Resetable semi-active damping also offers the unique opportunity to sculpt the structural ... Figure 1: Schematic of semi-active resetable actuators ... the initial pressures in both chambers are P0 with initial volumes V0 the resisting force is ...
Hybrid experimental analysis of semi-active rocking wall systems K.J. Mulligan, M. Fougere, J.B. Mander, J.G. Chase, B.L. Deam, G. Danton & R.B. Elliot Departments of Mechanical and Civil Engineering, University of Canterbury, Christchurch
2006 NZSEE Conference
ABSTRACT: Rocking walls are an effective method of dissipating seismic response energy and mitigating damage. Semi-active resetable devices have shown significant potential to dissipate energy, customize hysteretic behavior and reduce damage. Hence, the addition of a resetable device within a rocking wall can further improve the overall energy management during seismic events. A scaled semi-active rocking wall system, designed for a large open structure, is analysed using real-time, high-speed hybrid testing. The semi-active devices are controlled to provide supplementary resistance only for the upward rocking motion of the wall, providing semi-active energy dissipation over half of each cycle and relying on radiation damping for the other half. An validated model of the semi-active devices is used to examine the response of a full scale rocking wall system to a suite of earthquake ground motions to prove the overall concept. Overall, similar semi-active rocking walls could also be used as supplemental, lowfootprint response energy management systems in retrofitting a variety of structures.
1.
INTRODUCTION
Semi-active control is emerging as an effective method of mitigating structural damage from large environmental loads, with two main benefits over active and passive solutions. First, a large power/energy supply is not required. Second, they provide the broad range of control that a tuned passive system cannot, making them better able to respond to changes in structural behaviour due to non-linearity, damage or degradation. In this paper, a semi-active rocking wall is designed, analysed and validated in real-time, high-speed hybrid testing. Semi-active devices are particularly suitable where the device may not be required to be active for extended periods (Bobrow et al., 2000). The potential of semi-active systems to mitigate damage during seismic events is well documented (e.g. Barroso et al., 2003; Jansen and Dyke, 2000; Yoshida and Dyke, 2004). Instead of altering the damping of the system, resetable devices non-linearly alter the stiffness with the stored energy being released as the working fluid reverts to its initial pressure on resetting. In this case, semi-active resetable devices could be readily located in a hollow-core pre-cast rocking wall to provide added response energy management. Resetable semi-active damping also offers the unique opportunity to sculpt the structural hysteresis loop by actively controlling the device valve and reset times (Mulligan et al, 2005; Rodgers et al, 2006). For this rocking system the semi-active device provides an added restoring force on upwards rocking, restricting rotation. At each cycle’s peak rocking amplitude, the stored energy is released. The wall rocks back under self weight, dissipating further energy on impact.
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This process occurs for each cycle, rather than for only the 1-2 cycles of significant energy dissipation achieved using passive, pre-tensioned cable designs (Ajab et al, 2003). 2.
DEVICE DYNAMICS
Resetable devices are fundamentally hydraulic spring elements in which the un-stretched spring length can be reset to obtain maximum energy dissipation from the structural system (Bobrow et al. 2000). Energy is stored by compressing the working fluid as a piston is displaced. When the piston reaches its maximum displaced position, the stored energy is also at a maximum. At this point, the stored energy can be released by discharging the air to the non-working side of the device, thus resetting the un-stretched spring length, as shown in Figure 1a. Figure 1b shows a modified device design that treats each chamber independently. Independent valve control enables more diverse control laws and hysteresis loop re-shaping (Mulligan et al, 2005). Valve
Valve k0 Mass
Cylinder
a
Piston
b
Figure 1: Schematic of semi-active resetable actuators
Given that air is an ideal gas it obeys the law:
pV γ = c
(1) where γ is the ratio of specific heats, c is a constant and p and V are respectively the pressure and volume in a chamber of the device (Bobrow et al. 2000). Assuming the piston is centered and the initial pressures in both chambers are P0 with initial volumes V0 the resisting force is defined:
[
]
−γ −γ F ( x ) = ( p2 − p1 ) Ac = (V0 + Ax ) − (V0 − Ax ) Ac
(2)
Assuming small motions, Equation (2) can be linearized. F (x ) = −
2 A2γP0 x V0
(3)
where A is the piston area. Hence the effective stiffness of the resetable device is defined: k1 =
2. A2 .γ .P0 V0
(4)
Equations (1)-(4) are used to develop validated, non-linear device models (Mulligan et al, 2005). 2
3.
ANALYSIS METHODS
The rocking wall system analysed is a scaled version of a wall designed for a large, open structure. The basic dimensions are 0.45x5m with a mass of 2802kg . Each rocking wall supports a portion of a lumped roof mass, as shown in Figure 2. The resetable device can be located within hollow sections of the wall, with a schematic of the resulting forces shown in Figure 2. The semi-active device provides an added restoring force on upward rocking motion of the wall. When the wall reaches its peak rotation for any cycle, the energy stored in the device is released and the wall rocks downward purely under self weight. On impact with the ground 15% of the walls rocking energy is assumed to be absorbed, based on velocity. Hence, allowing free return without added semi-active resistance enables maximum energy dissipation from radiation damping on impact. If the remaining energy is sufficient the wall rocks up in the opposite direction where the semi-active device once again provides a semi-active restoring force. Roof Wr
Wr
R
H
Fact
α B
O’
θ
O
O Fact Wr
Figure 2: Schematic of rocking system showing semi-active device.
The linearised equation of motion of the wall rotating about rocking point O or O’ is defined: ••
I θ − MgHθ m MgB + Fact B = F (t )2 H
(5)
••
where I is the mass moment of inertia, θ is the rotational acceleration about the rotation point, M is the total mass of the system, g is acceleration due to gravity, H is the height to the effective centre of mass, θ is the rotation about O or O’, B is the width of the wall, Fact is the semi-active force, and F(t) is the applied force due to ground motion. Analysis is carried out using the real-time high-speed hybrid testing procedure where the rocking wall is represented by a computational model and the semi-active device is a physical substructure in a dynamic test rig. Real-time control and physical-virtual interface management is provided by a dSpace control prototyping system utilising Simulink. The hybrid test analysis procedure has the following steps, which are also illustrated in Figure 3, for any time step.
3
•
Wall model calculations determine the rotation of the wall depending on the ground motion and other forces
•
Rotation of the wall is converted into the linear displacement the actuator would experience when contained within the wall and this signal is sent to the dynamic test rig
•
Valve control for the semi-active devices is determined based on the current time step displacement and control law defined
•
Dynamic test rig supplies the displacement to the physical semi-active device
•
Force developed in the device is returned to the virtual system to be used in the subsequent time-step calculation.
The process is repeated step-wise for the complete ground motion record. Absolute Newmark–β with constant average acceleration integration was employed. A time step of 0.001 seconds was chosen for the entire process, as the error in each time step is small enough that no equilibrium iteration is required, enabling a rapid and simple test procedure. Finally, by using a physical fullscale, or near full-scale, device should provide a more realistic set of results.
Physical system Displacement command
Virtual System
Valve Control Measured Force and Displacement
Figure 3. Representation of major components and steps in hybrid testing procedure
Response of the wall system was examined for the odd half of the medium suite of ground motions from the SAC project, which are scaled for a 10% in 50 years probability of occurrence in the Los Angeles area (Sommerville et al, 1997). This suite is used because it contains a range of near and far field ground motions of significant size to cause significant rocking. This choice thus represents a compromise over the range of magnitudes and ground motion types possible.
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Finally, a non-linear analytical semi-active device model was developed and verified by comparing the results to the experimental results. This model captures all the important dynamics of the devices allowing further simulation and analysis for a full size wall (1.2x8m) with appropriately scaled semi-active devices. These results can then be performed for much larger sets of ground motions and statistically summarized.
4.
RESULTS AND DISCUSSION
In general, the addition of the semi-active devices to the rocking wall system decreases the angle of rotation of the wall for each cycle as illustrated in Figure 4. This result indicates the efficacy of the semi-active devices to provide supplementary damping to the structural system. Of note in the rocking wall application, is the ability to provide damping on each rocking cycle as opposed to only the peak rocking cycles when more traditional pre-stressed tendons are used (Ajab et al, 2004). Figure 5 shows a typical force-displacement response for the semi-active device where the lower flat portions are areas of minimal resistance due to friction. Rotation about point 0
0.03
Uncontrolled uncontrolled
0.02
Controlled controlled
theta (rad)
0.01 0 -0.01
Rotation about point 0’
-0.02 -0.03
0
5
10
15 time (sec)
20
25
30
Figure 4. Uncontrolled and controlled hybrid test responses for the Imperial Valley (1979) ground motion. 1500
500 Actuator Force (N)
Device Force (N)
1000
0
-500
-1000
-1500 -0.015
-0.01
-0.005
0 0.005 linear displac ement (m)
0.01
0.015
0.02
Displacement (m)
Figure 5. Typical Force-Displacement response of the hybrid tested semi-active device illustrating ability of response to be manipulated to only provide large forces when required in specific parts of the response cycle.
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However, Figure 6 shows that the amplitude of rocking is not always reduced. The result can be highly dependent on the relative timing of the input ground motion and the wall rocking motion. As the addition of the semi-active devices changes the frequency of rocking, the large peaks of the ground motion will occur when the wall is in rotational different positions for the uncontrolled and semi-active cases. Hence, if a large pulse occurs when the wall is returning to its neutral (stationary) position, the pulse will return the wall more rapidly to its centre position. Alternatively, if the wall is rocking away from the centre position when the pulse occurs, the subsequent rocking cycles will be larger. Loma Prieta, Gilroy 3 2
m/s
2
1 0 -1 -2 large pulse -3 15
20
25 time (sec)
30
35
0.015
theta (rad)
0.01 0.005
uncontrolled rocking towards centre position
c ontrolled
0 -0.005 rocking away from centre position
-0.01 -0.015 15
subsequent c ycles are larger for controlled c ase compared to unc ontrolled case 25 30 time (sec)
20
35
Figure 6. Effect of semi-active devices changing the period of rocking motion in hybrid testing The plot only shows the portion when the wall is rocking, prior to which the acceleration was insufficient to cause the wall to rock.
Similar results were obtained for the remainder of the medium suite ground motions examined using the hybrid test procedure. In most cases, reductions, similar to Figure 4, were observed. However, as illustrated in Figure 6, the non-linear effect of adding semi-active devices must be considered. Hence, the suite-based analysis approach offers the opportunity to classify the system response to a wider variety of ground motions and to use statistics amenable to use in currently used performance-based design methods. The non-linear analytical semi-active device model developed exhibits close correlation with the actual device. Figure 7 shows the hybrid test response and the simulated analytical response using the non-linear model. The accuracy of the model allows scaling of the semi-active devices for a variety of applications and analyses. Similarly accurate results with errors less than 1-5% on
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peak or average values indicate that the model is a very accurate representation of the device. Thus, a full scale rocking wall was examined and a summary of results is presented in Table 1. Three different sized devices, represented by different stiffness values, were examined. All the reduction factors (uncontrolled / controlled) are above 1.0, indicating that over the entire earthquake suite the amplitude of rocking is reduced. In addition, the geometric mean of the additional equivalent damping added by the device shows an increase of at least 5% damping to the rocking wall system for all three devices. Loma Prieta, Gilroy
m/s
2
5 0
theta (rad)
-5
0.02
5
10 time (sec )
15
20
5
10 time (sec )
15
20
10 time (sec )
15
20
a) hybrid result
0 -0.02
theta (rad)
0
0
0.02
b) analytical result
0 -0.02
0
5
Figure 7. Comparison of hybrid and non-linear analytical model test results for the Loma Prieta ground motion.
Table 1: Summary of reduction factors (uncontrolled/controlled) for rocking system response to medium suite
Metric
K=1000 kN/m
K = 5000 kN/m
K= 10000 kN/m
R.F geometric mean
1.01
1.14
1.21
R.F multiplicative variance
1.10
1.27
1.43
ξ geometric mean
5.11
5.47
7.12
ξ multiplicative variance
1.15
2.13
2.30
R.F – reduction factor, ξ – additional equivalent damping provided by devices
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5.
CONCLUSIONS
Semi-active devices can be used to significantly reducing the peak rotations of seismically excited rocking wall systems, thus enhancing the performance and energy dissipation. Analysis using a suite of ground motions that were probabilistically scaled for an equal likelihood of occurrence indicates the added energy dissipation is available for a realistic range of near and far field seismic events of significant magnitude. Importantly, the variation in results over this suite also indicates that, for a given rocking wall, the results are very dependent on the specific ground motion. Hence, it is critical that suites of events be used in designing or developing similar passive or semi-active system to better identify the overall efficacy of the entire semi-active system and ensure desired performance. For semi-active system this result occurs due to way the addition of semi-active devices to a rocking wall alters the period of rocking. Hence, the relative timing of the ground and rocking motion can be very different for the uncontrolled and controlled cases, leading to increases in the rocking angle for some periods of some ground motions, which is a novel and interesting result. Finally, the non-linear analytical device model developed was verified by comparing the hybrid test and analytical results over a suite of ground motions. As a result, scaled models can be applied to develop and analyze a wider variety of semi-active rocking wall structural applications.
REFERENCES Abdul Hamid N.,Y., (2006) “The Seismic Performance of Rocking Hollowcore Wall Structures” PhD Thesis, University of Canterbury. Ajab, JJ, Pekcan, G, Mander, JB, 2004. “Rocking Wall–Frame Structures with Supplemental Tendon Systems,” ASCE J. Eng. Mechanics, vol. 130, pp. 895-903. Barroso, L R, Chase, J G and Hunt, S J, 2003. "Resetable Smart-Dampers for Multi-Level Seismic Hazard Mitigation of Steel Moment Frames," J. of Structural Control, vol. 10(1), pp. 41-58. Bobrow, J E, Jabbari, F, Thai, K, 2000. “A New Approach to Shock Isolation and Vibration Suppression Using a Resetable Actuator,” ASME Transactions on Dynamic Systems, Measurement and Control, vol 122, pp. 70-573. Jansen, L M and Dyke, S J, 2000. “Semiactive Control Strategies for MR Dampers: Comparative Study,” J. of Eng. Mechanics, vol. 126(8), pp. 795-803. Mulligan, KJ, Chase, JG, Gue, A, Alnot, T, Rodgers, GW, Mander, JB, Elliott, RE, Deam, BL, Cleeve, L and Heaton, D, 2005. “Large Scale Resetable Devices for Multi-Level Seismic Hazard Mitigation of Structures,” Proc. 9th International Conference on Structural Safety and Reliability (ICOSSAR 2005), Rome, Italy, June 1922. Sommerville, P, Smith, N, Punyamurthula, S, and Sun, J, 1997. "Development of Ground Motion Time Histories for Phase II of the FEMA/SAC Steel Project." SAC Background Document Report No. SAC/BD-97/04. Yoshida, O and Dyke, S J, 2004. “Seismic Control of a Nonlinear Benchmark Building Using Smart Dampers,” J. of Eng. Mechanics, vol 130(4), pp. 386-392.
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