively large amplitude motion is representative
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20   of trends in typical earthquake records. Note that the experimental results of Figure 4 correspond
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     to a prototype device with shorter cylinder length to the one used to create Figure 3, and as such
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30   Total system delays including sensing response, valve reset and energy dissipation are also an
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     shows the load cell data and the valve control commands for an experimental test. Introducing a
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37   valve reset during a stroke shows the delays in the system from the time the valve reset command
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42   view of the valve reset in Figure 5b indicates that the valve reset command is given at 14.664
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44   seconds, and the force begins to drop at approximately 14.688 seconds. This results corresponds
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46   to a total delay between valve reset command and pressure drop of approximately 24ms, which is
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49   in the range of the valve reset times quoted with reference to the work of Barroso et al [4], and
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51   Hunt [1]. Furthermore, noting that the valve is closed again shortly after this reset, the total time
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                valve is closed again (~14,71s) is less than 50ms. This time includes the lag between the valve
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6               opening command being sent, the valve opening, pressure equilibrating, the valve closure
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8               command being sent, the valve closing, and pressure again beginning to increase within the
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11              cylinder. Hence, there is a total cycle for two valve operations of 50ms, which offers significant
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18              3.0 Analysis Methods
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21              This paper investigates the relative effectiveness of the three different control laws in Figures
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26              excited single degree of freedom structure fitted with a semi-active damper. The model structure
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28              includes internal structural damping of 5%. This value is commonly adopted by design codes and
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31              standards, is used in well-known textbooks [14], and has also been used in previous studies on
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33              damping systems [15]. The research utilises three earthquake suites from the SAC project [16],
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35              with 10 different time histories and two orthogonal directions for each history. The three suites
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38              represent ground motions having probabilities of exceedance of 50% in 50 years, 10% in 50
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40              years, and 2% in 50 years in the Los Angeles region, and are referred to as the low, medium and
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                high suites, respectively. Response statistics can thus be generated from the results of each
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50              Response spectra are generated for the three suites of ground motions, and spectral response plots
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52              were generated for the structural displacement, the structural force, the total base shear, and the
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                area under the response spectra in a seismically important T = 0.5 – 2.5 sec range. The structural
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     force is defined as the base shear for a linear, un-damped structure, whereas the total base shear
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6    is defined as the sum of the structural force and the resisting forces from the semi-active
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8    resetable device. The structural force is thus an indication of the required column strength, and
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11   the base shear is an indication of the required foundation strength.
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16   The reductions achieved by the addition of semi-active resetable devices are represented by
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18   reduction factors, normalised to the uncontrolled case results. These factors enable easy
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20   comparison of the different control laws and are a multiplicative factor. Each response spectra is
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     generated from 0.1 to 5.0 seconds in 0.1 second increments. These spectra were created for each
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     degree of freedom structure analysed.
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35   by a log-normal probability density function. Thus, the spectra can be analysed using the
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37   appropriate lognormal statistics for each suite and for all ground motions together [17]. Variables
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     within each ground motion suite may be represented by a median (the log-normal mean) and log-
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42   normal standard deviation [18], referred to herein as the dispersion factor ( ). For a log-normal
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     distribution of n samples xi , the log-normal geometric mean (median) x is defined:
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                                              1
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                                    ˆ                    ln( xi )                               (5)
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                Similarly, the log-normal-based standard deviation, or dispersion factor    is defined:
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                                                       1     n
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                                                                    (ln( xi / x))2
                                                                              ˆ                             (6)
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15              The statistical values used to indicate the change in response reduction factors are the median, or
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17              50th percentile, in Equation (5), and dispersion factor of Equation (6). The first metric indicates
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20              the expected value of the response while the second indicates the relative spread (or dispersion)
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22              over the suite of ground motions. Utilising complete earthquake suites rather than individual
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23
24              earthquakes eliminates the likelihood of erroneous conclusions being drawn about the viability of
                                                Pe

25
26
27              a control law due to atypical performance for a single earthquake. Thus, the two statistical
28
                                                           er

29                           ˆ
                parameters ( x , ) are plotted against each natural period increment to investigate overall trends.
30
31
32
                                                                    Re

33
34              Finally, empirical equations are derived to approximate the reduction factors for a given amount
35
36              of additional stiffness. These equations allow future applications of resetable devices to be
                                                                               vi

37
38
39              extended to accepted structural design analysis methods. The systematic bias introduced by
                                                                                     ew

40
41              incorporating these empirical equations into structural design analysis is also investigated.
42
43
44
45
46              Although this investigation focuses on single degree of freedom (SDOF) systems, the use of
47
48              semi-active resetable dampers can be easily extended for use in multiple degree of freedom
49
50
51              (MDOF) structures with configurations using single and multiple devices for suites of
52
53              probabilistically scaled ground motions [3, 4, 9, 19-21]. Specifically, research has shown that
54
55              placing resetable devices in different configurations throughout the SAC3 structure with total
56
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     authority of approximately 2000kN, representing 13.8% of building weight, can provide
4
5
6    geometric mean reductions in peak drift ranging from approximately 3-20%, and geometric mean
7
8    reductions in permanent drift as large as 83.5% [4]. Further research could investigate the
9
10
11   relative effects of the different resetable device control laws in MDOF structures. However, this
12
13   study focuses on their analysis for inclusion in fundamental design methods.
14
15
16
17
18   4.0 Results and Discussion
                         Fo
19
20
21   Each earthquake record within a given suite was used to simulate the structural response and the
22
                            r
23   maximum response value recorded for each period to generate the response spectra for every
24
                                      Pe

25
26   earthquake record at 0, 50, and 100% additional stiffness. The maximum structural force and
27
28   maximum total base-shear was also recorded and normalised to the uncontrolled state to give
                                             er

29
30
31   reduction factors. Therefore, the multiplicative structural force reduction factors give an
32
                                                     Re

33   indication of the change in the demand on the columns, whereas the total base-shear reduction
34
35   factors indicate the change in foundation demand.
36
                                                              vi

37
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39
                                                                   ew

40   4.1 Structural Force and Base Shear Results
41
42
43   The geometric mean structural force reduction factors for each suite are presented for the 1-3, 2-
44
45   4, and 1-4 control laws in Figures 6a-c respectively. The results indicate that the reductions are
46
47
48   largely independent of the suite used. Therefore, in Figure 6d, the structural force reduction
49
50   factors are averaged across all three suites and plotted for all three device types. These overall
51
52   average reduction factors indicate the relative ability of the different control laws at reducing the
53
54
55   seismic demand on the columns. Figure 6d indicates that the 1-4 device is about twice as
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                effective as the 1-3 and 2-4 devices. Specifically, the reductions factors are approximately 0.65-
4
5
6               0.8 for the 1-3 and 2-4 devices, which are very similar, and 0.4-0.5 for the 1-4 device. This result
7
8               is expected as the 1-3 and 2-4 devices only operate over two quadrants, whereas the 1-4 device
9
10
11              operates over all four quadrants, as shown in Figure 1. As such, the 1-3 and 2-4 devices provide
12
13              resistive forces for a smaller percentage (~50%) of each cycle and will consequently have shorter
14
15              active strokes, and thus store and dissipate less energy.
16
17
18
                                     Fo
19
20              The total base-shear reduction factors for 100% additional stiffness were obtained at every period
21
22
                for each record and device type. As with the structural force, the geometric mean base-shear
                                        r
23
24
                                                 Pe

25              reduction factor for each suite is plotted against natural period for each control law in Figures 7a-
26
27              c and the results averaged across all suites are given in Figure 7d. Figures 7a-c indicate the
28
                                                         er

29
30              relative reduction in total base shear across the three earthquake suites, whereas Figure 7d
31
32              indicates the relative ability of the three control laws at reducing the overturning demands on the
                                                                 Re

33
34
35
                foundation system. In this case, only the 2-4 device achieves significant reductions, with the 1-3
36
                                                                            vi

37              and 1-4 devices resulting in substantial increases across a majority of the spectrum. Again, the
38
39              results indicate that the reductions achieved for each device are largely suite invariant.
                                                                               ew

40
41
42
43
44              4.2 Displacement Spectral Area Reduction Factors
45
46
47              The structural force and base-shear reduction factors were observed to be largely suite invariant
48
49              and approximately constant within the natural period range of 0.5 to 2.5 seconds. The area under
50
51
52
                the displacement response spectra from structural period of 0.5 to 2.5 seconds is therefore
53
54              examined as an indication of the magnitude of the response over this range. This range is a
55
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     critical region for earthquake resistant design, above which the predominant design factor is wind
4
5
6    loading. The area under the displacement response spectra was numerically integrated for each
7
8    earthquake and normalised to the uncontrolled case to give area reduction factors. The geometric
9
10
11   mean of these area reduction factors was taken for each suite and then averaged across all three
12
13   suites due to the previous results displaying invariance to the type of ground motion. This
14
15   procedure was performed for the three control laws, with values of 0, 20, 50, 80, and 100%
16
17
18   additional resetable device stiffness, as shown in Figure 8. These area reduction factors can be
                         Fo
19
20   interpreted as an average reduction factor across the range of natural periods from 0.5 to 2.5 sec,
21
22
     representing a majority the constant velocity region of the spectra.
                            r
23
24
                                     Pe

25
26
27   The area reduction factors in Figure 8 show similar reductions for the 1-3 and 2-4 control laws.
28
                                             er

29
30   Both the 1-3 and the 2-4 fall well short of the reductions achieved for the 1-4 control law,
31
32   consistent with the other metrics. The reduction factors for the 1-3 and 2-4 control laws range
                                                     Re

33
34
35
     from 1.0 to approximately 0.65 as the additional stiffness is increased from 0% to 100%, with the
36
                                                             vi

37   2-4 control law slightly outperforming the 1-3 control law across the entire range. The 1-4
38
39   control law shows the greatest reduction in the displacement spectral areas, with a reduction
                                                                   ew

40
41
42   factor of approximately 0.4 for 100% additional stiffness.
43
44
45
46   4.3 Empirical Area Reduction Factors
47
48
49   To extend resetable devices to design analysis, empirical equations are fitted to the results to
50
51
52
     approximate the reduction factors for a given percentage of additional resetable device stiffness.
53
54   The area reduction factors calculated from the empirical equations are also plotted in Figure 8.
55
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                The form of the empirical equations is:
4
5
6
7
8                       R = 1/ B                                                                                    (7)
9
10
11              where R is the multiplicative reduction factor, and          is the divisive reduction factor, defined as:
12
13
14
15
16                                       K resetable
17
                        B = 1+ C                                                                                    (8)
                                         K structural
18
                                           Fo
19
20
21
22              where Kresetable is the additional stiffness provided by the resetable device, Kstructural is the
                                              r
23
24
                structural stiffness, and C is a constant dependant on the control law, and has a value of 1.43,
                                                         Pe

25
26
27              1.59, and 5.75 for the 1-3, 2-4, and 1-4 control laws respectively.
28
                                                                er

29
30
31
32              Furthermore, the empirical reduction factors can be utilised to quantify the additional stiffness in
                                                                       Re

33
34              terms of equivalent viscous damping, and compare the reductions achieved by the semi-active
35
36
                system to those that can be achieved by the use of a simple viscous damper. The overall effective
                                                                               vi

37
38
39              damping can be represented by an intrinsic damping component, and a hysteretic damping
                                                                                    ew

40
41              component, defined as an equivalent viscous damping [15]. The intrinsic damping is the inherent
42
43
44              structural damping, typically defined as 5% of critical for a spectral investigation and the
45
46              hysteretic component can be defined as:
47
48
49
50
51                                C K resetable
52                       eq   =                                                                                     (9)
53
                                  10 K structural
54
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     where   eq   is the equivalent viscous damping of the added resetable device. Equation (9) can be
4
5
6    used to show that 100% additional stiffness will produce 14.3%, 15.9%, and 57.5% equivalent
7
8    viscous damping for the 1-3, 2-4 and 1-4 control laws respectively.
9
10
11
12
13   4.4 Displacement Reduction Factor Results
14
15
16   The maximum displacement response was recorded for every simulation to obtain the
17
18   displacement spectra for every earthquake record and all three control laws at 0, 50 and 100%
                           Fo
19
20
21
     additional resetable stiffness. These displacement spectra are normalised to the uncontrolled case
22
                              r
23   to obtain the displacement reduction factors at every natural period. These reduction factors are
24
                                      Pe

25   plotted against natural period, for the 1-3, 2-4, and 1-4 control laws at 50% and 100% added
26
27
28   stiffness and presented in Figure 9. The reduction factor values given by Equations (7)-(8) are
                                              er

29
30   also plotted as horizontal lines in Figure 9, to graphically indicate how accurately these equations
31
32
     represent the actual displacement reduction factors. Although some variation from the empirical
                                                     Re

33
34
35   equation can be seen above T = 3.0 second period, Figure 9 shows that the empirical equations
36
                                                             vi

37   are appropriate over the entire constant velocity region of the spectra, between natural periods of
38
39
                                                                  ew

40   0.4 and 3.0 seconds.
41
42
43
44   To investigate the suite dependence of the reduction factors, the normalised reduction factors are
45
46
47   calculated. These normalised reduction factors are defined as the geometric mean reduction
48
49   factor derived from each suite divided by the overall geometric mean reduction factors from all
50
51
52
     ground motion records in all three suites. The normalised reduction factors give an indication of
53
54   the relative performance of the semi-active resetable devices across each suite, where a value
55
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                close to 1.0 indicates little deviation from the overall average. The results from this analysis are
4
5
6               presented in Figure 10, and show that the displacement reduction factors are largely suite
7
8               invariant, with values close to unity for most periods, for all device types and stiffness values.
9
10
11
12
13              4.5 Analysis of Bias and Spread
14
15
16              To investigate the systematic bias introduced by the use of the empirical equations to
17
18              approximate the displacement reduction factors, the multiplicative error factors are calculated.
                                     Fo
19
20
21
                These factors are determined by dividing the actual reduction factor by the reduction factor from
22
                                        r
23              Equation (7), for the corresponding control law and additional stiffness, at every period. The
24
                                                 Pe

25              results of this analysis are presented for all three control laws, at 50 and 100% additional
26
27
28              stiffness, in Figure 11. The multiplicative error factors show that over the constant velocity
                                                         er

29
30              region of T = 0.4-3.0 seconds the systematic bias introduced by the use of the empirical equations
31
32
                is generally less than ±20%, with the 2-4 control showing variations generally less than ±10%.
                                                                 Re

33
34
35
36
                                                                         vi

37              To give a further indication of the spread of the results, the dispersion factor, , of the normalised
38
39
                                                                               ew

40              displacement spectra are plotted in Figure 12 for each suite and device type at both 50% and
41
42              100% added stiffness. This figure gives an indication of the level of variation of these normalised
43
44              average results, and indicates the consistency with which the different control laws can mitigate
45
46
47              the response for a range of ground motion records. Dispersions ( ) of the displacement reduction
48
49              factors, shown in Figure 12, show that the 1-4 control law leads to noticeably larger record-to-
50
51
52              record variability than the 1-3 and 2-4 devices. Over the typical range of use, dispersion factors
53
54              in the order of   = 0.2-0.25 can be expected for the 1-3 and 2-4 control laws, whereas for the 1-4
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     damper this increases to     = 0.3-0.35. Again, the results are seen to be largely suite invariant.
5
6
7
8
9
10
11   5.0 Summary of Relative Performance
12
13   5.1 Performance of the 1-4 Control Law
14
15
16   Overall, the results show several, perhaps expected, phenomena. The 1-4 control law shows the
17
18   biggest reduction in both displacement response and structural force. However, a possible
                         Fo
19
20
21
     shortfall of the 1-4 control law is that, on average, it results in only minimal reductions, and in
22
                            r
23   many cases substantial increases, to the total base-shear force. This result is clearly evident in
24
                                       Pe

25   Figure 7, with reduction factors of approximately 0.9 at low structural natural periods, but at
26
27
28   higher periods the reduction factors increase to approximately 1.6. This peak value represents a
                                               er

29
30   60% increase in the total base shear and is a potentially undesirable result due to the increase in
31
32
     force and overturning demands on the foundation system. Such a result may be of little
                                                       Re

33
34
35   consequence for new, purpose designed structures, as the foundation can be strengthened to
36
                                                               vi

37   account for the use of a 1-4 device, which consequently lowers the demand on the columns
38
39
                                                                     ew

40   during seismic events to minimise structural damage.
41
42
43
44   5.2 Performance of the 1-3 Control Law
45
46
47   Figure 7d shows similar trends in base shear demands for the 1-3 and 1-4 control laws with
48
49
50
     slightly larger increases for the 1-3 device across a majority of the spectrum. As with the 1-4
51
52   device, the peak reduction factor is 1.6 representing a 60% increase in foundation demand.
53
54   Hence, the 1-3 device has the same undesirable impact as the 1-4 device with reductions in
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                displacement response coming at the expense of an increase in total base shear. More generally,
4
5
6               it may be less desirable to semi-actively resist only motion away from equilibrium for these
7
8               reasons.
9
10
11
12
13              5.3 Performance of the 2-4 Control Law
14
15
16              However, increased foundation loads would be of potentially high importance for retrofit
17
18              applications where it is neither practical, nor feasible, to strengthen the foundations. Such retrofit
                                     Fo
19
20
21
                applications are particularly important as a substantial majority of the built environment in 2025
22
                                        r
23              has likely already been constructed. The 2-4 control law only resists motion towards the
24
                                                 Pe

25              equilibrium position and releases energy at the equilibrium position. This approach is also very
26
27
28              effective at reducing displacement response, as seen in Figure 9, with reduction factors of
                                                         er

29
30              approximately 0.7-0.85, and 0.55-0.75 for 50 and 100% added stiffness respectively, similar to
31
32
                the 1-3 device. The spread of the results across the three suites in Figure 10 shows only minimal
                                                                 Re

33
34
35              deviations across the spectrum with values close to 1.0. The 2-4 control law is the most
36
                                                                         vi

37              consistent of the three control laws across all ground motion records, and indicates that the
38
39
                                                                               ew

40              performance of the 2-4 device, in particular, is suite invariant and is equally appropriate for both
41
42              the near field or far field ground motions described by these suites.
43
44
45
46
47              The significant advantage of the 2-4 control law over its counterparts is its ability to provide
48
49              displacement and structural force reductions, while also reducing the total base-shear transmitted
50
51
52
                to the foundation, as seen in Figures 7a-d. These figures show that while the 1-3 and 1-4 control
53
54              laws increase total base shear across almost the entire spectrum, the 2-4 control law reduces the
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     base shear across the same range of structural natural periods. This result is of particular interest
4
5
6    if considering retrofit of semi-active resetable devices to existing buildings where it is not
7
8    generally feasible to strengthen the foundation system to cope with the increased demand. More
9
10
11   specifically, 2-4 resetable damping offers unique opportunities to add supplemental damping that
12
13   are not readily available via more traditional, passive supplemental damping and retrofit devices.
14
15
16
17
18   Overall, the most applicable approach to sculpting hysteretic behaviour, by any means, depends
                            Fo
19
20   on the critical design parameters. If the foundations can be strengthened appropriately, and the
21
22
     reduction in column demand is the critical parameter, then it is best to implement the 1-4 control
                               r
23
24
                                       Pe

25   law that resists all motion. However, if the strength of the foundation is limited, or important in a
26
27   retrofit application, a 2-4 control law that resists only motion toward the centre is better. Finally,
28
                                              er

29
30   for any given percentage of additional stiffness the geometric mean or log-normal standard
31
32   deviation of the results within each suite is not notably affected by the suite for any of the three
                                                      Re

33
34
35
     control laws investigated.
36
                                                              vi

37
38
39   6.0 Capacity Spectrum Implementation for Simplified Seismic Analysis and Design
                                                                    ew

40
41
42   The results of the analysis in design lend themselves towards use in the capacity spectrum
43
44   method. The capacity spectrum method is useful where a realistic evaluation of the earthquake
45
46   hazard can be analysed based on the statistical analysis of past earthquake records. The seismic
47
48
49   demand spectrum is given as the lesser of:
50
51
52
53
                            Fa S s
54                   Cd =                                                                          (9)
55                           B
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                                      Fv S1
4                              Cd =                                                                        (10)
5                                     TB
6
7                                     Fv S1Td
8                              Cd =                                                                        (11)
9                                      T 2B
10
11              where Fa and Fv are adjustments on spectral acceleration for short and long periods at different
12
13
14
                soil classes; Ss and S1 are spectral acceleration at short periods and the one-second period; B is
15
16              the approximate damping reduction factor defined in Equations (7)-(8); T = the period of the
17
18              structure; and Td = the period at the junction of the constant spectral velocity and displacement
                                      Fo
19
20
21              portions of the spectra. For convenience, setting the value of Design Acceleration Intensity to 1g,
22
                                         r
23              and assuming normal soil, then FvS1 = 1g, and setting FaSs = 2.5FvS1 enables Equations (9)-(12)
24
                                                                  Pe

25
                to be simplified to the lesser of: Cd = 2.5/B, Cd = 1/TB, and Cd = Td/T2B. The customary code-
26
27
28              based acceleration-period design spectra can be transformed into Acceleration-Displacement
                                                                            er

29
30              Response Spectra (ADRS) by using the following transformation:
31
32
                                                                             Re

33
34
35                                                g
36                               = Cd T 2             2
                                                                                                           (12)
                                              4
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38
39
                                                                                          ew

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41              Thus Equations (10) and (11) respectively become:
42
43
44
45
46                                    Fv S1
                                                  2
                                                              g        1
47                               =                                2
                                                                                                           (13)
48                                     B                  4            Cd
49
50
                                      Fv S1               g
51                               =                            2
                                                                      Td                                   (14)
52                                     B              4
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     The corresponding ADRS are presented for the three control laws with 0, 20, 50, 80, and 100%
4
5
6    additional resetable stiffness in Figure 13. These spectra show the largest reductions for the 1-4
7
8    control law due to the larger values of the reduction factors, B, matching the spectral analyses.
9
10
11
12
13   7.0 Conclusions
14
15
16   Overall, this research demonstrates that the novel new two chamber, resetable device design
17
18
     allows a broader range of control laws than conventional semi-active devices. These broader
                          Fo
19
20
21   control laws provide the ability to semi-actively re-shape hysteretic behaviour to provide the
22
                             r
23   most appropriate hysteresis loop for a given application. Based upon the investigation described
24
                                      Pe

25
26   herein, the following conclusions can be drawn:
27
28
                                              er

29
30
31     •   The hysteretic behaviour can be re-shaped to minimise structural force or minimise base-
32
                                                      Re

33         shear depending on design requirements.
34
35     •   Analysis of displacement response and structural force indicate that all three device types
36
                                                             vi

37
38         reduce structural demand effectively. The 1-4 device achieves the highest reductions due to
39
                                                                   ew

40         its longer active strokes and consequently higher energy absorption and dissipation, but is
41
42
43
           also the most variable over all three suites.
44
45     •   All three control law devices are largely suite invariant indicating a robustness to the type
46
47         of ground motion encountered.
48
49
50     •   Displacement spectral area reduction factors showed substantial reductions achieved by the
51
52         use of up to 100% additional stiffness, with similar results between the 1-3 and 2-4
53
54
55
           devices. The largest reductions were again seen for the 1-4 device.
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                  •   The systematic bias introduced by use of empirical equations to the approximate the area
5
6                     reduction factors was generally less than 20%. The analysis is easily extended to the
7
8                     capacity spectrum method to incorporate resetable devices into future design analyses.
9
10
11                •   A major distinction between the control laws is that the 2-4 device is the only device that
12
13                    reduces total base shear, as well as structural force and displacement response. These
14
15
                      devices are unique in their ability to add damping only in the 2nd and 4th quadrants, and
16
17
18                    provide a unique opportunity for retrofit of structures with limited ability to manage
                                    Fo
19
20                    increased base shear.
21
22
                  •   The ability to actively re-shape hysteretic behaviour for given applications holds significant
                                       r
23
24
                                                 Pe

25                    promise in the ability to mitigate seismic structural damage. Finally, it is important to note
26
27                    that it is not the exact reductions that are important, but rather the unique characteristics of
28
                                                         er

29
30                    the device types and their corresponding effect on structural response parameters.
31
32
                                                                 Re

33
34
35
                Overall, the methods presented can be generalised as an effective analysis approach to other
36
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37              semi-active or passive devices. The resetable device control laws used can also be readily
38
39              generalised to other working fluids or even to similar devices, such as viscous dampers. Most
                                                                               ew

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41
42              importantly, the approach presented is amenable to direct use in performance based design
43
44              methods, increasing the ability to consider these devices and methods in more practical
45
46
47
                experiments.
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4    References
5
6    [1]    S. Hunt, "Semi-active smart-dampers and resetable actuators for multi-level seismic
7           hazard mitigation of steel moment resisting frames," Master of Engineering (ME) thesis,
8
9           Dept of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand.,
10          2002.
11   [2]    K. J. Mulligan, J. G. Chase, A. Gue, T. Alnot, G. W. Rodgers, J. B. Mander, R. B. Elliott,
12          B. L. Deam, L. Cleeve, and D. Heaton, "Large Scale Resetable Devices for Multi-Level
13          Seismic Hazard Mitigation of Structures," Proc. 9th International Conference on
14
15
            Structural Safety and Reliability (ICOSSAR 2005), Rome, Italy, June 19-22., 2005.
16   [3]    J. E. Bobrow, F. Jabbari, and K. Thai, "A New Approach to Shock Isolation and
17          Vibration Suppression Using a Resetable Actuator," ASME Transactions on Dynamic
18          Systems, Measurement, and Control., vol. 122, pp. 570-573, 2000.
                        Fo
19   [4]    L. R. Barroso, J. G. Chase, and S. Hunt, "Resettable smart dampers for multi-level
20
21
            seismic hazard mitigation of steel moment frames," Journal of Structural Control, vol.
22          10(1), pp. 41-58, 2003.
                           r
23   [5]    L. M. Jansen and S. J. Dyke, "Semiactive Control Strategies for MR Dampers:
24          Comparative Study," ASCE Journal of Engineering Mechanics, vol. 126(8), pp. 795-803,
                                    Pe

25          2000.
26
27
     [6]    O. Yoshida and S. J. Dyke, "Seismic Control of a Nonlinear Benchmark Building Using
28          Smart Dampers," ASCE Journal of Engineering Mechanics, vol. 130, pp. 386-392, 2004.
                                            er

29   [7]    F. Jabbari and J. E. Bobrow, "Vibration Suppression with a Resetable Device," ASCE
30          Journal of Engineering Mechanics, vol. 128(9), pp. 916-924, 2002.
31   [8]    K. Mulligan, J. Chase, A. Gue, J. Mander, T. Alnot, B. Deam, G. Rodgers, L. Cleeve, and
32
            D. Heaton, "Resetable Devices with Customised Performance for Semi-Active Seismic
                                                   Re

33
34          Hazard Mitigation of Structures," Proc of NZ Society for Earthquake Engineering 2005
35          Conference (NZSEE 2005), March 11-13, Wairakei, New Zealand., 2005.
36   [9]    J. G. Chase, L. Barroso, and S. Hunt, "The impact of total acceleration control for semi-
                                                           vi

37          active earthquake hazard mitigation," Journal of Engineering Structures, Elsevier
38          Science, vol. 26(2), pp. 201-209, 2004.
39
                                                                ew

40   [10]   K. J. Mulligan, M. Fougere, J. B. Mander, J. G. Chase, B. Deam, G. Danton, and R. B.
41          Elliott, "Semi-Active Rocking Wall Systems for Enhanced Seismic Energy Dissipation,"
42          8th US National Conference on Earthquake Engineering (8NCEE), San Francisco, April
43          18-21, 10-pages., 2006.
44   [11]   G. W. Rodgers, J. B. Mander, J. G. Chase, K. J. Mulligan, B. Deam, and A. J. Carr, "Re-
45
46          Shaping Hysteretic Behaviour Using Resetable Devices to Customise Structural Response
47          and Forces," 8th US National Conference on Earthquake Engineering (8NCEE), San
48          Francisco, April 18-21, 10-pages., 2006.
49   [12]   S. J. Dyke and B. F. Spencer, "Modeling and control of magnetorheological dampers for
50          seismic response reduction," Smart Materials and Structures, vol. 5, pp. 565-575, 1996.
51
52
     [13]   B. F. Spencer, S. J. Dyke, M. K. Sain, and J. Carlson, "Phenomenological Model of
53          Magnetorheological Damper," ASCE Journal of Engineering Mechanics, vol. 123, pp.
54          230-238, 1997.
55   [14]   A. K. Chopra, "Dynamics of Structures : Theory and Applications to Earthquake
56          Engineering," Prentice-Hall Inc, New Jersey, USA, 729pp, 1995.
57
58                                                 26
59
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                [15]   G. Pekcan, J. B. Mander, and S. S. Chen, "Fundamental Considerations for The Design of
4
5                      Non-linear Viscous Dampers," Earthquake Engineering and Structural Dynamics, vol.
6                      28, pp. 1405-1425, 1999.
7               [16]   P. Sommerville, N. Smith, S. Punyamurthula, and J. Sun, "Development of Ground
8                      Motion Time Histories For Phase II Of The FEMA/SAC Steel Project, SAC Background
9
                       Document Report SAC/BD-97/04," 1997.
10
11              [17]   R. P. Kennedy, C. A. Cornell, R. D. Campbell, S. Kaplan, and H. F. Perla, "Probabilistic
12                     Seismic Safety Study Of An Existing Nuclear Power Plant.," Nuclear Engineering and
13                     Design, vol. 59, pp. 315-338, 1980.
14              [18]   E. Limpert, W. A. Stahel, and M. Abbt, "Log-normal distributions across the sciences:
15                     keys and clues.," Bioscience 2001; vol 51(5), pp. 341-352., 2001.
16
17              [19]   K. Mulligan, J. Chase, L. Barroso, and S. Hunt, "Impact of Control Architecture in the
18                     Reliability of Resetable Device Controlled Tall Structures," Proc. 9th Intl Conf on
                                   Fo
19                     Structural Safety and Reliability (ICOSSAR 2005) Rome, Italy, June 19-22, 8-pages,
20                     ISBN 90-5966-040-4., 2005.
21              [20]   J. G. Chase, K. J. Mulligan, L. R. Barroso, and S. J. Hunt, "Actuator-Actuator Interaction
22
                       and Instability in Decentralised Semi-Active Control of Non-Linear Seismically Excited
                                      r
23
24                     Tall Structures,," Proc. 9th International Conference on Structural Safety and Reliability
                                               Pe

25                     (ICOSSAR 2005), Rome, Italy, June 19-22, 8-pages, ISBN 90-5966-040-4., 2005.
26              [21]   J. G. Chase, L. Barroso, and S. Hunt, "Impact of Decentralized Semi-Active Control on
27                     the Stability of Tall Structures Under Seismic Loading,," Proc. 2003 Pacific Earthquake
28
                       Engineering Conference (PCEE), CD-ROM, 8 pages, Christchurch, NZ, February 13-15.,
                                                       er

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30                     2003.
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5                             F       FS
                                                                            F     FB>FS
                                                      F
6                       a)                                              4           1

7
                                           +                      =
8
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10                                                                      3           2

11
                                                                            F     FB>FS
12                            F       FS              F
                        b)
13
14                                         +                      =
15
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18                                                                                FB>FS
                         Fo
19                            F       FS               F                    F
                        c)
20
21                                         +                      =
22
                            r
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26                            F       FS               F                    F     FB=FS
                        d)
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28                                         +                      =
                                               er

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                                                     Re

33   Figure 1: Schematic hysteresis for a) viscous damping, b) a 1-4 device, c) a 1-3 device, and d) a
34   2-4 device. Quadrants are labelled in the first panel, and FB = total base shear, FS = base shear for
35       a linear, undamped structure. FB > FS indicates an increase due to the additional damping.
36
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6               a)
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8                                                                         Valve
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                                     k0
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15                                                  Mass
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                                   Fo
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                b)
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                                                        er

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40              Figure 2: Schematic representation of semi-active devices attached to a single degree-of-freedom
41               system a) Schematic of conventional resetable device using an external plumbing system with a
42              single valve to connect the two sides of the piston. b) Schematic of independent chamber design.
43
44
                               Each valve vents to atmosphere for a pneumatic, or air-based device.
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       b)                                                                         c)
                                                            Pe

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26                4000                                                                        2500

27                3000
                                                                                              2000
28                                                                                            1500
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29                2000
                                          Model                                                                                  Experiment
                                                                                              1000
30
                                                                                  Force (N)
     Force (N )




                  1000
31                                                                                             500
32                   0                                                                           0
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33
                                                                                               -500
34                -1000
35                                                                                            -1000
                  -2000                               Experiment                                        Model
36                                                                                            -1500
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37                -3000                                                                       -2000
38                    -15     -10        -5        0        5        10     15                    -20   -15      -10     -5       0       5       10    15
                            Piston Displacement from Centre Position (mm)                               Piston Displacement from Centre Position (mm)
39
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41
42       Figure 3: a) Dimensioned drawing of the prototype resetable device, and the Force-displacement
43         curves for actuator in a single degree of freedom structure showing both the analytical model
44         prediction and experimental result. Ground motion is a 2 m/s2 sine wave of frequency 0.1Hz.
45
46
                   Figure 3 b) shows a 1-3 control law, and Figure 3 c) shows a 2-4 control law.
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                Figure 4: Experimental hysteresis loops for a prototype device tested with harmonic inputs of a)
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52                  10 mm amplitude and 2 Hz Frequency, and b) 16.5 mm amplitude and 1 Hz frequency.
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5              a)
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9                             Initial valve                     Force drop
10                                reset                         from valve
11                             command                             reset
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14                                                              Force begins
15                                                              to increase as
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                                                                 valve closes
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                       Fo
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                                              er

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35                       Force begins
36                          to drop
                           ~14.688s
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38                       Valve reset                      Force begins
39                       command                           to increase
                                                                ew

40                        ~14.664s                            again
41                                                           ~14.71s
42
43                                                  Command
44                                                  for valve
                                                     to close
45                                                   ~14.69s
46
47
48
49
50
51   Figure 5: Total system delays shown by introducing a valve reset during compression, a) shows
52
53
          the overall force behaviour and b) shows an enlargement of the critical reset region.
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                                      r
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                                                       er

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                Figure 6: Reduction factors for the structural force for a linear, undamped structure, where (a-c)
                                                                            ew

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41                 show each suite and device and (d) shows the three control laws averaged over all suites.
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                                            er

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38
     Figure 7: Reduction factors for the total base shear of a structure where (a-c) show the reduction
39
                                                                 ew

40    factors for each suite and device, and (d) shows the averages across all suites for each device.
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34              Figure 8: Reduction factors for the area under the displacement response spectra between 0.5
35                and 2.5 second periods normalised to the uncontrolled case and averaged across all suites.
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                                                            1-3 Control Law - 50% added stiffness                                                                                   1-3 Control Law - 100% added stiffness
6
7                                                                                                       Low
                                                                                                        Medium
                                                                                                                                                                                                                                 Low
                                                                                                                                                                                                                                 Medium
8                                        1
                                                                                                        High
                                                                                                                                                              1
                                                                                                                                                                                                                                 High
                                    d




                                                                                                                                                         d
     Displacement Reduction factor B




                                                                                                                          Displacement Reduction factor B
9
10                                      0.8                                                                                                                  0.8

11
12                                      0.6                                                                                                                  0.6

13
14                                      0.4                                                                                                                  0.4

15
16                                      0.2                                                                                                                  0.2

17
18                                       0
                                              0   0.5   1     1.5     2      2.5     3     3.5      4    4.5     5
                                                                                                                                                              0
                                                                                                                                                                   0   0.5      1      1.5     2     2.5      3     3.5      4      4.5   5
                                                                      Fo
19                                                                  Period (seconds)                                                                                                         Period (seconds)
20
21                                                          2-4 Control Law - 50% added stiffness                                                                                   2-4 Control Law - 100% added stiffness

22                                                                                                      Low                                                                                                                      Low
                                                                         r
23                                                                                                      Medium
                                                                                                        High
                                                                                                                                                                                                                                 Medium
                                                                                                                                                                                                                                 High
                                         1                                                                                                                    1
                                    d




                                                                                                                                                         d
24
     Displacement Reduction factor B




                                                                                                                          Displacement Reduction factor B
                                                                                             Pe

25
                                        0.8                                                                                                                  0.8
26
27
                                        0.6                                                                                                                  0.6
28
                                                                                                           er

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                                        0.4                                                                                                                  0.4
30
31
                                        0.2                                                                                                                  0.2
32
                                                                                                                     Re

33
                                         0                                                                                                                    0
34                                            0   0.5   1     1.5     2      2.5     3     3.5      4    4.5     5                                                 0   0.5      1      1.5     2     2.5      3     3.5      4      4.5   5
                                                                    Period (seconds)                                                                                                         Period (seconds)
35
36
                                                                                                                                                               vi

                                                            1-4 Control Law - 50% added stiffness                                                                            Displacement - 1-4 Control Law -100% added stiffness
37
                                                                                                        Low                                                                                                                      Low
38                                                                                                      Medium                                                                                                                   Medium
                                                                                                        High                                                                                                                     High
39                                       1                                                                                                                    1
                                    d




                                                                                                                                                         d




                                                                                                                                                                             ew
     Displacement Reduction factor B




                                                                                                                          Displacement Reduction factor B




40
41                                      0.8                                                                                                                  0.8

42                                                                                                                                                                                                         Equation (7)
43                                      0.6                                                                                                                  0.6

44
45                                      0.4                                                                                                                  0.4

46
47                                      0.2                                                                                                                  0.2

48
49                                       0
                                              0   0.5   1     1.5     2      2.5     3     3.5      4    4.5     5
                                                                                                                                                              0
                                                                                                                                                                   0   0.5      1      1.5     2     2.5      3     3.5      4      4.5   5
50                                                                  Period (seconds)                                                                                                         Period (seconds)
51
52                                      Figure 9: Displacement Reduction Factors for 50% and 100% added stiffness for all three
53                                                 control laws. Also shown are the reduction factors given by Equation (7).
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                                                                          ew

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50              Figure 10: Displacement Reduction Factors for each suite normalised to the average value across
51                          all ground motion records for 50% and 100% added stiffness, for 1-3, 2-4 and 1-4
52                                                           control laws.
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                                                             Earthquake Engineering and Structural Dynamics                                                                                              Page 38 of 40


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3                                          1-3 Control Law - 50% added stiffness                                                               1-3 Control Law - 100% added stiffness
4                       2
                                                                                       Low
                                                                                                                            2
                                                                                                                                                                                            Low
5                      1.8                                                             Medium                              1.8                                                              Medium
                                                                                       High                                                                                                 High
6                      1.6                                                                                                 1.6
7                      1.4                                                                                                 1.4
8
     Ractual/Rtheory




                                                                                                         Ractual/Rtheory
                       1.2                                                                                                 1.2
9
                        1                                                                                                   1
10
11                     0.8                                                                                                 0.8

12                     0.6                                                                                                 0.6

13                     0.4                                                                                                 0.4
14                     0.2                                                                                                 0.2
15
                        0                                                                                                   0
16                           0   0.5   1     1.5     2      2.5     3     3.5      4    4.5     5                                0   0.5   1     1.5      2     2.5     3      3.5      4    4.5     5

17                                                 Period (seconds)                                                                                    Period (seconds)

18                      2
                                           2-4 Control Law - 50% added stiffness
                                                                                                                            2
                                                                                                                                               2-4 Control Law - 100% added stiffness
                                                     Fo
19                                                                                     Low                                                                                                  Low
                       1.8                                                             Medium                              1.8                                                              Medium
20                                                                                     High                                                                                                 High

21                     1.6                                                                                                 1.6

22                     1.4                                                                                                 1.4
                                                        r
23
     Ractual/Rtheory




                                                                                                         Ractual/Rtheory
                       1.2                                                                                                 1.2

24                      1                                                                                                   1
                                                                            Pe

25
                       0.8                                                                                                 0.8
26
                       0.6                                                                                                 0.6
27
28                     0.4                                                                                                 0.4
                                                                                          er

29                     0.2                                                                                                 0.2

30                      0                                                                                                   0
                             0   0.5   1     1.5     2      2.5     3     3.5      4    4.5     5                                0   0.5   1     1.5      2     2.5     3      3.5      4    4.5     5
31                                                 Period (seconds)                                                                                    Period (seconds)
32
                                                                                                    Re

                                           1-4 Control Law - 50% added stiffness                                                               1-4 Control Law - 100% added stiffness
33                      2                                                                                                   2
                                                                                       Low                                                                                                  Low
34                     1.8                                                             Medium                              1.8                                                              Medium
                                                                                       High                                                                                                 High
35                     1.6                                                                                                 1.6
36
                       1.4                                                                                                 1.4
                                                                                                                            vi

37
     Ractual/Rtheory




                                                                                                         Ractual/Rtheory




                       1.2                                                                                                 1.2
38
39                      1                                                                                                   1
                                                                                                                                       ew

40                     0.8                                                                                                 0.8

41                     0.6                                                                                                 0.6
42                     0.4                                                                                                 0.4
43
                       0.2                                                                                                 0.2
44
45                      0
                             0   0.5   1     1.5     2      2.5     3     3.5      4    4.5     5
                                                                                                                            0
                                                                                                                                 0   0.5   1     1.5      2     2.5     3      3.5      4    4.5     5
46                                                 Period (seconds)                                                                                    Period (seconds)
47    Figure 11: Displacement reduction factors normalized to the theoretical value given by Equation (7).
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                                                                   ew

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50              Figure 12: Coefficients of variation, , for the displacement reduction factors.
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                                                       vi

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                                                            ew

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55   Figure 13: Acceleration Displacement Response Spectra (ADRS) for the three control laws.
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