Apparent Weakening in SDOF Yielding Structures Usinga Negative Stiffness Device: Experimental

and Analytical StudyD. T. R. Pasala1; A. A. Sarlis2; A. M. Reinhorn, F.ASCE3; S. Nagarajaiah, M.ASCE4;

M. C. Constantinou, M.ASCE5; and D. Taylor6

Abstract: The peak deformation, acceleration, and the base shear experienced by the structures can be reduced by simulating yielding in anelastic system—also referred as apparent weakening. The negative stiffness device (NSD), used in this study, exhibits nonlinear-elasticnegative stiffness behavior; by adding NSD to the elastic structure (primary structure), the resulting structure-device assembly behaves likea bilinear-elastic structure. In an elastic structure, the acceleration and base shear experienced by the structure can be reduced by adding thenegative stiffness device, and the additional deformations caused from the reduced stiffness can be contained by adding viscous dampers.Previously, the authors have carried out experimental studies to demonstrate the effectiveness of apparent weakening in elastic structures, butlittle is known about the behavior of these systems when the primary structure itself yields. This paper focuses on the issues that may emanatewith the addition of NSD to the systems prone to yielding while highlighting the response reduction achieved by proper design of the NSD.Shake-table studies of a three-story fixed-base structure (3SFS) braced in top two stories, that is acting as a single degree of freedom system(SDOF) are presented in this paper. Two NSDs and a viscous damper are installed in the first story of the SDOF-3SFS. The structure issubjected to strong ground motions, with and without the NSDs, so that the columns of the SDOF-3SFS in the first story yield. Analyticalmodels for all the components, that can capture the observed experimental behavior, are also presented in this paper. To demonstrate theadvantages of apparent-weakening in yielding SDOF structures, experimental and analytical responses of four different systems, (1) 3SFS,(2) 3SFS with damper, (3) 3SFS with NSD, and (4) 3SFS with NSD and damper, are compared for a suite of ground motions. Shake-tabletests on SDOF-3SFS have confirmed that by adding the NSD and damper, the acceleration and base shear and deformation of the bilinearinelastic structure and NSD assembly will be reduced by more than 20% for moderate ground motions, and the collapse of structure can beprevented for severe ground motions. DOI: 10.1061/(ASCE)ST.1943-541X.0001077. © 2014 American Society of Civil Engineers.

Author keywords: Yielding structures; Apparent weakening; Negative stiffness device (NSD); Passive control; Earthquake protection;Shake-table tests; Weakening and damping; Adaptive stiffness; Seismic effects.

Introduction

To reduce the acceleration and base shear of the structure alongwith the deformations, Reinhorn et al. (2005) and Viti et al. (2006)proposed the concept—weakening and damping of structures. Byweakening the structure (reducing its strength), the accelerationscan be reduced, and by introducing the additional supplementaryviscous damping, the interstory drifts can also be simultaneouslycontrolled. Nagarajaiah et al. (2010) have proposed the concept

of apparent weakening to emulate a yielding system without alter-ing the structure properties. An apparent weakening is introducedin the structural system using a complementary negative stiffnessdevice (NSD) that mimics yielding of the global system, thusattracting it away from the main structural system. Pasala et al.(2013) demonstrated the concept of apparent weakening in elasticsystems through experimental studies on a fixed-base single degreeof freedom (SDOF) system for a suite of ground motions. Pasalaet al. (2014) experimentally tested the effectiveness by simulatingbilinear-elastic behavior in a SDOF elastic structure using NSD.Because it is unrealistic to presume that the primary structure willremain elastic during a severe ground motion, the prospect ofadapting apparent weakening in yielding SDOF structures isexplored in this paper.

Experimental results presented in this paper are a continuationof the previous work, presented by Pasala et al. (2014), which willbe referred to as the companion paper hereafter. The objectiveof the companion paper was to demonstrate the effectiveness ofapparent weakening in elastic SDOF structures by emulating a bi-linear elastic behavior using NSD. This paper focuses on extendingthe apparent weakening concept to yielding SDOF systems andanalysis of the complex nonlinear inelastic behavior of the NSDand yielding SDOF structure assembly. The experiments were con-ducted on a three-story fixed base structure (3SFS) braced in thetop two stories; hence, tested as a SDOF-3SFS system at theNEES equipment site at University at Buffalo, SUNY. Yielding in

1Former Ph.D. Student, Dept. of Civil and Environmental Engineering,Rice Univ., Houston, TX 77005.

2Former Ph.D. Student, Dept. of Civil and Environmental Engineering,Univ. at Buffalo, Buffalo, NY 14260.

3Clifford C. Furnas Professor, Dept. of Civil and EnvironmentalEngineering, Univ. at Buffalo, Buffalo, NY 14260.

4Professor, Dept. of Civil and Environmental Engineering and Mechan-ical Engineering, Rice Univ., Houston, TX 77005 (corresponding author).E-mail: Satish.Nagarajaiah@rice.edu

5Professor, Dept. of Civil and Environmental Engineering, Univ. atBuffalo, Buffalo, NY 14260.

6President, Taylor Devices, North Tonawanda, NY 14120.Note. This manuscript was submitted on October 19, 2012; approved on

March 20, 2014; published online on July 16, 2014. Discussion period openuntil December 16, 2014; separate discussions must be submitted for in-dividual papers. This paper is part of the Journal of Structural Engineer-ing, © ASCE, ISSN 0733-9445/04014130(8)/$25.00.

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SDOF-3SFS is achieved by increasing the amplitude of the groundexcitation input to the shake table, when compared with thecompanion paper (Pasala et al. 2014) wherein the structure waselastic. The first story columns are kept identical in elastic(companion paper) and inelastic studies (current paper); in thispaper, the yielded columns are replaced after each damaging earth-quake test.

A brief description on the NSD and yielding structure behavioris presented next. The experimental results (time history responseand hysteresis loops) of SDOF-3SFS (termed as 3SFS henceforthfor brevity) are compared with 3SFS equipped with (1) NSD,(2) damper, and (3) NSD and damper. Comparisons with analyticalpredictions are also presented. Full details of the NSD can be foundin the companion paper (Pasala et al. 2014) and (Sarlis et al. 2013;Sarlis 2013). Because of space limitations, it is not described againin this paper.

Effect of Apparent Weakening in Yielding Structures

Assume a primary structure with a yield displacement uy, yieldforce Fy, and apparent yield displacement of the assembly is u 0

yas shown in Fig. 1. The behavior of primary structure itself is in-elastic; adding a nonlinear-elastic NSD to the yielding structure willresult in a more involved system. To understand the permanentdeformations in the structure and NSD assembly, it is studied intwo different regimes: (1) mild yielding systems; and (2) severelyyielding systems.

Mild Yielding Systems

The response of the structure and the NSD assembly in mild yield-ing case is shown in Fig. 1(a). Say, for a given ground motion, theassembly first deforms in the negative direction, but the deforma-tion is less than uy [shown in Fig. 1(a)]; so the assembly will remainnonlinear elastic and retraces its path. Next, the assembly deformsin the positive direction and reaches a peak deformation at point-5,u5 > uy [shown in Fig. 1(a)]. Because the peak deformation isgreater than uy, the primary structure will undergo a permanent in-elastic deformation and will have a residual drift (plastic deforma-tion). If there are no significant peaks in the subsequent load, theassembly will oscillate at low amplitudes and reaches point-6[shown in Fig. 1(a)] at the end of loading cycle. The plastic defor-mation in the primary structure is ud6 ¼ u5 − uy; subscript d refersto permanent drift. Because the NSD exerts no stiffness for juj < u 0

y,

the permanent deformation in the assembly is the same as the plas-tic deformation in the primary structure (since jud6j < u 0

y) as shownin Fig. 1(a). If the permanent yield deformation in the structure, ud,is less than u 0

y, then the structure is assumed to have undergonemild yielding. The assembly always reaches the point where theassembly force is zero, following a loading cycle. Next, the con-dition in which, judj > u 0

y, is studied.

Severely Yielding Systems

Assume a particular load in which there are two dominant half-pulses, one in the positive direction and the other in negativedirection. For the first pulse, say, the assembly deforms in the neg-ative direction with mild yielding as shown in Fig. 1(b). Then forthe next pulse, the assembly deforms in the positive direction andreaches a peak deformation u7. Since there are no other domi-nant pulses, the assembly oscillates at low amplitudes and reachespoint-9, where the assembly force is zero as shown in Fig. 1(b). Thepermanent deformation in the assembly is ud9. It should be notedthat the assembly force is zero at point-9, but the force in the pri-mary structure is Fs9, and the force in NSD is FNSD9 as shown inFig. 1(b). The forces in structure and NSD have the same magni-tude but with different signs, so they nullify each other to resultin zero assembly force. The actual plastic deformation in the pri-mary structure is u 0

d9; which is less than ud9 as shown in Fig. 1(b).By disconnecting the NSD, the primary structure will move tothe displacement u 0

d9. The difference of the displacements ud9 andu 0d9 is called recoverable permanent-drift or simply recoverable-

drift, urd.In brief, for severely yielded systems, the permanent drift in

the assembly (ud9) is larger than the plastic deformation in the pri-mary structure (u 0

d9). Deformation, urd can be recovered from theassembly to reach u 0

d9. In the experiments carried on the 3SFS, theconnectors between the NSD and primary structure have a slottedhole to disengage the NSD easily and recover urd. The concept ex-plained using schematic diagrams shown in Fig. 1 can be extendedto multiple loading cycles. Detailed experimental setup is describedin the companion paper by the authors (Pasala et al. 2013); a con-cise description is presented in the next section.

Experimental Setup

The top two stories are braced in the 3SFS, so the structure behavesas a SDOF and columns in the first story will yield first when the

(a) (b)

Fig. 1. Drift and residual drift in NSD and bilinear inelastic structure assembly (solid gray line: bilinear inelastic structure; dashed gray line: NSD;dashed dark line: assembly): (a) F–D behavior of mildly yielded system with NSD; (b) F–D behavior of heavily yielded system with NSD

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frame is subjected to severe ground motion. The strain in thecolumns of the first floor is monitored during the tests, and theyare replaced after they yield. Four standard ground motions (seeTable 1) from the pacific earthquake engineering research (PEER)database have been used for the shake-table tests. Because replac-ing the columns after every test is expensive and time-consuming,shake-table tests have only been conducted on (1) 3SFS with NSD(termed as NS) and (2) 3SFS with NSD and damper assembly(termed as AS) for different peak ground accelerations (PGA) val-ues. To highlight the advantages achieved by using the AS over theexisting passive energy dissipation devices, experimental/analyticalresults of four different systems are compared in this study for allground motions: (1) 3SFS; (2) 3SFS with NSD (NS); (3) 3SFS withdamper (PS); and (4) 3SFS with NSD and damper (AS). Analyticalmodels, based on principles of mechanics, describing the behaviorof all four systems are presented in the companion paper (Pasalaet al. 2013), thus, not described in this paper. Linear model is usedfor the viscous damper in the companion paper, because the damperforce-velocity exhibits linear behavior. Because nonlinear force-velocity behavior of the damper is observed in tests, only a modi-fied damper model is presented in this paper.

Modified Analytical Model for Nonlinear ViscousDamper

The damping ratio of the optimal supplemental damper required forAS is 20% (Pasala et al. 2014). The observed force-velocity behav-ior of the viscous damper is linear if the amplitude of the stroke isless than 1 inch (in.), but for larger amplitudes the force-velocitybehavior observed was nonlinear as shown in Fig. 2. To capture thebehavior of damper for all amplitudes, a higher order term is addedto the linear term as shown in Eq. (1). In simulations, the forceexerted by the damper, FPD, is calculated using

FPD ¼ 0.312 × u̇þ 1.17 × 10−6 × u̇5 ð1Þ

Velocity, u̇, in Eq. (1) is in in:=s and force, FPD, is in kipswhereas the coefficients have suitable compatible dimensions.Force-velocity behavior of the damper comparing the experimental

results with the analytical results [Eq. (1)] is shown in Fig. 2.Readers should refer to Pasala (2012) and Pasala et al. (2014) forparameter values and the details pertaining to the analytical modelsfor the 3SFS and NSD used in this paper.

The analytical model developed for the 3SFS in the companionpaper (Pasala et al. 2013) are meant for an elastic system. The sameanalytical model is applicable for the yielding 3SFS structure.Along with the information provided in the companion paper,values for yield displacement, uy, and post yielding stiffness ofeach story in 3SFS are needed. These parameters in the model areestimated by minimizing the error between experimental andanalytical hysteresis loop in each story.

The next section contains the experimental and analytical resultsof 3SFS and NSD assembly demonstrating the concept of apparentweakening in yielding SDOF systems.

Experimental and Simulation Results

In this section, apparent weakening in mild and severely yieldingsystems is analyzed through shake-table studies on SDOF-3SFS

Table 1. Summary of the Peak Responses from the Shake-Table Tests

Serialnumber

Groundmotion System

PGA(g)

First story response Permanent drift

Base shear(kips)

Deformation(in.) Acceleration (g) Total (in.)

Recovered(in.)

Residual(in.)

1 Kobe 1995 BSa 0.65 2.84 0.79 0.05 — 0.05 17.722 PSa 0.65 1.93 (32%) 0.73 (7%) 0.36 — 0.36 22.36 (−26%)3 NSa 0.65 2.49 (12%) 0.77 (2%) 1.67 0.80 0.87 21.74 (−23%)4 AS 0.65 1.39 (51%) 0.58 (26%) 0.17 0.00 0.17 13.65 (23%)5 Pacoima 1997 BS 0.75 1.92 0.56 0.20 — 0.20 15.896 PSa 0.81 1.16 (40%) 0.49 (13%) 0.00 — 0.00 14.47 (9%)7 NS 0.81 2.83 (−47%) 0.46 (18%) 2.11 0.97 1.14 14.45 (9%)8 AS 0.75 1.65 (14%) 0.37 (35%) 0.32 0.12 0.20 10.10 (36%)9 Sylmar 1994 BS 0.68 1.29 0.47 0.00 — 0.00 13.3710 PSa 0.68 0.87 (32%) 0.38 (19%) 0.02 — 0.02 11.78 (12%)11 NS 0.63 1.71 (−32%) 0.34 (27%) 0.91 0.58 0.33 9.08 (32%)12 AS 0.63 1.18 (8%) 0.34 (28%) 0.10 0.00 0.10 8.55 (36%)13 Newhall 1994 BS 0.44 1.72 0.50 0.03 — 0.03 15.9114 PSa 0.48 1.16 (33%) 0.44 (13%) 0.00 — 0.00 14.56 (8%)15 NS 0.48 1.36 (21%) 0.44 (12%) 0.43 0.30 0.13 10.02 (37%)16 AS 0.44 1.30 (24%) 0.39 (22%) 0.16 0.00 0.16 10.71 (33%)

Note: Percentage decrease in the response with respect to the BS is shown in parentheses (negative % = “increase”).aThe response is calculated using analytical model.

Fig. 2. Comparison of analytical and experimental force-velocitybehavior of damper

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(termed as 3SFS henceforth for brevity) and NSD assembly (NS).Because the damper has a maximum stroke of only two in., andthe yield displacement is 1.5 in., the severe yielding tests couldnot be carried on the 3SFS, NSD, and damper assembly. The3SFS and NSD assembly is subjected to Pacoima ground motionat two different amplitudes (peak ground acceleration [PGA] of0.57 grams [g] and 0.81 g).

Mildly Yielding Structure

When the assembly is subjected to the ground motion at PGAof 0.57 g, there is mild yielding in the primary structure. Theforce-deformation behavior of the primary structure (3SFS), andNSDs (NSD-East and NSD-West) is shown in Figs. 3(a and b).The experimental and analytical force-deformation behaviorof the assembly with the components is shown separately inFigs. 3(c and d). Analytical and experimental first floor dis-placement and acceleration history response are compared inFigs. 4(b and c), respectively. From force-deformation behaviorof the primary structure shown in Figs. 3(a) and 4(b), it is clearthat the 3SFS has yielded at 1.8 seconds (s) when the peak defor-mation in the structure is 1.7 in. The structure has undergone plasticdeformation, and it subsequently vibrates about 0.11 in. dis-placement. Although the primary structure has yielded, since theplastic deformation is less than the apparent yield displacementdue to the addition of NSD (uy ¼ 0.25 in:), the permanent driftin the assembly is the same as the permanent drift in the primarystructure. Force-deformation behavior, shown in Figs. 3(c and d),confirms that the base shear of the assembly is reduced by morethan 50% by adding the NSD to the primary structure. This behav-ior is consistent in both the analytical results and observed exper-imental results.

The trace of the force-deformation curves of 3SFS, NSD, andassembly [similar to the schematic curves shown in Fig. 1(a)] areshown in Fig. 4(a) for experimental results. The trace of force-deformation curves is obtained by removing the friction in theNSDs [Fig. 3(b)], assembly [Fig. 3(c)], and discarding the vibra-tions after 2.5 s. From the experimental results in Fig. 4(a), the per-manent drift in the assembly and the plastic deformation in theprimary structure are the same, 0.11 in. The plastic deformation in3SFS, and the permanent drift in assembly are the same because theapparent yield displacement in NSD is 0.25 in. and the permanentdrift in the primary structure is less than 0.25 in. (explained inprevious section).

Severely Yielding Structure

When the structure is subject to the ground motion with PGAof 0.81 g, the primary structure in NS has yielded significantly.The force-deformation behavior of the primary structure (3SFS),and NSDs (NSD-East and NSD-West, is shown in Figs. 5(aand b). The experimental and analytical force-deformation behav-ior of the assembly with the components is shown separately inFigs. 5(c and d) for the severe yielding case. Analytical and exper-imental first floor deformation and acceleration response are com-pared in Figs. 6(b and c), respectively. The overall trend in theanalytical and experimental behavior depicted in Figs. 5 and 6 issimilar, but the peak deformation in the experiments is 2.83 in., andthe peak deformation in analytical studies is 2.51 in.

From the experimental force-deformation behavior of primarystructure shown in Fig. 5(c) and displacement response shownin Fig. 6(b), 3SFS has yielded at 2.3 s when the peak deformationin the structure is 2.83 in. The plastic deformation of the primarystructure is 1.14 in. as shown in Fig. 5(a). Since the apparent yielddisplacement due to the addition of NSD (0.25 in.) is significantlyless than the plastic deformation in the 3SFS, the permanent de-formation of the assembly will be higher than the plastic deforma-tion of the 3SFS because of the presence of NSD as explained inthe previous section. The assembly starts stiffening at 2 in. dis-placement resulting in the increase in base shear as shown inFigs. 5(c and d). Once the assembly starts stiffening, the excessivedeformation caused from the initial reduction in stiffness is con-tained, and the structure is prevented from collapsing. Essentially,the accelerations and base shear of the assembly are reduced be-tween the displacements, 0.5 and 3 in. by removing a segment ofthe force deformation loop of primary structure.

The trace of the force-deformation curves of 3SFS, NSD, andassembly (similar to the schematic curves shown in Fig. 1) areshown in Fig. 6(a) for experimental results. The trace of force-deformation curves is obtained by removing the friction in theNSDs [Fig. 5(b)], assembly [Fig. 5(c)], and discarding the vibra-tions after 2.5 s. From the experimental results in Fig. 6(a), the per-manent drift in the assembly is 2.1 in., and the plastic deformationin the primary structure, shown in Fig. 7(a), is 1.14 in. Because theplastic deformation in the primary structure is much larger than theapparent yield displacement, the permanent drift in the assemblywill be higher than the plastic deformation in the primary structure(explained in the previous section). So, in the experimental results,the residual drift in the NS will be 1.14 in. and the recoverable driftis 0.97 in. In the deformation plot shown in Fig. 6(b), the NSD is

Fig. 3. Force-deformation behavior of 3SFS and NSD assembly (NS) in mildly yielding structures for Pacoima ground motion (PGA ¼ 0.57 g);comparison of experimental and analytical response of (a) 3SFS; (b) NSD east and west; (c) components from experimental results; (d) componentsfrom analytical predictions

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Fig. 4. (a) Force-deformation behavior of 3SFS and NSD assembly (NS) for Pacoima ground motion (PGA ¼ 0.57 g) depicting the residual andrecoverable drift in mildly yielding structures; comparison of experimental and analytical response of (b) first floor displacement and (c) first flooracceleration

Fig. 5. Force-deformation behavior of 3SFS and NSD assembly (NS) in heavily yielding structures for Pacoima ground motion (PGA ¼ 0.81 g);comparison of experimental and analytical response of (a) 3SFS; (b) NSD east and west; (c) components from experimental results; (d) componentsfrom analytical predictions

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Fig. 6. (a) Force-deformation behavior of 3SFS and NSD (NS) in heavily yielding structures for Pacoima ground motion (PGA ¼ 0.81 g) depictingthe residual and recoverable drift (square-marker: residual drift; circular-marker: permanent drift of the structure and NSD assembly); comparison ofexperimental and analytical response of (b) first floor displacement; (c) first floor acceleration

Fig. 7. Force-deformation behavior of components for Kobe-FN ground motion (PGA ¼ 0.65 g): (a) base structure*; (b) base structure and passivedamper*; (c) base structure and NSD*; (d) base structure, NSD, and damper (* denotes simulation results)

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disengaged from the structure at 9 s, so the drift in the structure hasreduced to 1.14 in.

In the next section, the advantages of using NSD in conjunctionwith the damper to attenuate the acceleration, displacement, baseshear, and permanent drift response is demonstrated by comparingwith three other systems.

Apparent-Weakening in Yielding Systems

Response of four different types of systems is compared to justifythe need for using damper in conjunction with the NSD. The foursystems are: (1) 3SFS or the base structure (BS); (2) 3SFS and NSDassembly (NS); (3) 3SFS, NSD, and damper assembly (AS); and(4) 3SFS and damper (PS). Because the columns were yielding sig-nificantly after a strong ground motion, only one yielding experi-ment for each ground motion has been conducted. For Newhall,Pacoima, and Sylmar ground motions, the NS is tested and forKobe ground motion, the AS is tested. In this section, the exper-imental results of AS and the simulation results of BS, PS, andNS are compared to demonstrate the effectiveness of apparent-weakening and damping in yielding structures. The results fromother shake-table tests are summarized in Table 1.

Force-deformation loops of all the four systems with compo-nents are shown in Fig. 7 for Kobe-FN ground motion with PGAof 0.65 g. Superscript * denotes that the results are simulatedusing the analytical model. From Fig. 7(a), the BS has undergoneplastic deformation in the negative direction first and then inthe positive direction. The peak deformation, acceleration, andbase shear of the first story in the BS are 2.84 in., 0.79 g,and 17.7 kips, respectively, shown in Table 1. In the case with3SFS and NSD assembly (NS), the assembly has undergone sig-nificant deformation until it gets into the stiffening region, dueto the reduction in assembly stiffness beyond 0.5 in., shown inFig. 7(c). Once the assembly enters the stiffening region, due tothe high assembly stiffness and large plastic deformation of the3SFS, the assembly will remain in the stiffening region. Perma-nent drift in the assembly after the ground motion is 1.67 in.,and the plastic deformation in the 3SFS is 0.87 in. as shown inFig. 7(c). So, by adding the NSD, the accelerations and baseshears can be reduced for deformation range between 0.5 and2.3 in., and collapse of structure can be prevented for deforma-tions larger than 2.3 in.

By adding a viscous damper to the structure along with theNSD, the experimental force-deformation loops of the assemblyand components are shown in Fig. 8. With the addition of viscousdamper, the excessive deformations caused due to the reduction inassembly stiffness are contained. From Fig. 7(d) and Table 1, thepeak deformation, acceleration, and base shear of the AS are re-duced to 1.39 in., 0.58 g, and 13.7 kips, respectively. The responseof 3SFS with the viscous damper (PS) is also presented to demon-strate the role of NSD in the AS. Although the deformations arereduced with the addition of viscous damper alone, the accelerationand the base shear of the assembly will significantly increase dueto the damper force, as shown in Figs. 7(b) and 8 and Table 1.The peak deformation, acceleration, and base shear of the PSare 1.93 in., 0.73 g, and 22.4 kips, respectively. Clearly, by addingthe NSD along with the damper, the base shear and accelerationscan be reduced. Assembly force-deformation loops of all the foursystems, shown in Fig. 8, confirms that by adding NSD and damperto the bilinear inelastic structure, the displacement, acceleration,and base shear can be reduced.

Table 1 summarizes the peak response characteristics of all thefour systems for four ground motions. The key observations are:

1. Because of the presence of NSD, base shear of the NS andAS is consistently reduced up to 35% compared with theBS in the mild yielding cases. In the severe yielding caseswith NS, the base shear in the assembly will be similar tothe BS because of the stiffening. However, collapse of thestructure can be prevented due to the higher stiffness at largerdeformations.

2. Addition of only viscous damper (PS) reduces the deformationof the assembly by more than 30%, but the reduction in baseshear is less than 10%. In the case of Kobe ground motion, thebase shear in PS has increased by 23%.

3. By adding the viscous damper in conjunction with NSD (AS),the displacements, accelerations, and base shears can be con-sistently reduced by more than 20%.

4. Permanent drift in the assembly after the ground motion isequal to the plastic deformation in the primary structure, if theplastic-deformation is less than apparent yield displacement.If the plastic deformation in primary structure is larger thanthe apparent yield-displacement, then, the permanent drift inthe assembly will be higher than the plastic deformation; how-ever, the drift in excess of plastic-deformation is recoverable(by resetting the NSD to zero position).

5. The residual drift in AS is comparable with that of BS and PS.It should be noted that in the case of BS and PS, the 3SFS isyielding in both the directions, so some of the plastic defor-mation is recovered. In the case of 3SFS and NSD assembly(NS), since the force-deformation behavior of NSD is veryasymmetric, the assembly has the tendency to remain in theregion in which it has yielded first; while, in the case of 3SFS,NSD, and damper assembly (AS), such a tendency is generallyabsent (see Fig. 8).

6. The apparent weakening and simulated yielding from theaddition of NSD reduces the forces, and the damper reducesthe displacements and interstory drifts. With reduced inters-tory drift, the columns attract less force and damage.

Conclusions

The concept of apparent weakening is evaluated in yieldingSDOF systems experimentally and analytically. Comprehensive

Fig. 8. Assembly’s force-deformation behavior of first story forKobe-FN ground motion (PGA ¼ 0.65 g) (* denotes simulationresults)

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experimental and simulation studies have been carried on SDOF-3SFS yielding structure, to study the behavior of bilinear inelasticstructure and NSD assembly during severe ground motions. Withthe addition of NSD to the bilinear inelastic structure, the baseshear and accelerations of the assembly are reduced by more than30% for ground motions in which there is mild yielding in the pri-mary structure. For more severe ground motions, the deformationsof the assembly will be larger than the stiffening point of NSD,resulting in very high assembly stiffness, thereby increasing thebase shear and accelerations of the assembly, but preventing col-lapse of structure. In the case where there is severe yielding in theprimary structure, the permanent drift in the assembly is larger thanthe plastic deformation in the primary structure. The permanentdrift in excess of the plastic-deformation in the primary structurecan be completely recovered by disengaging the NSD from the pri-mary structure.

Experimental results have confirmed that the consistent re-ductions in displacements, accelerations, and base shear can beachieved by adding the NSD and viscous fluid damper to thebilinear inelastic structure (AS). Analogous to the inelastic design,the acceleration and base shear and deformation of the structure andNSD assembly can be reduced by more than 20% for moderateground motions, and the collapse of structure can be prevented forsevere ground motions. Additionally by disconnecting the NSD, agood part of the inelastic excursions incurred after a severe groundmotion can be recovered.

Acknowledgments

Funding by National Science Foundation, grant NSF- NEESR-CMMI-0830391, for this project with Dr. Joy Pauschke as programdirector, is gratefully acknowledged.

References

Nagarajaiah, S., Reinhorn, A. M., Constantinou, M. C., Taylor, D.,Pasala, D. T. R., and Sarlis, A. A. (2010). “True adaptive negative stiff-ness: A new structural modification approach for seismic protection.”Proc., 5th World Conf. on Structural Control and Monitoring,International Association of Structural Control and Monitoring(IASCM), Univ. of Southern California, Los Angeles, CA.

Pasala, D. T. R. (2012). “Seismic response control of structures using noveladaptive passive and semi-active variable stiffness and negative stiffnessdevices.” Ph.D. thesis, Dept. of Civil and Environmental Engineering,Rice Univ., Houston, TX.

Pasala, D. T. R., Sarlis, A. A., Nagarajaiah, S., Reinhorn, A. M.,Constantinou, M. C., and Taylor, D. (2013). “Adaptive negative stiff-ness: A new structural modification approach for seismic protection.”J. Struct. Eng., 10.1061/(ASCE)ST.1943-541X.0000615, 1112–1123.

Pasala, D. T. R., Sarlis, A. A., Reinhorn, A. M., Nagarajaiah, S.,Constantinou, M. C., and Taylor, D. (2014). “Simulated bilinear-elasticbehavior in a SDOF elastic structure using negative stiffness device:Experimental and analytical study.” J. Struct. Eng., 10.1061/(ASCE)ST.1943-541X.0000830, 04013049.

Reinhorn, A. M., Viti, S., and Cimellaro, G. P. (2005). “Retrofit of struc-tures: Strength reduction with damping enhancement.” Proc., 37thUJNR Panel Meeting on Wind and Seismic Effects, Public WorksResearch Institute (PWRI), Tsukuba-shi, Japan, 16–21.

Sarlis, A. A. (2013). “Adaptive seismic protection systems.” Ph.D. thesis,Dept. of Civil and Environmental Engineering, Univ. at Buffalo, SUNY,Buffalo, NY.

Sarlis, A. A., Pasala, D. T. R., Constantinou, M. C., Reinhorn, A. M.,Nagarajaiah, S., and Taylor, D. (2013). “Negative stiffness device forseismic protection of structures.” J. Struct. Eng., 10.1061/(ASCE)ST.1943-541X.0000616, 1124–1133.

Viti, S., Cimellaro, G. P., and Reinhorn, A. M. (2006). “Retrofit of a hos-pital through strength reduction and enhanced damping.” Smart Struct.Syst., 2(4), 339–355.

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