Hybrid base-isolation of a nonlinear building using a passive resettable stiffness damper

Engineering Structures 178 (2019) 206–211

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Hybrid base-isolation of a nonlinear building using a passive resettable stiffness damper

T

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Kenneth K. Walsh , Grace Sallar, Eric P. Steinberg Department of Civil Engineering, Ohio University, Athens, OH 45701, USA

A B S T R A C T

The resetting semi-active stiffness damper (RSASD) has been shown to be effective at protecting civil structures subject to near-field earthquake ground motions. Recently, a modified version of the RSASD was proposed in which the semi-active components of the damper were replaced by a novel mechanism for achieving resetting of the damper force. The resulting resetting passive stiffness damper (RPSD) represents a more reliable and robust alternative to its semi-active counterpart. The objective of the research presented herein is to investigate the RPSD for hybrid base-isolation of a nonlinear building subject to a suite of near-field earthquake ground motions. The performance of the RPSD is benchmarked through a comparison with other passive, semi-active, and hybrid seismic protective systems. It is shown that the RPSD is effective in reducing the peak base drifts while maintaining the building response within the elastic range. Furthermore, it is shown that the performance of the RPSD generally compared well with the other protective systems in terms of reducing the peak base drifts, but was not as effective at reducing the peak base absolute accelerations or floor drifts.

1. Introduction Over the last two decades, research on the resetting semi-active stiffness damper (RSASD) has shown it to be effective at controlling the response of building and bridge structures in the presence of near-field ground motions [1–10]. Within the last decade, researchers have sought to improve upon the RSASD. In order to address the increase in base shear resulting from the displacement-dependent control forces (i.e. large damper forces at large structure displacements), the 2–4 semiactive resettable stiffness damper was proposed [11–14]. In this variation of the RSASD, separate valves for each chamber allow for independent control of the pressure on each side of the piston, and a wider variety of control laws may be imposed. In the 2–4 control law, the valves are closed when the piston moves toward the equilibrium position, thereby providing damping only in the 2nd and 4th quadrants of the force-deflection plot. As a result, energy can be dissipated without increasing the base shear force in the structure. More recently, a modified version of the RSASD in which all of the feedback components were replaced by a simple mechanism for resetting the damper force was presented [15]. The resulting damper, the resetting passive stiffness damper (RPSD), represents a more reliable and robust alternative to the RSASD for rate-independent damping of buildings and bridges. To date, the RPSD resetting mechanism has been validated through small-scale laboratory studies, and a model for predicting the displacement-dependent resetting of the damper force has been developed. The results of numerical simulations have shown that

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the RPSD can achieve a control performance similar to that of the RSASD when used in the isolation layer of a seismically-excited baseisolated building undergoing long period vibratory motion. Recent research also indicates that the RPSD is effective for seismically-excited short-period structures when motion amplification is incorporated into the resetting mechanism [16]. Past numerical studies on the RPSD were performed for linear structures, and the damper performance was evaluated solely through comparison with the RSASD. However, research has shown that the nonlinear response of fixed-base and base-isolated buildings in the presence of near-fault ground motions should also be considered [17,18]. Therefore, an investigation of the RPSD performance for more realistic structures and with comparison to a wider range of control devices is warranted to evaluate its potential as a seismic protective system. In the present study, the RPSD is investigated for hybrid control of a nonlinear base-isolated building subject to a suite of near-field earthquake ground motions, and its performance is compared with that previously reported for other seismic protective systems for the same structure and ground motions. In the following sections of this paper, the RPSD and nonlinear base-isolated building models, along with the example structure and ground motions, will be presented. Then, the response of the structure equipped with the RPSD, along with the responses previously reported for other seismic protective systems, will be compared.

Corresponding author. E-mail addresses: [email protected] (K.K. Walsh), [email protected] (G. Sallar), [email protected] (E.P. Steinberg).

https://doi.org/10.1016/j.engstruct.2018.10.037 Received 23 February 2018; Received in revised form 2 October 2018; Accepted 15 October 2018 0141-0296/ © 2018 Elsevier Ltd. All rights reserved.

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end is located at an initial vertical distance yo from the rack. A mechanically operated valve is positioned at a height h above the lever and has a piston with stroke ls. The distance de the lever must travel to engage the valve piston is referred to as the engaging distance. The maximum vertical distance the lever travels is termed d (d = de + ls). A model for predicting the displacement-dependent resetting of the damper force has been previously reported [15]. For convenience, the equations are summarized here as follows:

External Loop

Double-Acting Piston Pneumatic Cylinder Fig. 1. RPSD construction.

Fs (t ) = −ks·x r (t );

(1)

2. RPSD

x r (t ) = x e (t )−x p (t );

(2)

2.1. Description

x e (t )=x p (t ) if x1 ≤|xL (t )|≤x2

A schematic of the RPSD is shown in Fig. 1. Similar to the RSASD, it consists of a pneumatic cylinder separated into two chambers by a double-acting piston, and an external loop fitted with a valve for regulating gas flow between the chambers. However, in the RPSD all the semi-active components of the RSASD (i.e. sensor, microcontroller, electromechanical valve, power supply) are replaced by a simple racklever mechanism for achieving resetting of the damper force. The racklever mechanism consists of a grooved rack connected to the damper piston. Mounted above the rack is a vertical channel containing a spring-loaded lever (spring not shown). The lever is free to translate in the vertical direction and rotate about the end in the channel. The starting position of the lever is one where the lever is oriented at some angle with respect to the rack, and with its free end resting in the grooves of the rack. Mounted above the lever in the channel is a mechanically operated, normally-closed valve with spring return. For each change in direction of the damper piston, the rack forces the lever to move up the vertical channel and engage the valve piston, thereby opening the valve. The valve remains open until the lever moves back down the channel and disengages the valve piston, at which time the valve closes. Through this process the valve is pulsed open and closed each time the damper piston changes direction, and the RPSD is capable of similar operation to the RSASD without the need for any semi-active components.

x ṗ (t0−Δt ) ⎤ d 2d y∗ (t ) = −⎡ x (t )2 + xL (t ); ⎢ |x ṗ (t0−Δt )| ⎥ x 02 L x0 ⎣ ⎦

(4)

xL (t ) = xl−x p (t );

(5)

xl = x p (t ) if x ṗ (t ) = 0

x ṗ (t0−Δt )·y∗ (t ) > 0

and

and

x ṗ (t0−Δt )·y∗ (t ) < 0

(3)

(6)

In Eqs. (1)–(6), Fs(t) is the damper force, ks is the damper effective stiffness [4,9], xr(t) is the resetting displacement, xe(t) is the engaging displacement, xp(t) is the piston displacement, x ṗ (t ) is the piston velocity, to is the time at the instant there is a change in piston direction, Δt is the time increment, xL(t) is the horizontal displacement of the lever tip (initialized to zero after each change in piston direction), xl is the piston position at the instant the last change in direction of the piston occurred, and x1 and x2 are the horizontal lever tip displacements at which the lever engages and disengages the valve piston, respectively. Small-scale laboratory experiments were performed to validate the resetting mechanism of the RPSD (see Fig. 3a). The damper was subjected to a sinusoidal displacement input with an amplitude of 50.8 mm and a frequency of 0.19 Hz, as shown in Fig. 3b. The time-histories of the damper forces, as well as hysteresis plots, obtained from three experimental trials and using Eqs. (1)–(6) are shown in Fig. 3c-d. Comparison of the theoretical and experimental results shows generally good agreement. Differences in the plots at the instant of resetting are due to the model assumption that the damper force drops to zero the instant the valve is opened. The results shown in Fig. 3c-d demonstrate that the RPSD works as intended. At each change in piston direction, the lever engages the valve and the damper force drops to zero. The damper force then remains zero until the lever disengages the valve and the valve closes, at which time the force in the damper begins to build again.

2.2. Force resetting Resetting of the RPSD force depends on the piston displacement after each change in direction, which in turn depends on the drift of the structure in which the damper is installed. Consider the rack-lever representation shown in Fig. 2. The lever is of length L and is oriented at an initial angle θo to the rack. The tip of the lever rests on the rack at an initial horizontal distance xo from the vertical channel, and the other

h

h ls

ls

de

de

y(t) d L

L

L yo

yo (t)

o

xo

xL(t) xo

(a)

(b)

Fig. 2. RPSD rack-lever mechanism (a) before and (b) during lever displacement. 207

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60

Trial 1

Trial 2

Trial 3

40

xp (mm)

20 0 -20 -40 -60

0

5

10

(a)

25

(b)

Trial 1

Trial 2

Trial 3

Theoretical

Trial 1 60

40

40

20

20

Fs (N)

60

Fs (N)

20

80

80

0

-20

-40

-40

-60

-60 0

5

10

15

20

-80 -60

25

Trial 2

Trial 3

Theoretical

0

-20

-80

15

Time (s)

-40

-20

0

20

40

60

xp (mm)

Time (s)

(c)

(d)

2

2

1

1 Fr1(t)/Fy1

Frb(t)/Fyb

Fig. 3. Experimental testing of RPSD: (a) laboratory setup, (b) displacement input, (c) damper force time-history, and (d) damper hysteresis.

0

-1

-2 -2

0

-1

-1

0 xdb(t)/Dyb

1

-2 -2

2

(a)

-1

0 xd1(t)/Dy1

1

2

(b)

Fig. 4. Restoring force at the (a) isolator and (b) first floor of the nonlinear base-isolated building.

where M and C are the n x n mass and damping matrices of the structure, respectively. The n x 1 vectors ẋ (t) and x¨ (t) represent the horizontal velocities and accelerations (relative to the ground) of each mass at time t. The matrix E is an n x r excitation location matrix for the r x 1 vector of excitation forces Fe(t), the matrix G is an n x l location matrix for the l x 1 vector of restoring forces Fr(t), and the matrix D is an n x m location matrix for the m x 1 vector of control forces Fs(t)

3. Nonlinear base-isolated building model For an n degree-of-freedom nonlinear base-isolated building model equipped with m RPSDs at the isolator level, the governing equation of motion is:

Mx¨ (t ) + Cẋ (t ) = EFe (t )−GFr (t )−DFs (t )

(7) 208

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Northridge earthquake recorded at the Santa Monica and Sylmar County stations (NR-SM and NR-SC), and the east-west components of the Chi-Chi earthquake (CC-068 and CC-084). The uncontrolled responses of the base-isolated building were then compared with those previously reported [5]. Percent differences in the peak base drift and absolute acceleration responses, and the maximum peak drift over all floors of the building, were calculated. The results (not shown) showed excellent agreement between the present and previous studies, with the largest percent difference being less than 3%, the next largest being less than 1%, and the remaining percent differences being close to zero.

produced by the RPSDs. The nonlinear restoring forces at each floor of the building, and in the isolator, are modeled according to a method adopted from the literature [5]. The restoring force, Fri (t), at the ith level of the base-isolated building is given by:

Fri (t ) = αi ki x di (t ) + (1−αi ) ki Dyi νi (t )

(8)

where xdi is the drift displacement, ki is the elastic stiffness, αi is the ratio of post-yielding to pre-yielding stiffness, and Dyi is the yield deformation, at the ith level. The hysteretic variable νi(t), subject to |νi (t )| ≤ 1, is determined from:

̇ (t )−βi |x di ̇ (t )||νi (t )|ni − 1 νi (t )−γi x di ̇ (t )|νi (t )|ni ] νi̇ (t ) = Dyi−1 [Ai x di

4.2. Seismic protective systems

(9) In order to benchmark the performance of the RPSD for seismic protection of the five-story base-isolated building, it was compared with other passive, semi-active, and hybrid control systems. In all, the RPSD was compared with the uncontrolled base-isolated building, and the base-isolated building with the following eight systems: (1) passive viscous fluid damper (PVFD) with a damping coefficient of 80.65 kN s/ m (ζ = 47.5%), (2) passive friction damper (PFD) with a constant friction force of 14 kN, (3) PFD with a constant friction force of 14 kN in combination with PVFD with a damping coefficient of 76.05 kN s/m (ζ = 47.5%), (4) PFD with a constant friction force of 28 kN in combination with PVFD with a damping coefficient of 57.61 kN s/m (ζ = 35%), (5) RSASD with damper stiffness of 400 kN/m and simple control law (i.e. resetting at peak damper displacement), (6) RSASD with damper stiffness of 400 KN/m and simple control law combined with PVFD with a damping coefficient of 20.49 kN s/m (ζ = 15%), (7) semi-active electromagnetic friction damper (SAEMFD), and (8) SAEMFD in combination with PVFD with a damping coefficient of 39.09 kN s/m (ζ = 25%). Additional details on the SAEMFD may be found in [5]. The RPSD was modelled with a lever length of L = 12.7 mm, initial horizontal position of xo = 11 mm, and engaging distancede = 2 mm, leading to values of x1 and x2 equal to 1.4 mm and 20.6 mm, respectively. These parameters were selected considering practical limits on the minimum lever length (L) and orientation (θo) such that the device would be constructible, while also providing sufficient vertical travel of the lever to actuate the valve. The effective stiffness of the RPSD was taken to be equal to that of the RSASD (ks = 400 kN/m).

where Ai, βi, and γi influence the size and shape of the hysteresis loop, and ni affects the smoothness of the transition from the pre-yielding to post-yielding stiffness. The isolation restoring force and the restoring force at the first floor of the base-isolated building are shown in Fig. 4. The restoring force and drift displacement are normalized by the corresponding yield force and yield displacement, respectively. 4. Numerical example 4.1. Five-story base-isolated building The performance of the RPSD is investigated for a five-story baseisolated building adopted from the literature [5] (see Fig. 5). The fundamental period of the fixed-base building is 0.3 s with a first mode damping ratio of 2%. It is assumed that the building is isolated using low damping laminated rubber-bearings, resulting in a fundamental isolation period of 2.5 s with 4% damping in the first mode. More details on the non-linear base-isolated building model may be found in [5]. To validate the model used in the present study, the uncontrolled responses were obtained for a suite of near-field earthquake ground motions that included the north-south and east-west components of the Kobe earthquake (KB-NS and KB-EW), the north-south and east-west components of the Takatori earthquake (TT-NS and TT-EW), the

5. Results and discussion The peak base drift and absolute acceleration, as well as the maximum peak drift over all floors of the building, are presented in Table 1 for the seismic protective systems and ground motions considered. The results show that the RPSD was capable of significantly reducing the peak base drift of the isolator compared to the uncontrolled case, with the smallest reduction being 33% for the TT-NS ground motion, and the largest reduction being 74% for the NR-SM ground motion. Table 1 also shows that the RPSD was able to keep the peak base drifts within 40 cm, which has been reported as the acceptable limit for design practice [5]. The favorable performance of the RPSD in terms of reducing the peak base drifts came at the cost of increased peak absolute base accelerations and floor drifts, which were both found to be greater than those for the uncontrolled base-isolated building for all ground motions considered. The increase in the peak base accelerations were in the form of spikes in the acceleration response resulting from the discontinuity in the damper force at the instant of resetting. Meanwhile, the increase in the peak floor drifts did not lead to yielding at any floor level, as the largest peak floor drift for all ground motions was found to be 0.81 cm, which was less than the minimum value for the yield deformation of 1.36 cm over all floor levels. The data in Table 1 may be used to benchmark the performance of the RPSD through a comparison with other seismic protective systems. In terms of reducing peak base drifts, Table 1 shows that the RPSD was:

Fig. 5. Five-story base-isolated building model with RPSD installed at the isolation level. 209

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Table 1 Peak responses of nonlinear base-isolated building. Earthquake ground motion System

Response

KB-NS (0.82 g) (3)

KB-EW (0.62 g) (4)

(1)

(2)

Uncontrolled

xb (cm) ab (g) xmax (cm)

34.4 0.16 0.14

RPSD

xb (cm) ab (g) xmax (cm)

PVFD (47.5%)

TT-NS (0.61 g)

NR-SM (0.88 g) (7)

NR-SC (0.61 g)

(5)

TT-EW (0.65 g) (6)

CC-084 (1.01 g)

(8)

CC-068 (0.51 g) (9)

25.6 0.13 0.11

51 0.23 0.2

60.3 0.27 0.23

31.9 0.15 0.13

50.4 0.22 0.19

99 0.41 0.36

61.4 0.27 0.23

17.6 1.40 0.39

12.7 1.00 0.28

34.4 2.98 0.81

31.9 2.26 0.67

8.4 0.70 0.20

14.2 1.07 0.32

38.3 1.37 0.44

32.3 2.68 0.69

xb (cm) ab (g) xmax (cm)

14.5 0.22 0.2

11.8 0.19 0.18

33.3 0.37 0.31

31.3 0.35 0.31

12.2 0.21 0.12

18.6 0.2 0.16

40.3 0.3 0.25

30.6 0.34 0.31

PFD (14 kN)

xb (cm) ab (g) xmax (cm)

23.1 0.44 0.14

15.9 0.5 0.14

54.8 0.46 0.25

48.2 0.45 0.22

11.6 0.51 0.16

28.5 0.41 0.16

79.3 0.5 0.33

49.8 0.42 0.23

PVFD (45%) PFD (14 kN)

xb (cm) ab (g) xmax (cm)

11.1 0.48 0.22

10.7 0.45 0.2

32.4 0.43 0.33

29.8 0.55 0.31

5.2 0.58 0.23

11.8 0.42 0.18

35.7 0.51 0.27

26.7 0.4 0.31

PVFD (35%) PFD (28 kN)

xb (cm) ab (g) xmax (cm)

11.2 0.87 0.24

10.6 0.85 0.21

34.4 0.87 0.33

30.6 1.01 0.3

4 0.9 0.32

11.6 0.8 0.19

33.5 0.85 0.28

25.1 0.82 0.3

SAEMFD

xb (cm) ab (g) xmax (cm)

25.2 1.15 0.27

16.4 0.62 0.17

35.7 1.59 0.35

28 1.32 0.29

11.7 0.51 0.1

16.4 0.73 0.19

38.1 1.94 0.37

67 3.43 0.54

SAEMFD PVFD (25%)

xb (cm) ab (g) xmax (cm)

18.4 0.96 0.23

13.3 0.47 0.16

31.7 1.23 0.35

30.6 1.34 0.29

8.3 0.38 0.13

13.4 0.59 0.16

36.5 1.6 0.31

45.6 2.38 0.38

RSASD

xb (cm) ab (g) xmax (cm)

15.3 1.43 0.38

11.6 1.09 0.29

30 3.05 0.81

29.5 2.39 0.65

8.5 0.68 0.17

13.7 1.25 0.32

34.8 1.47 0.44

27.3 2.43 0.66

RSASD PVFD (15%)

xb (cm) ab (g) xmax (cm)

14.2 1.25 0.38

10.9 1.04 0.3

26.5 2.71 0.72

26.4 2.12 0.6

7.2 0.6 0.16

12.9 1.14 0.31

32.4 1.29 0.42

24.4 2.23 0.61

(10)

active, and hybrid seismic protective systems. It was found that the RPSD was capable of substantially reducing the peak base drifts of the isolator compared to the uncontrolled base-isolated building, while maintaining the building response within the elastic range. However, the peak base absolute accelerations and maximum peak floor drifts were increased relative to the uncontrolled system. Comparison of the RPSD with other seismic protective systems revealed that, in general, it was capable of a similar performance in terms of reducing peak base drifts, but not as effective in reducing the peak base absolute accelerations and floor drifts.

(1) more effective than the PFD (14kN) for all ground motions considered; (2) more effective than the SAEMFD for all but the TT-NS, TTEW, and CC-068 ground motions, for which it yielded a similar response; (3) as effective as the remaining protective systems for all ground motions, with the exception of the PVFD (45%) with PFD (14 kN) and PVFD (35%) with PFD (28 kN) for the KB-NS, NR-SM, and CC-084 ground motions, for which it was found to be not as effective. In terms of reducing the absolute acceleration of the base, Table 1 shows that the RPSD was: (1) more effective than the PVFD (35%) with PFD (28 kN), the SAEMFD, and the SAEMFD with PVFD (25%), for the NRSM, CC-068 and CC-084, and CC-068 ground motions, respectively, but not as effective as these protective systems for the remaining ground motions; (2) as effective as the RSASD and RSASD with PVD (15%) for all ground motions; (3) not as effective as the remaining protective systems for all ground motions. In terms of reducing the maximum peak drift over all floors of the building, the RPSD was as effective as the RSASD and RSASD with PVD (15%), but not as effective as the remaining protective systems, for all ground motions. Exceptions to the latter occurred for the PVFD (45%) with PFD (14 kN) and PVFD (35%) with PFD (28 kN) systems for the NR-SM ground motion.

Acknowledgements This work was partially supported by the National Science Foundation, United States, Grant No. 1235273. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.engstruct.2018.10.037. References

6. Conclusions

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The aim of the present study was to investigate the performance of a new type of passive seismic damper, the resetting passive stiffness damper (RPSD), for protecting a nonlinear base-isolated building subject to near-field earthquake ground motions. In doing so, the performance of the RPSD was benchmarked against other passive, semi210

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