# Stochastic Optimization and Sensitivity Analysis of the Combined Negative Stiffness Damped Outrigger and Conventional Damped Outrigger Systems Subjected to Nonstationary Seismic Excitation

Authors: Zhihao Wang; Yuxiang Zhou; Chuangjie Fang; Jingwen Zhang
Journal: Structural Control and Health Monitoring
 (2023)

---

## Abstract

In recent years, various kinds of damped outrigger systems, particularly with the negative stiffness devices, have been shown to effectively suppress excessive vibration induced by earthquake and wind. To gain insight into such systems, this paper proposes a stochastic optimization method to investigate optimal configurations of combined negative stiffness damped outriggers (NSDOs) and conventional damped outriggers (CDOs) subjected to nonstationary stochastic seismic excitation. The simplified analysis model of various combinations of damped outriggers is developed, and the state-space representation of outrigger systems is then formulated. The nonstationary seismic excitation is modeled as a uniformly modulated stationary Gaussian process with a time-modulating function following Clough–Penzien spectrum, and combining equations of motions of outrigger systems and seismic excitation gives rise to the augmented state-space representation of structure-damper-excitation and subsequently the differential Lyapunov equation. The optimal designs of damped outriggers are defined via the solution of the differential Lyapunov equation. The multiobjective optimization with Pareto optimal fronts is adopted to deal with conflicting objectives between harmful interstory drift and floor absolute acceleration. The sensitivity of negative stiffness devices on optimal objectives is also investigated. The optimal designs are further examined under real typical and near-fault earthquake records. The results demonstrate the efficacy of the proposed method and provide insight into the systems subjected to nonstationary seismic excitation. While the purely NSDO systems could have better performance with adequate negative stiffness devices, the proposed combined NSDO and CDO systems prove to be more effective with limited negative stiffness devices and thus provide more design flexibilities tailored to different application situations.


---

## Body

## 1. Introduction

The conventional outrigger (CO) systems that configure outriggers between the perimeter columns and the core have shown to be effective to reduce the overall lateral displacement of the building and the base bending moment of the core and thus ensure the safety of tall buildings under static lateral loads such as mean wind load [1, 2]. However, the majority of the external excitations that tall buildings encounter are dynamic loadings [3], and the CO systems are difficult to provide sufficient damping to suppress excessive vibration for high-rise structures [4–8]. Therefore, extra energy dissipation means are generally needed for tall buildings to improve the seismic safety.Smith and Willford first proposed the novel concept of damped outrigger systems by introducing viscous dampers vertically between perimeter columns and the end of outriggers (referred to as conventional damped outrigger (CDO) systems hereafter, see Figure1 [9]) and demonstrated the excellent control performance of such systems under wind loadings [9, 10]. Soon some researchers applied the CDO systems to improve the seismic performance of tall buildings [11–13]. Subsequently, the control performance of the damped outrigger systems was investigated subjected to multiple hazards, e.g., earthquake and wind [14, 15]. These investigations have shown superior performance of the damped outrigger systems over the conventional outrigger systems by providing significant supplementary damping to the structures.Figure 1
Conceptual details of the outrigger damper system.Because the CDO systems in essence are to fully utilize the large vertical relative deformation between outrigger ends and perimeter columns and provide additional damping for the whole structures, a good number of studies were aimed to achieve maximum supplementary damping and the associated with optimal design [16, 17].To achieve satisfactory modal damping ratios, however, the perimeter columns in the CDO systems were required to have adequate axial stiffness [18–20]. Such finding was consistent with previous result of the damping effect of Maxwell damping element (MDE) if the assembly of outrigger viscous damper and perimeter column was viewed as a MDE and the damping effect would be limited with insufficient MDE stiffness [21–23]. Further improvement of such systems on adding damping could be beneficial when increasing the size of perimeter columns might not be desirable.To relax the restriction of perimeter column and improve the damping behavior of damped outrigger systems, one of the promising solutions is the introduction of negative stiffness devices which have proven to amplify the relative motion between their two terminals as with inerter dampers [24–28]. For instance, Shi and Zhu [29] conducted a comparative study of using negative stiffness damper and inerter damper for vibration isolation, while Li et al. [30] incorporated negative stiffness element with controllable damping element to achieve quasiactive control effect under earthquake. Wang et al. [31] proposed a negative stiffness damped outrigger (NSDO) system within which negative stiffness devices were incorporated in parallel with viscous dampers in the CDO system, and significant improvement on maximum damping ratio for structures with only one damped outrigger. Due to the interaction of outriggers, however, the NSDO systems with multiple damped outriggers might not always have the maximum damping level compared to the corresponding CDO systems (see [32] for more details). Yet the introduction of NSDO presented inspirational development for damped outrigger systems as damped outriggers could be applied for wider scenarios without the need of significantly increasing the size of perimeter columns.While design based on maximum modal damping ratio generally produces acceptable control effect, such design could not guarantee optimal control results for stochastic excitations. By modeling the stochastic seismic excitation as filtered white noise, Fang et al. [33] proposed a stochastic optimization procedure and applied it to the design of the CDO systems with the seismic excitation modeled as Kanai–Tajimi spectrum. Sun et al. [34] further extended the above procedure to parametrically investigate the optimum parameters of the NSDO system where the Clough–Penzien (CP) model was used to model seismic excitation. These studies investigated the optimal designs of damped outrigger systems with the seismic excitation modeled as a stationary stochastic process. To date, engineering applications of damped outriggers in buildings can be found in [9, 35, 36], and an interesting application scenario is to mitigate vibration of long-span bridges [37].Despite the advancement thus far, five main issues exist. First, as with modal damping analysis mentioned above, the structures with multiple NSDOs might not always be the optimal solution due to the interaction of outriggers, and further examination of the design and performance of multiple NSDO systems is necessary. Second, although the previous studies combined the CO either with the CDO [19] or with NSDO [15, 31, 34], the potential benefit of combining the CDO and NSDO has not been explored. Third, the stochastic analysis of damped outrigger systems considering the uncertainty of earthquake previously adopted the assumption of stationary, but the natural seismic excitations often exhibits nonstationary characteristics [38–40]; unless the analysis results obtained from stationary excitation assumption for such systems are formally justified, designing the outrigger systems and evaluating the performance subjected to nonstationary seismic excitation are more challenging and of great importance from practical perspective. Fourth, the parameters of installed devices obviously affect the system seismic performance, and this is particularly the case for the NSDO system where the negative stiffness devices play a significant role in structural dynamic characteristic, but the sensitivity on desired objectives has not been studied. Finally, to make the stochastic optimization more intuitive and align with engineering practice, the obtained optimal design needs to be further examined under both real far and near-field earthquake records.This paper proposes a stochastic optimization method to investigate optimal configurations of combined negative stiffness damped outriggers (NSDOs) and conventional damped outriggers (CDOs) subjected to nonstationary stochastic seismic excitation. In this study, three types of damped outrigger systems considered are (1) the conventional viscously damped outrigger placed above the negative stiffness outrigger, referred to as C&N hereafter; (2) the NSDO placed above the CDO, referred to as N&C hereafter, and (3) the double NSDOs, referred to as DN hereafter. First, the simplified analysis model of various combinations of damped outriggers is developed, and the state-space representation of outrigger systems is formulated. Second, the nonstationary seismic excitation is modeled as a uniformly modulated stationary Gaussian process with a time-modulating function following CP spectrum. Third, combining equations of motions of outrigger systems and seismic excitation gives rise to the augmented state-space representation of structure-damper-excitation and subsequently the differential Lyapunov equation. Fourth, the optimal designs of damped outriggers are defined via the solution of the differential Lyapunov equation, and the multiobjective optimization with Pareto optimal fronts is adopted to deal with conflicting objectives, followed by the sensitivity of negative stiffness ratio on design objectives. Finally, the efficacy of the proposed method is demonstrated through a tall building designed with various damped outriggers and the optimal designs are further examined under real earthquake records.


## 2. Problem Formulation

This section presents the stochastic analysis for the mixed NSDO and CDO systems subject to nonstationary seismic excitation. First, a simplified structural model of these outrigger systems is developed. With the motion equations of structure-damper-excitation, the state-space representation is formulated. The nonstationary seismic excitation is then modeled as uniformly modulated stationary Gaussian stochastic process, where the CP model is adopted to deal with low frequency issue. Finally, an augmented state-space representation of such systems with seismic excitation considered is formulated and subsequently the differential Lyapunov equation governing the stochastic structural responses is formulated, the root-mean-square (RMS) values of which will be used as optimization objectives.


### 2.1. Simplified Model of the NSDO Systems

Inspired by active controllers that exhibit negative slope of force-deformation relationship, negative stiffness dampers (NSDs) could be realized through variable orifice dampers [41], semiactive dampers via linear quadratic regulator [42], precompressed springs [43], static and moving magnets integrated with eddy current [44], snap-through effect of a prebuckled beam [45], and inverted friction pendulum sliding dampers [46]. A well-developed NSD using a precompressed spring and a pivot plate is adopted to connect both perimeter columns and outrigger ends (see [34] for more details).Figure2 shows the equivalent models of different outriggers. For the CDO, shown in Figure 2(a), the force at the outrigger end produced by the viscous damper is(1)FCDO=cdu˙−u˙c,where u and uc are the vertical displacements of the ends of the outrigger and the perimeter column relative to the ground, respectively, the overdot represents derivative with respect to time, and cd is the damping coefficient.Figure 2
Equivalent model for (a) conventional damped outrigger (CDO) and (b) negative stiffness damped outrigger (NSDO).
(a)(b)In Figure2(b), a negative stiffness device is incorporated in parallel with a viscous damper in the NSDO, and the force at the outrigger end generated by the NSDO is(2)FNSDO=kNSu−uc+cdu˙−u˙c,where kNS is the negative stiffness. Obviously, the CDO can be viewed as the special case of the NSDO when kNS=0.The simplified structural model for the NSDO system is shown in Figure3, where different outrigger systems can be obtained with adjustment of the design parameters. EI, m, and H, respectively, indicate flexural rigidity, the mass per unit length, and the height of the core, which is modeled as a uniform Bernoulli–Euler beam.Figure 3
Simplified model of tall buildings with multiple NSDOs.The outriggers are assumed to have infinite flexural rigidity with the span of 2r. The axial stiffness of the perimeter column is EcAc. Outriggers are located at a height of α1H, …, αjH, …, αnH from the bottom of the building, respectively. For convenience, the following derivation considers the NSDO system with two damped outriggers and can be easily extended to tall buildings with any number of outriggers.The equilibrium equation between the NSDO and the perimeter column at the first outrigger is(3)Fr1=kNS1u1−uc1+cd1u˙1−u˙c1,kNS1uc1−u1+cd1u˙c1−u˙1+kc1uc1−kc2uc2−uc1=0,where u1 is the vertical displacements of the first outrigger end; uc1 is the axial deformation of perimeter column below the first outrigger; kNS1 and cd1 are the negative stiffness and damping coefficient of the first NSDO, respectively; and the axial stiffness of perimeter columns split by outriggers is kc1=EcAc/α1H,kc2=EcAc/α2−α1H.The equilibrium equation between the NSDO and the perimeter column at the second outrigger is(4)Fr2=kNS2u2−uc2+cd2u˙2−u˙c2,kNS2uc2−u2+cd2u˙c2−u˙2+kc2uc2−uc1=0,where u2 is the vertical displacements of the second outrigger end; uc2 is the axial deformation of perimeter column between the second and the first outriggers; and kNS2 and cd2 are the negative stiffness and damping coefficient of the second NSDO, respectively.Combining equations (1)–(4) leads to(5)Fr=KNSu−uc+Cdu˙−u˙c=Kcuc,where Fr=Fr1t,Fr2tT, u=u1,u2T, uc=uc1,uc2T, and(6)KNS=kNS1kNS2,Cd=cd1cd2,Kc=kc1+kc2−kc2−kc2kc2.It should be pointed out that properly adjusting the parameters of viscous dampers and the negative stiffness in the NSDO systems results in various kinds of outrigger systems. For instance, the NSDO systems reduce to the CDO systems if all negative stiffness is set to zero and further reduce to either the CO systems with the viscous coefficient set to infinite or the bare structure (structure without outriggers, referred to as BS hereafter) if the viscous coefficient is set to zero. For multiple outrigger structures, the mixed NSDO and CO systems can be obtained when some of diagonal elements inthe negatives stiffness K_NS are set to zeros and some diagonal components in the viscous coefficient C_d infinite, while the mixed NSDO and CDO systems result from the diagonal entries of negative stiffness K_NS being partly set to zeros.


### 2.2. State-Space Representation of the NSDO

As in benchmark study, the tall building can be simplified as a cantilevered beam and the structural responses can be derived from it [47]. In this study, the finite element analysis method is used to derive the motion equation of the structural seismic response. The dynamic equation can be expressed as(7)Mx¨+Cx˙+Kx=Fext+Fθr,where x=u1,θ1,⋯uj,θj⋯un,θnT is the displacement vector with uj and θj representing the transverse and rotational components at the j-th story and n being the number of beam elements and M, C, and K are the global mass, damping, and stiffness matrices with M and K assembled from their corresponding element-wise mass and stiffness matrices modeled as a Bernoulli–Euler beam [48].(8)Me=mL042015622L054−13L022L04L0213L0−3L025413L0156−22L0−13L0−3L02−22L04L02,where L0 stands for the element length. Similarly, the stiffness matrix for one element is(9)Ke=EIL03126L0−126L06L04L02−6L02L02−12−6L012−6L06L02L02−6L04L02.Also, the global damping matrixC can be constructed by Rayleigh assumption.Substitutingu=rθn=rΓTx into equation (7), Fθr can be expressed as(10)Fθr=−2Frr=−2rKNSrΓTx−uc+CdrΓTx˙−u˙c,where Γ is the position vector of the outriggers. In addition, the inertial force caused by earthquake on one Bernoulli–Euler beam element is given by [49](11)Fej=−mx¨gL0126L06−L0,where x¨g is the acceleration of horizontal ground motion and L0=H/n is the length of the discretized element.The seismic excitationFext can be formulated as(12)Fext=δx¨g,where δ is the earthquake action vector assembled from Fej.In this study, three parameters design variables considered: dimensionless negative stiffness ratioη, dimensionless outrigger damping coefficient c, and outrigger location N0. Also, the dimensionless parameters c, β, and η are defined as [34](13)c=cdr2HmEI,β=EI2EcAcr2,η=kNSHr2EI.Substituting equations (10)–(13) into equation (7) yields the following equation:(14)Mrx¨r+Crx˙r+Krxr=Pr,with Mr, Cr, Kr, and xr as follows:(15a)Mr=M00me,Cr=C+Γ2Cdr2ΓT−2CdrΓCdrΓT−Cd,(15b)Kr=K+Γ2KNSr2ΓT−2KNSrΓKNSrΓT−Kc+KNS,xr=xuc,Pr=Fext0,where me is the virtual mass of the connection between the perimeter column and the damper, and its value is set very small to avoid the singularity of the mass matrix Mr [50].(16)Cd=c1HmEIr2c2HmEIr2,KNS=EIη1Hr2EIη2Hr2,Kc=EI2r2βα1H+EI2r2βα2−α1H−EI2r2βα2−α1H−EI2r2βα2−α1HEI2r2βα2−α1H.From equation (14), the state-space representation of the NSDO systems can be expressed as(17)Z˙r=AbZr+Bb,Yr=CbZr+Db,where Zr=xrx˙r,Ab=0I−Mr−1Kr−Mr−1Cr,Bb=0Mr−1Pr, and Cb and Db are output matrices determined by output variable Yr. If the displacement and acceleration are selected to form the output vector, we will have the following expression:(18)Cb=I0−Mr−1Kr−Mr−1Cr,Db=0Mr−1Pr.


### 2.3. Nonstationary Stochastic Seismic Excitation

Assuming that the evolution of the frequency content with time can be neglected, the earthquake excitation is considered as a uniformly modulated stationary Gaussian stochastic process with a zero mean described by the two-side evolutionary power spectral density (PSD) function, that is [51],(19)Sx¨gω=ϕt2Sx¨fω,where ϕt is the time-modulating function expressed as(20)ϕt=tt1α0≤t≤t1,1t1≤t≤t2,e−t−t2/τt2≤t≤TD,where t1 and t2 are the start and end time of the strong motion duration, respectively; α and τ are the constant shape coefficients; and TD is the strong motion duration. The envelope function is shown in Figure 4.Figure 4
Nonstationary amplitude envelope function.The Clough–Penzien (CP) model is introduced to correct the unrealistically high values of the power spectral density function at low frequency in the Kanai–Tajimi spectrum [52].(21)Sx¨fω=S0ωg4+4ξg2ωg2ω2ωg2−ω22+4ξg2ωg2ω2ω4ωf2−ω22+4ξf2ωf2ω2,where S0 is the two-sided power spectral density of the input white noise, ωg and ξg are the site predominant frequency and damping ratio for the high-frequency constant, respectively, and ωf and ξf are the corresponding parameters for the low-frequency constant, respectively.Thus, the earthquake accelerationx¨gt is expressed as(22)x¨gt=ϕtx¨ft,where x¨ft is a stochastic stationary process and can be considered as the response of two linear filters subject to white-noise excitation w with a constant PSD as follows:(23)x¨ft+2ξfωfx˙ft+ωf2xft=−x¨0t−wt,(24)x¨0t+2ξgωgx˙0t+ωg2x0t=−wt,where x0 (t) is the response of the first filter.DefiningZg=xfx0x˙fx˙0T, equations (23) and (24) can be then expressed in the state-space form as(25)Z˙g=AgZg+Bgwt,x¨gt=CgZg,where Ag, Bg, and Cg represent the characteristics of the excitation, which are given by(26)Ag=00100001−ωf2ωg2−2ξfωf2ξgωg0−ωg20−2ξgωg,Bg=000−1,Cg=ϕt−ωf2ωg2−2ξfωf2ξgωg.


### 2.4. RMS Values of Structural Response

An augmented state vector,Za, is defined as Za=ZrTZgTT to yield a combined representation of a controlled structure and excitation:(27)Z˙a=AaZa+Bawt,ya=CaZa,Aa=ABCg0Ag,Ba=0Bg,Ca=CDCg.The covariance of the structural responses is employed to assess structural performance. The covariance matrixΓXa=Exa−μxaxa−μxaT of the vector Za is determined through the solution of(28)Γ˙xa=AaΓxa+ΓxaAaT+2πBaS0BaT.Assuming that the initial conditions are deterministic,Γxa0=Γ0=0, to solve equation (28). Γ˙xa at a given time (t) is described using a backward difference scheme [51]:(29)Γ.xat=Γxat−Γxat−1∆t,where ∆t is the time interval.At each time interval, structural responses are obtained using the solution of(30)∆tAa−IΓxat+∆tΓxatAaT+2π∆tBaS0BaT+Γxat−1=0.In the case ofϕt=1, as ∆t ⟶ ∞, the seismic loading became a stationary stochastic process. The stationary solution can be obtained directly as a solution of the Lyapunov equation:(31)Γ˙xa=AaΓxa+ΓxaAaT+2πBaS0BaT=0.Finally, the RMS response of the output structural responses is given by(32)YaRMS=diagCaΓxaCaT,where diag (·) is the diagonal term of a square matrix.


## 2.1. Simplified Model of the NSDO Systems

Inspired by active controllers that exhibit negative slope of force-deformation relationship, negative stiffness dampers (NSDs) could be realized through variable orifice dampers [41], semiactive dampers via linear quadratic regulator [42], precompressed springs [43], static and moving magnets integrated with eddy current [44], snap-through effect of a prebuckled beam [45], and inverted friction pendulum sliding dampers [46]. A well-developed NSD using a precompressed spring and a pivot plate is adopted to connect both perimeter columns and outrigger ends (see [34] for more details).Figure2 shows the equivalent models of different outriggers. For the CDO, shown in Figure 2(a), the force at the outrigger end produced by the viscous damper is(1)FCDO=cdu˙−u˙c,where u and uc are the vertical displacements of the ends of the outrigger and the perimeter column relative to the ground, respectively, the overdot represents derivative with respect to time, and cd is the damping coefficient.Figure 2
Equivalent model for (a) conventional damped outrigger (CDO) and (b) negative stiffness damped outrigger (NSDO).
(a)(b)In Figure2(b), a negative stiffness device is incorporated in parallel with a viscous damper in the NSDO, and the force at the outrigger end generated by the NSDO is(2)FNSDO=kNSu−uc+cdu˙−u˙c,where kNS is the negative stiffness. Obviously, the CDO can be viewed as the special case of the NSDO when kNS=0.The simplified structural model for the NSDO system is shown in Figure3, where different outrigger systems can be obtained with adjustment of the design parameters. EI, m, and H, respectively, indicate flexural rigidity, the mass per unit length, and the height of the core, which is modeled as a uniform Bernoulli–Euler beam.Figure 3
Simplified model of tall buildings with multiple NSDOs.The outriggers are assumed to have infinite flexural rigidity with the span of 2r. The axial stiffness of the perimeter column is EcAc. Outriggers are located at a height of α1H, …, αjH, …, αnH from the bottom of the building, respectively. For convenience, the following derivation considers the NSDO system with two damped outriggers and can be easily extended to tall buildings with any number of outriggers.The equilibrium equation between the NSDO and the perimeter column at the first outrigger is(3)Fr1=kNS1u1−uc1+cd1u˙1−u˙c1,kNS1uc1−u1+cd1u˙c1−u˙1+kc1uc1−kc2uc2−uc1=0,where u1 is the vertical displacements of the first outrigger end; uc1 is the axial deformation of perimeter column below the first outrigger; kNS1 and cd1 are the negative stiffness and damping coefficient of the first NSDO, respectively; and the axial stiffness of perimeter columns split by outriggers is kc1=EcAc/α1H,kc2=EcAc/α2−α1H.The equilibrium equation between the NSDO and the perimeter column at the second outrigger is(4)Fr2=kNS2u2−uc2+cd2u˙2−u˙c2,kNS2uc2−u2+cd2u˙c2−u˙2+kc2uc2−uc1=0,where u2 is the vertical displacements of the second outrigger end; uc2 is the axial deformation of perimeter column between the second and the first outriggers; and kNS2 and cd2 are the negative stiffness and damping coefficient of the second NSDO, respectively.Combining equations (1)–(4) leads to(5)Fr=KNSu−uc+Cdu˙−u˙c=Kcuc,where Fr=Fr1t,Fr2tT, u=u1,u2T, uc=uc1,uc2T, and(6)KNS=kNS1kNS2,Cd=cd1cd2,Kc=kc1+kc2−kc2−kc2kc2.It should be pointed out that properly adjusting the parameters of viscous dampers and the negative stiffness in the NSDO systems results in various kinds of outrigger systems. For instance, the NSDO systems reduce to the CDO systems if all negative stiffness is set to zero and further reduce to either the CO systems with the viscous coefficient set to infinite or the bare structure (structure without outriggers, referred to as BS hereafter) if the viscous coefficient is set to zero. For multiple outrigger structures, the mixed NSDO and CO systems can be obtained when some of diagonal elements inthe negatives stiffness K_NS are set to zeros and some diagonal components in the viscous coefficient C_d infinite, while the mixed NSDO and CDO systems result from the diagonal entries of negative stiffness K_NS being partly set to zeros.


## 2.2. State-Space Representation of the NSDO

As in benchmark study, the tall building can be simplified as a cantilevered beam and the structural responses can be derived from it [47]. In this study, the finite element analysis method is used to derive the motion equation of the structural seismic response. The dynamic equation can be expressed as(7)Mx¨+Cx˙+Kx=Fext+Fθr,where x=u1,θ1,⋯uj,θj⋯un,θnT is the displacement vector with uj and θj representing the transverse and rotational components at the j-th story and n being the number of beam elements and M, C, and K are the global mass, damping, and stiffness matrices with M and K assembled from their corresponding element-wise mass and stiffness matrices modeled as a Bernoulli–Euler beam [48].(8)Me=mL042015622L054−13L022L04L0213L0−3L025413L0156−22L0−13L0−3L02−22L04L02,where L0 stands for the element length. Similarly, the stiffness matrix for one element is(9)Ke=EIL03126L0−126L06L04L02−6L02L02−12−6L012−6L06L02L02−6L04L02.Also, the global damping matrixC can be constructed by Rayleigh assumption.Substitutingu=rθn=rΓTx into equation (7), Fθr can be expressed as(10)Fθr=−2Frr=−2rKNSrΓTx−uc+CdrΓTx˙−u˙c,where Γ is the position vector of the outriggers. In addition, the inertial force caused by earthquake on one Bernoulli–Euler beam element is given by [49](11)Fej=−mx¨gL0126L06−L0,where x¨g is the acceleration of horizontal ground motion and L0=H/n is the length of the discretized element.The seismic excitationFext can be formulated as(12)Fext=δx¨g,where δ is the earthquake action vector assembled from Fej.In this study, three parameters design variables considered: dimensionless negative stiffness ratioη, dimensionless outrigger damping coefficient c, and outrigger location N0. Also, the dimensionless parameters c, β, and η are defined as [34](13)c=cdr2HmEI,β=EI2EcAcr2,η=kNSHr2EI.Substituting equations (10)–(13) into equation (7) yields the following equation:(14)Mrx¨r+Crx˙r+Krxr=Pr,with Mr, Cr, Kr, and xr as follows:(15a)Mr=M00me,Cr=C+Γ2Cdr2ΓT−2CdrΓCdrΓT−Cd,(15b)Kr=K+Γ2KNSr2ΓT−2KNSrΓKNSrΓT−Kc+KNS,xr=xuc,Pr=Fext0,where me is the virtual mass of the connection between the perimeter column and the damper, and its value is set very small to avoid the singularity of the mass matrix Mr [50].(16)Cd=c1HmEIr2c2HmEIr2,KNS=EIη1Hr2EIη2Hr2,Kc=EI2r2βα1H+EI2r2βα2−α1H−EI2r2βα2−α1H−EI2r2βα2−α1HEI2r2βα2−α1H.From equation (14), the state-space representation of the NSDO systems can be expressed as(17)Z˙r=AbZr+Bb,Yr=CbZr+Db,where Zr=xrx˙r,Ab=0I−Mr−1Kr−Mr−1Cr,Bb=0Mr−1Pr, and Cb and Db are output matrices determined by output variable Yr. If the displacement and acceleration are selected to form the output vector, we will have the following expression:(18)Cb=I0−Mr−1Kr−Mr−1Cr,Db=0Mr−1Pr.


## 2.3. Nonstationary Stochastic Seismic Excitation

Assuming that the evolution of the frequency content with time can be neglected, the earthquake excitation is considered as a uniformly modulated stationary Gaussian stochastic process with a zero mean described by the two-side evolutionary power spectral density (PSD) function, that is [51],(19)Sx¨gω=ϕt2Sx¨fω,where ϕt is the time-modulating function expressed as(20)ϕt=tt1α0≤t≤t1,1t1≤t≤t2,e−t−t2/τt2≤t≤TD,where t1 and t2 are the start and end time of the strong motion duration, respectively; α and τ are the constant shape coefficients; and TD is the strong motion duration. The envelope function is shown in Figure 4.Figure 4
Nonstationary amplitude envelope function.The Clough–Penzien (CP) model is introduced to correct the unrealistically high values of the power spectral density function at low frequency in the Kanai–Tajimi spectrum [52].(21)Sx¨fω=S0ωg4+4ξg2ωg2ω2ωg2−ω22+4ξg2ωg2ω2ω4ωf2−ω22+4ξf2ωf2ω2,where S0 is the two-sided power spectral density of the input white noise, ωg and ξg are the site predominant frequency and damping ratio for the high-frequency constant, respectively, and ωf and ξf are the corresponding parameters for the low-frequency constant, respectively.Thus, the earthquake accelerationx¨gt is expressed as(22)x¨gt=ϕtx¨ft,where x¨ft is a stochastic stationary process and can be considered as the response of two linear filters subject to white-noise excitation w with a constant PSD as follows:(23)x¨ft+2ξfωfx˙ft+ωf2xft=−x¨0t−wt,(24)x¨0t+2ξgωgx˙0t+ωg2x0t=−wt,where x0 (t) is the response of the first filter.DefiningZg=xfx0x˙fx˙0T, equations (23) and (24) can be then expressed in the state-space form as(25)Z˙g=AgZg+Bgwt,x¨gt=CgZg,where Ag, Bg, and Cg represent the characteristics of the excitation, which are given by(26)Ag=00100001−ωf2ωg2−2ξfωf2ξgωg0−ωg20−2ξgωg,Bg=000−1,Cg=ϕt−ωf2ωg2−2ξfωf2ξgωg.


## 2.4. RMS Values of Structural Response

An augmented state vector,Za, is defined as Za=ZrTZgTT to yield a combined representation of a controlled structure and excitation:(27)Z˙a=AaZa+Bawt,ya=CaZa,Aa=ABCg0Ag,Ba=0Bg,Ca=CDCg.The covariance of the structural responses is employed to assess structural performance. The covariance matrixΓXa=Exa−μxaxa−μxaT of the vector Za is determined through the solution of(28)Γ˙xa=AaΓxa+ΓxaAaT+2πBaS0BaT.Assuming that the initial conditions are deterministic,Γxa0=Γ0=0, to solve equation (28). Γ˙xa at a given time (t) is described using a backward difference scheme [51]:(29)Γ.xat=Γxat−Γxat−1∆t,where ∆t is the time interval.At each time interval, structural responses are obtained using the solution of(30)∆tAa−IΓxat+∆tΓxatAaT+2π∆tBaS0BaT+Γxat−1=0.In the case ofϕt=1, as ∆t ⟶ ∞, the seismic loading became a stationary stochastic process. The stationary solution can be obtained directly as a solution of the Lyapunov equation:(31)Γ˙xa=AaΓxa+ΓxaAaT+2πBaS0BaT=0.Finally, the RMS response of the output structural responses is given by(32)YaRMS=diagCaΓxaCaT,where diag (·) is the diagonal term of a square matrix.


## 3. Multiobjective Optimization

In engineering practice, the specific design of a building at hand is supposed to meet various kinds of requirements with limited available resources, and how to achieve the goal while minimizing the use of resources is challenging. In this section, two main objectives associated with the safety and serviceability of tall buildings, harmful interstory drift and floor acceleration, are first defined and the definition of the former is briefly introduced. The Pareto optimal front is then used to provide trade-off between conflicting objectives. Finally, a procedure to implement multiobjective optimization approach is outlined.


### 3.1. Structural Optimization Formulation

The optimization problem can be defined as follows:(33)designvariables:v=ηcN0Tminimize:Jdualvsubjectto:GYa,v≤0vmin≤v≤vmax,where v is the design variable, Jdual indicates the objective function related to harmful interstory drift and floor absolute acceleration, GYa,v is defined as the constrained function of design variables and responses, which could be empty if not specified explicitly, and vmin and vmax are the lower and upper bounds of design variables, respectively.


### 3.2. Optimization Objectives

Because different objectives might have several orders of difference in magnitude and cause numerical issues during calculation, the two objectives used herein are defined as the maximum structural responses of the damped outrigger systems normalized by the corresponding responses of the bare structure (BS), which are given as(34)Jhdr=maxThdrmaxThdr0,(35)Jacc=maxTaccmaxTacc0,where Thdr and Tacc are the RMS values of the harmful interstory drift ratio and floor absolute acceleration of the damped outrigger systems under nonstationary stochastic excitation obtained from equation (32), respectively, whereas the subscript “0” indicates the corresponding response quantity of the BS, and max· denotes the maximum value inside the bracket. The harmful interstory drift ratio in equation (34) will be briefly revisited below.Traditionally, the interstory drift ratioθi of the i-th story has been adopted as a performance index and can be expressed as [53](36)θi=∆uihi,where ∆ui is interstory drift of the i-th story, respectively, and hi the height of the i-th story.Examination of the interstory drift in equation (36), however, reveals that the total drift consists of three components: interstory shear drift ∆usi, bending-induced flexural drift ∆ubi, and drift ∆uri due to rigid body rotation of the floor, as shown in Figure 5 [54]. While the first two components could cause damage to the structures and hence the name harmful interstory drift, the last one is considered harmless to the related story. Recently, researchers argued that the harmless drift should be removed and the focus should be placed on harmful interstory drift for design purpose [33]. As such, the harmful interstory drift is adopted as the seismic performance index and can be calculated by the secant method [54]:(37)θ˜i=∆uihi−∆ui−1hi−1=θi−θi−1.Figure 5
Illustration of interstory drift components.


### 3.3. Pareto Optimal Fronts

For multiobjective optimization, the ideal situation is to optimize all the objectives simultaneously, which is typically not the case in the real world; some objectives are conflicting, i.e., minimization of one objective leads to amplification of other objectives. Such problem can be solved by using Pareto optimal fronts, which associate each objective a weight to reflect the trade-off between conflicting objectives and transform the multiobjective optimization into single objective design. In particular, the Pareto optimal frontJdual balancing the objectives of harmful interstory drift and floor acceleration is defined as [34](38)Jdual=ζJacc+1−ζJhdr,where ζ ∈ [0, 1] is the weight value for the dual-response design function. When ζ=1,Jdual=Jacc, which represents the optimization based on harmful interstory drift, whereas ζ=0,Jdual=Jhdr, which refers to the optimization based on floor acceleration.The steps to implement the multiobjective optimization method proposed in this paper are shown in the flowchart in Figure6. During optimization, the generalized pattern search (GPS) algorithm is adopted because of its capability of dealing with noncontinuous and nondifferentiable objective functions [55, 56], which is available with the functions “patternsearch” and “fminsearch” in MATLAB toolbox.Figure 6
Dual-response design flowchart.


## 3.1. Structural Optimization Formulation

The optimization problem can be defined as follows:(33)designvariables:v=ηcN0Tminimize:Jdualvsubjectto:GYa,v≤0vmin≤v≤vmax,where v is the design variable, Jdual indicates the objective function related to harmful interstory drift and floor absolute acceleration, GYa,v is defined as the constrained function of design variables and responses, which could be empty if not specified explicitly, and vmin and vmax are the lower and upper bounds of design variables, respectively.


## 3.2. Optimization Objectives

Because different objectives might have several orders of difference in magnitude and cause numerical issues during calculation, the two objectives used herein are defined as the maximum structural responses of the damped outrigger systems normalized by the corresponding responses of the bare structure (BS), which are given as(34)Jhdr=maxThdrmaxThdr0,(35)Jacc=maxTaccmaxTacc0,where Thdr and Tacc are the RMS values of the harmful interstory drift ratio and floor absolute acceleration of the damped outrigger systems under nonstationary stochastic excitation obtained from equation (32), respectively, whereas the subscript “0” indicates the corresponding response quantity of the BS, and max· denotes the maximum value inside the bracket. The harmful interstory drift ratio in equation (34) will be briefly revisited below.Traditionally, the interstory drift ratioθi of the i-th story has been adopted as a performance index and can be expressed as [53](36)θi=∆uihi,where ∆ui is interstory drift of the i-th story, respectively, and hi the height of the i-th story.Examination of the interstory drift in equation (36), however, reveals that the total drift consists of three components: interstory shear drift ∆usi, bending-induced flexural drift ∆ubi, and drift ∆uri due to rigid body rotation of the floor, as shown in Figure 5 [54]. While the first two components could cause damage to the structures and hence the name harmful interstory drift, the last one is considered harmless to the related story. Recently, researchers argued that the harmless drift should be removed and the focus should be placed on harmful interstory drift for design purpose [33]. As such, the harmful interstory drift is adopted as the seismic performance index and can be calculated by the secant method [54]:(37)θ˜i=∆uihi−∆ui−1hi−1=θi−θi−1.Figure 5
Illustration of interstory drift components.


## 3.3. Pareto Optimal Fronts

For multiobjective optimization, the ideal situation is to optimize all the objectives simultaneously, which is typically not the case in the real world; some objectives are conflicting, i.e., minimization of one objective leads to amplification of other objectives. Such problem can be solved by using Pareto optimal fronts, which associate each objective a weight to reflect the trade-off between conflicting objectives and transform the multiobjective optimization into single objective design. In particular, the Pareto optimal frontJdual balancing the objectives of harmful interstory drift and floor acceleration is defined as [34](38)Jdual=ζJacc+1−ζJhdr,where ζ ∈ [0, 1] is the weight value for the dual-response design function. When ζ=1,Jdual=Jacc, which represents the optimization based on harmful interstory drift, whereas ζ=0,Jdual=Jhdr, which refers to the optimization based on floor acceleration.The steps to implement the multiobjective optimization method proposed in this paper are shown in the flowchart in Figure6. During optimization, the generalized pattern search (GPS) algorithm is adopted because of its capability of dealing with noncontinuous and nondifferentiable objective functions [55, 56], which is available with the functions “patternsearch” and “fminsearch” in MATLAB toolbox.Figure 6
Dual-response design flowchart.


## 4. An Illustrated Example

This section demonstrates the stochastic optimization approach with an illustrated example. This section starts with a description of tall building with two damped outriggers in the forms of C&N, N&C, and DN. Stochastic optimization of three damped outrigger systems based on single objective subjected to stationary and nonstationary seismic excitation is compared. As the objectives are conflicting, the Pareto optimal fronts are used to provide trade-off, and sensitivity of negative stiffness ratio on design objectives is investigated.


### 4.1. Building Information

The numerical example used herein to demonstrate the stochastic optimization procedure is a 60-story building. The main model parameters are as follows: the height of the buildingH = 200 m, the flexural rigidity of the core EI = 1.47 × 1013 Nm2, distributed mass along the height of the core m = 1.08 × 105 kg/m, and the span of outrigger r = 15 m. The dimensionless stiffness ratio of core to perimeter column β = 2 [48]. The first three natural frequencies of the bare structure (structure without outriggers, BS) are 0.1632 Hz, 1.0228 Hz, and 2.8940 Hz. As shown in Figure 3, the two damped outriggers are designed as the mixed CDO and NSDO, i.e., either the CDO over the NSDO (C&N) or the NSDO over CDO (N&C), and double NSDOs (DN), where outriggers are assumed to have infinite flexural rigidity as described in Section 2.1. The finite element method is used to analyze the damped outrigger systems, and the building is divided into 60 elements with each story consisting a beam element; the damping matrix C is constructed by Rayleigh assumption, and the damping ratio of the first and third modes is assumed as 0.02.In the random seismic response analysis, the selected parameters of CP spectrum model areS0 = 4.62 × 10−4 m2/s3, ωg = 15 rad/s, ξg = 0.6, ωf = 1.5 rad/s, and ξf = 0.6. These parameters are selected from [57] to reflect firm-soil conditions. The parameters for the nonstationary envelope function given in equation (20) are taken to be t1 = 4.78 s, t2 = 8.96 s, α = 2.60, and τ = 0.13; the earthquake duration is chosen as TD = 30 s [52]. During the process of optimization, the upper level of damped outriggers is fixed at the top as the conventional configuration in the outrigger systems, and the elevation of the lower outrigger is considered as design variable. Meanwhile, the dimensionless viscous coefficient is within the range of [0, 1], and the dimensionless negative stiffness coefficient η is set within the range of [−0.1, 0] because too small value of it will cause instability of system [31]. During analysis, the main structure is assumed to remain linear.


### 4.2. Single-Objective Optimization

Before getting into the insight of the damped outrigger systems, the nonstationary and stationary response envelopes of the harmful interstory drift and floor absolute acceleration of the BS as well as the corresponding maximum values evolving with time are shown in Figure7. As observed in Figure 8, the peak RMS structural responses under nonstationary stochastic excitation occur after the strong motion duration ends. Moreover, the stationary responses of the BS overestimate the responses by 15.62% in maximum harmful interstory drift ratio and 16.52% in maximum floor absolute acceleration, i.e., the stationary optimization might lead to over-conservative results. This is mainly because the time-modulating function φ (t) of the stationary stochastic excitation is 1 (see equation (20)). Furthermore, while the structural maximum acceleration occurs at the top, the maximum harmful interstory drift ratio of structure appears at the base, as opposed to the commonly used interstory drift ratio, which typically shows up at the upper levels in the tall buildings.Figure 7
RMS response values of BS under nonstationary and stationary stochastic excitation. (a) Envelope of RMS harmful interstory drift ratio. (b) Envelope of RMS peak acceleration. (c) Bottom RMS harmful drift responses under both excitations. (d) Top RMS acceleration responses under both excitations.
(a)(b)(c)(d)Figure 8
Comparisons of RMS response values of different outrigger systems. (a) RMS harmful interstory drift ratio under nonstationary excitation. (b) RMS peak acceleration under nonstationary excitation. (c) Bottom RMS drift under nonstationary and stationary stochastic excitation based on harmful interstory drift optimization. (d) Top RMS acceleration under nonstationary and stationary stochastic excitation based on floor acceleration optimization.
(a)(b)(c)(d)With the proposed stochastic optimization, the optimal results of different outrigger systems are listed in Table1, and the response envelopes of different outrigger systems based on different objectives under nonstationary stochastic excitation are compared in Figures 8(a) and 8(b). Also shown in Figures 8(c) and 8(d) are the comparisons of the RMS values of the maximum harmful drift ratio and floor absolute acceleration under both kinds of excitations, which appear on the bottom and roof levels, respectively, where the responses of the structure under stationary stochastic excitation in dash lines are denoted by “(s).” As with the case of BS, the peaks of the RMS response under the nonstationary excitation occur after the strong motion duration, with the RMS stationary responses values of up to 6.82% and 9.24% overestimation on harmful interstory drift and floor acceleration, respectively. From Table 1, it can be observed that compared to the BS structure, all three damped outrigger systems have significant response reduction; specifically, around 36%–50% reduction in harmful interstory drift ratio and above 30% reduction in floor acceleration can be achieved when such systems are optimized based on harmful drift and floor acceleration, respectively. Among three damped outrigger systems, the DN system performs the best, while the C&N system has the compatible performance with the N&C system, and the latter performs slightly better than the former. Also observed is that these two objectives are conflicting. Take the N&C system as an example, and the optimal normalized harmful interstory drift ratio is 0.5966, but with acceleration as objective, it increases to 0.6169; likewise, the optimal normalized floor acceleration is 0.6658, but it becomes 0.6809 when using harmful drift as optimization objective. To deal with the competing objectives, the Pareto optimal fronts are adopted as a multiobjective optimization method to balance the two performance objectives and will be elaborated in the next section.Table 1
Detailed parameters and optimal results.
ObjectiveParameterC&NN&CDNHarmful interstory drift ratioη−0.09977−0.09902−0.09387c0.009840.009990.00791N0464940Jhdr0.63680.59660.4958Jacc0.69770.68090.6497Peak accelerationη−0.09858−0.09899−0.09378c0.009850.009980.00787N0343230Jhdr0.71170.61690.5837Jacc0.68150.66580.6355C&N represents the conventional damped outrigger (CDO) placed above the negative stiffness damped outrigger (NSDO), while N&C represents the NSDO placed above the CDO, and DN denotes double NSDOs. Also,Jhdr and Jacc represent the objective values of harmful interstory drift ratio and floor absolute acceleration of damped outrigger systems based on the objectives specified in the first column, respectively.


### 4.3. Multiobjective Optimization

As stated in the Section3.2, the common solution for multiobjective optimization with conflicting objectives is to assign each objective a weight representing its importance, and Pareto optimal fronts provide design flexibility with changing weights of objectives.To deal with the conflicting objectives and investigate the influence of stationarity of seismic excitation, Figures9–11(a) show the Pareto optimal fronts of different outrigger systems subjected to nonstationary and stationary seismic excitation and Figures 9(b)–11(d) the associated with optimal designs. The optimal designs based on stationary excitation assumption are also evaluated under the condition of nonstationary seismic excitation and the results are also plotted in Figures 9–11(a). In Figures 9–11, “Opt-NS” and “Opt-S” indicate structural optimizations under nonstationary and stationary random excitations, respectively, while “Opt-S-NS” indicates that the stationary-based optimal design is evaluated under the condition of nonstationary seismic excitation.Figure 9
Optimal results for C&N. (a) Pareto front of performance indexes. (b) Optimal outrigger position. (c) Optimal outrigger damping coefficient. (d) Optimal negative stiffness ratio.
(a)(b)(c)(d)Figure 10
Optimal results for N&C. (a) Pareto front of performance indexes. (b) Optimal outrigger position. (c) Optimal outrigger damping coefficient. (d) Optimal negative stiffness ratio.
(a)(b)(c)(d)Figure 11
Optimal results for DN. (a) Pareto front of performance indexes. (b) Optimal outrigger position. (c) Optimal outrigger damping coefficient. (d) Optimal negative stiffness ratio.
(a)(b)(c)(d)In the case of the DN system, as observed in the Figure11, the optimal damped outrigger locations coincide except for the weight being one with only one-story difference, and the maximum discrepancies of viscous coefficient and negative stiffness are 1.3% and 0.2%, respectively. Such tiny differences in the optimal design parameters have little influence for structures subjected to nonstationary earthquake, i.e., the optimization using stationary assumption is justified. As for both cases of combining conventional damped outrigger and negative stiffness damped outrigger in Figures 9 and 10, the optimal elevations of damped outrigger have the maximum two-story distiction, and the optimal viscous coefficient and negative stiffness obtained from stationary excitation have 1.4% and 1.8% discrepancies compared to those obtained from nonstationary excitation. Moreover, these minor differences in optimal designs only have 1.2% performance deterioration and generally is considered negligible when damped outrigger systems are subjected to more realistic nonstationary seismic excitation. These investigations justify the use of stationary seismic excitation for damped outrigger systems during optimization, i.e., the less time-consumingstationary-based seismic optimization is adequate for engineering design practice, which is consistent with the findings for traditional buildings in [58]. For the sake of scrutiny, nonstationary-based optimization is still adopted in the following unless specified explicitly.To see how the different configurations of damped outrigger systems behave, Figure12 shows the Pareto optimal fronts of three outrigger systems under stationary and nonstationary seismic excitations, respectively. Apparently, the DN system performs the best in reducing both harmful interstory drift and structural acceleration, followed by the combined system with the negative stiffness outrigger placing over viscous damper outrigger. In other words, the DN system could have the best performance among the three damped outrigger systems with sufficient negative stiffness devices available.Figure 12
Pareto front for different outrigger systems. (a) Stationary-based optimal design. (b) Nonstationary-based optimal design.
(a)(b)


### 4.4. Sensitivity Analysis of Negative Stiffness Ratio

However, the above comparison is not completely fair because the DN system is using almost as twice amount of the negative stiffness devices as the N&C and C&N systems, and the optimal negative stiffness ratios for three systems in Figures9–11(d)are close to the maximum allowable amount.Such suggest that the performance of three damped outrigger systems might be dominated by the amount of negative stiffness devices. Therefore, the sensitivity of system performance with respect to the total amount of negative stiffness ratio needs to be investigated and could be casted as constrained optimizations.Figure13 shows the Pareto optimal fronts of three damped outrigger systems with the maximum negative stiffness ratio varying from 0.05 to 0.1, and 0.13 only for the DN system. In Figure 13, the number inside the parentheses indicates the maximum total amount of negative stiffness ratio, e.g., “DN (0.05)” represents that Pareto front for the total normalized negative stiffness ratio (−η) in the range of 0–0.05. Comparing Figure 13 with Figures 9–12 shows that as the total amount of negative stiffness ratio available decreases (0.2 for the DN system and 0.1 for the other two), the systems decrease the reduction of the maximum harmful drift and floor absolute acceleration. More importantly, with the same total 0.5 negative stiffness ratio available, the C&N and N&C systems are superior to the DN system in reducing the harmful interstory drift and acceleration. When the maximum amount of negative stiffness ratio is 0.1, while the DN system performs slightly better than the C&N system, the N&C system still outperforms the other two and is still competitive against the DN system with −η = 0.13, where the former is slightly better than the latter on the right side of the intersection and less appealing on the left.Figure 13
Pareto optimal fronts with different total negative stiffness ratio (−η). (a) 0-0.05. (b) 0-0.1 (0.13 only for the DN system).
(a)(b)To see more clearly the sensitivity of negative stiffness ratio on the seismic performance, consider the three damped outrigger systems with the design objectives of balancing the mitigation on harmful interstory drift and floor acceleration (ζ = 0.5). Figure 14 shows the objectives of the maximum normalized harmful drift ratio and floor absolute acceleration of three damped outrigger systems against maximum allowable negative stiffness ratio, respectively. Again, the N&C and C&N systems, particularly the N&C system, have been shown to be more effective in reducing harmful interstory drift and floor acceleration within limited amount of negative stiffness ratio, while the DN system could have the best performance with sufficient amount of negative stiffness ratio.Figure 14
Influences of total normalized negative stiffness ratio (−η) at weight ζ = 0.5. (a) Harmful drift ratio. (b) Acceleration.
(a)(b)The above investigations suggest that the proposed combined NSDO and CDO damped outrigger system could provide valuable design flexibilities with damped outrigger configurations tailored for various objectives: (1) with adequate negative stiffness devices that do not cause the system to become unstable, the DN system would be the best design option by bringing down structure responses; (2) with limited amount negative stiffness devices, the N&C system would be the optimal configuration for its best utilization of devices, followed by the C&N and DN systems; and (3) when the available amount of negative stiffness devices lies in between, there might be a trade-offbetween the DN and N&C systems depending on the weightsassociated with harmful interstory drift and floor absolute acceleration.


## 4.1. Building Information

The numerical example used herein to demonstrate the stochastic optimization procedure is a 60-story building. The main model parameters are as follows: the height of the buildingH = 200 m, the flexural rigidity of the core EI = 1.47 × 1013 Nm2, distributed mass along the height of the core m = 1.08 × 105 kg/m, and the span of outrigger r = 15 m. The dimensionless stiffness ratio of core to perimeter column β = 2 [48]. The first three natural frequencies of the bare structure (structure without outriggers, BS) are 0.1632 Hz, 1.0228 Hz, and 2.8940 Hz. As shown in Figure 3, the two damped outriggers are designed as the mixed CDO and NSDO, i.e., either the CDO over the NSDO (C&N) or the NSDO over CDO (N&C), and double NSDOs (DN), where outriggers are assumed to have infinite flexural rigidity as described in Section 2.1. The finite element method is used to analyze the damped outrigger systems, and the building is divided into 60 elements with each story consisting a beam element; the damping matrix C is constructed by Rayleigh assumption, and the damping ratio of the first and third modes is assumed as 0.02.In the random seismic response analysis, the selected parameters of CP spectrum model areS0 = 4.62 × 10−4 m2/s3, ωg = 15 rad/s, ξg = 0.6, ωf = 1.5 rad/s, and ξf = 0.6. These parameters are selected from [57] to reflect firm-soil conditions. The parameters for the nonstationary envelope function given in equation (20) are taken to be t1 = 4.78 s, t2 = 8.96 s, α = 2.60, and τ = 0.13; the earthquake duration is chosen as TD = 30 s [52]. During the process of optimization, the upper level of damped outriggers is fixed at the top as the conventional configuration in the outrigger systems, and the elevation of the lower outrigger is considered as design variable. Meanwhile, the dimensionless viscous coefficient is within the range of [0, 1], and the dimensionless negative stiffness coefficient η is set within the range of [−0.1, 0] because too small value of it will cause instability of system [31]. During analysis, the main structure is assumed to remain linear.


## 4.2. Single-Objective Optimization

Before getting into the insight of the damped outrigger systems, the nonstationary and stationary response envelopes of the harmful interstory drift and floor absolute acceleration of the BS as well as the corresponding maximum values evolving with time are shown in Figure7. As observed in Figure 8, the peak RMS structural responses under nonstationary stochastic excitation occur after the strong motion duration ends. Moreover, the stationary responses of the BS overestimate the responses by 15.62% in maximum harmful interstory drift ratio and 16.52% in maximum floor absolute acceleration, i.e., the stationary optimization might lead to over-conservative results. This is mainly because the time-modulating function φ (t) of the stationary stochastic excitation is 1 (see equation (20)). Furthermore, while the structural maximum acceleration occurs at the top, the maximum harmful interstory drift ratio of structure appears at the base, as opposed to the commonly used interstory drift ratio, which typically shows up at the upper levels in the tall buildings.Figure 7
RMS response values of BS under nonstationary and stationary stochastic excitation. (a) Envelope of RMS harmful interstory drift ratio. (b) Envelope of RMS peak acceleration. (c) Bottom RMS harmful drift responses under both excitations. (d) Top RMS acceleration responses under both excitations.
(a)(b)(c)(d)Figure 8
Comparisons of RMS response values of different outrigger systems. (a) RMS harmful interstory drift ratio under nonstationary excitation. (b) RMS peak acceleration under nonstationary excitation. (c) Bottom RMS drift under nonstationary and stationary stochastic excitation based on harmful interstory drift optimization. (d) Top RMS acceleration under nonstationary and stationary stochastic excitation based on floor acceleration optimization.
(a)(b)(c)(d)With the proposed stochastic optimization, the optimal results of different outrigger systems are listed in Table1, and the response envelopes of different outrigger systems based on different objectives under nonstationary stochastic excitation are compared in Figures 8(a) and 8(b). Also shown in Figures 8(c) and 8(d) are the comparisons of the RMS values of the maximum harmful drift ratio and floor absolute acceleration under both kinds of excitations, which appear on the bottom and roof levels, respectively, where the responses of the structure under stationary stochastic excitation in dash lines are denoted by “(s).” As with the case of BS, the peaks of the RMS response under the nonstationary excitation occur after the strong motion duration, with the RMS stationary responses values of up to 6.82% and 9.24% overestimation on harmful interstory drift and floor acceleration, respectively. From Table 1, it can be observed that compared to the BS structure, all three damped outrigger systems have significant response reduction; specifically, around 36%–50% reduction in harmful interstory drift ratio and above 30% reduction in floor acceleration can be achieved when such systems are optimized based on harmful drift and floor acceleration, respectively. Among three damped outrigger systems, the DN system performs the best, while the C&N system has the compatible performance with the N&C system, and the latter performs slightly better than the former. Also observed is that these two objectives are conflicting. Take the N&C system as an example, and the optimal normalized harmful interstory drift ratio is 0.5966, but with acceleration as objective, it increases to 0.6169; likewise, the optimal normalized floor acceleration is 0.6658, but it becomes 0.6809 when using harmful drift as optimization objective. To deal with the competing objectives, the Pareto optimal fronts are adopted as a multiobjective optimization method to balance the two performance objectives and will be elaborated in the next section.Table 1
Detailed parameters and optimal results.
ObjectiveParameterC&NN&CDNHarmful interstory drift ratioη−0.09977−0.09902−0.09387c0.009840.009990.00791N0464940Jhdr0.63680.59660.4958Jacc0.69770.68090.6497Peak accelerationη−0.09858−0.09899−0.09378c0.009850.009980.00787N0343230Jhdr0.71170.61690.5837Jacc0.68150.66580.6355C&N represents the conventional damped outrigger (CDO) placed above the negative stiffness damped outrigger (NSDO), while N&C represents the NSDO placed above the CDO, and DN denotes double NSDOs. Also,Jhdr and Jacc represent the objective values of harmful interstory drift ratio and floor absolute acceleration of damped outrigger systems based on the objectives specified in the first column, respectively.


## 4.3. Multiobjective Optimization

As stated in the Section3.2, the common solution for multiobjective optimization with conflicting objectives is to assign each objective a weight representing its importance, and Pareto optimal fronts provide design flexibility with changing weights of objectives.To deal with the conflicting objectives and investigate the influence of stationarity of seismic excitation, Figures9–11(a) show the Pareto optimal fronts of different outrigger systems subjected to nonstationary and stationary seismic excitation and Figures 9(b)–11(d) the associated with optimal designs. The optimal designs based on stationary excitation assumption are also evaluated under the condition of nonstationary seismic excitation and the results are also plotted in Figures 9–11(a). In Figures 9–11, “Opt-NS” and “Opt-S” indicate structural optimizations under nonstationary and stationary random excitations, respectively, while “Opt-S-NS” indicates that the stationary-based optimal design is evaluated under the condition of nonstationary seismic excitation.Figure 9
Optimal results for C&N. (a) Pareto front of performance indexes. (b) Optimal outrigger position. (c) Optimal outrigger damping coefficient. (d) Optimal negative stiffness ratio.
(a)(b)(c)(d)Figure 10
Optimal results for N&C. (a) Pareto front of performance indexes. (b) Optimal outrigger position. (c) Optimal outrigger damping coefficient. (d) Optimal negative stiffness ratio.
(a)(b)(c)(d)Figure 11
Optimal results for DN. (a) Pareto front of performance indexes. (b) Optimal outrigger position. (c) Optimal outrigger damping coefficient. (d) Optimal negative stiffness ratio.
(a)(b)(c)(d)In the case of the DN system, as observed in the Figure11, the optimal damped outrigger locations coincide except for the weight being one with only one-story difference, and the maximum discrepancies of viscous coefficient and negative stiffness are 1.3% and 0.2%, respectively. Such tiny differences in the optimal design parameters have little influence for structures subjected to nonstationary earthquake, i.e., the optimization using stationary assumption is justified. As for both cases of combining conventional damped outrigger and negative stiffness damped outrigger in Figures 9 and 10, the optimal elevations of damped outrigger have the maximum two-story distiction, and the optimal viscous coefficient and negative stiffness obtained from stationary excitation have 1.4% and 1.8% discrepancies compared to those obtained from nonstationary excitation. Moreover, these minor differences in optimal designs only have 1.2% performance deterioration and generally is considered negligible when damped outrigger systems are subjected to more realistic nonstationary seismic excitation. These investigations justify the use of stationary seismic excitation for damped outrigger systems during optimization, i.e., the less time-consumingstationary-based seismic optimization is adequate for engineering design practice, which is consistent with the findings for traditional buildings in [58]. For the sake of scrutiny, nonstationary-based optimization is still adopted in the following unless specified explicitly.To see how the different configurations of damped outrigger systems behave, Figure12 shows the Pareto optimal fronts of three outrigger systems under stationary and nonstationary seismic excitations, respectively. Apparently, the DN system performs the best in reducing both harmful interstory drift and structural acceleration, followed by the combined system with the negative stiffness outrigger placing over viscous damper outrigger. In other words, the DN system could have the best performance among the three damped outrigger systems with sufficient negative stiffness devices available.Figure 12
Pareto front for different outrigger systems. (a) Stationary-based optimal design. (b) Nonstationary-based optimal design.
(a)(b)


## 4.4. Sensitivity Analysis of Negative Stiffness Ratio

However, the above comparison is not completely fair because the DN system is using almost as twice amount of the negative stiffness devices as the N&C and C&N systems, and the optimal negative stiffness ratios for three systems in Figures9–11(d)are close to the maximum allowable amount.Such suggest that the performance of three damped outrigger systems might be dominated by the amount of negative stiffness devices. Therefore, the sensitivity of system performance with respect to the total amount of negative stiffness ratio needs to be investigated and could be casted as constrained optimizations.Figure13 shows the Pareto optimal fronts of three damped outrigger systems with the maximum negative stiffness ratio varying from 0.05 to 0.1, and 0.13 only for the DN system. In Figure 13, the number inside the parentheses indicates the maximum total amount of negative stiffness ratio, e.g., “DN (0.05)” represents that Pareto front for the total normalized negative stiffness ratio (−η) in the range of 0–0.05. Comparing Figure 13 with Figures 9–12 shows that as the total amount of negative stiffness ratio available decreases (0.2 for the DN system and 0.1 for the other two), the systems decrease the reduction of the maximum harmful drift and floor absolute acceleration. More importantly, with the same total 0.5 negative stiffness ratio available, the C&N and N&C systems are superior to the DN system in reducing the harmful interstory drift and acceleration. When the maximum amount of negative stiffness ratio is 0.1, while the DN system performs slightly better than the C&N system, the N&C system still outperforms the other two and is still competitive against the DN system with −η = 0.13, where the former is slightly better than the latter on the right side of the intersection and less appealing on the left.Figure 13
Pareto optimal fronts with different total negative stiffness ratio (−η). (a) 0-0.05. (b) 0-0.1 (0.13 only for the DN system).
(a)(b)To see more clearly the sensitivity of negative stiffness ratio on the seismic performance, consider the three damped outrigger systems with the design objectives of balancing the mitigation on harmful interstory drift and floor acceleration (ζ = 0.5). Figure 14 shows the objectives of the maximum normalized harmful drift ratio and floor absolute acceleration of three damped outrigger systems against maximum allowable negative stiffness ratio, respectively. Again, the N&C and C&N systems, particularly the N&C system, have been shown to be more effective in reducing harmful interstory drift and floor acceleration within limited amount of negative stiffness ratio, while the DN system could have the best performance with sufficient amount of negative stiffness ratio.Figure 14
Influences of total normalized negative stiffness ratio (−η) at weight ζ = 0.5. (a) Harmful drift ratio. (b) Acceleration.
(a)(b)The above investigations suggest that the proposed combined NSDO and CDO damped outrigger system could provide valuable design flexibilities with damped outrigger configurations tailored for various objectives: (1) with adequate negative stiffness devices that do not cause the system to become unstable, the DN system would be the best design option by bringing down structure responses; (2) with limited amount negative stiffness devices, the N&C system would be the optimal configuration for its best utilization of devices, followed by the C&N and DN systems; and (3) when the available amount of negative stiffness devices lies in between, there might be a trade-offbetween the DN and N&C systems depending on the weightsassociated with harmful interstory drift and floor absolute acceleration.


## 5. Comparisons of the Efficiency of Different Outrigger Systems

To show the efficiency of the proposed stochastic optimization method for designing different outrigger systems in reducing earthquake-induced vibration as well as the potentials of the proposed combined NDO and CDO systems, the optimal designs of such systems obtained from last section are further examined under real earthquake records. Specifically, the optimal configurations of four damped outrigger systems with the weight equal to 0.5 and with total maximum absolute negative stiffness ratio being 0.1 (0.2 for the original DN system) are selected as examples for further evaluating their performance regarding time history response and energy distribution under typical real earthquake records as well as near-fault earthquake response envelopes. Note that the weight being 0.5 means that the structural acceleration is assumed to be as important as harmful drift ratio. The detailed parameters based on dual-response criterion are listed in Table2, where DN (0.1) represents the results of the DN system with the maximum total absolute negative stiffness ratio being 0.1, which is the same level as the C&N and N&C systems in the table. Such amount of limited negative stiffness ratio is chosen to show the competitiveness of the C&N and N&C systems, which could outperform the DN system with smaller negative stiffness ratio (see Figure 14), i.e., the proposed C&N and N&C systems are competitive with the DN system given the same negative stiffness ratio even in the worst scenario, and consequently the potentials of applying the C&N and N&C systems are evidenced.Table 2
Detailed parameters based on dual-response criterion.
ParameterC&NN&CDNDN (0.1)η−0.09822−0.09995−0.09379−0.04945c0.009930.009960.007910.00989N040423542


### 5.1. Time History Response

In this section, the obtained optimal designs of different outrigger systems are further examined for structures under El Centro and Kobe records. Figure15 shows the time history records of El Centro and Kobe on the left column and the corresponding power spectrum densities on the right with the peak ground acceleration adjusted to 0.35 g.Figure 15
Acceleration and frequency-amplitude relationships for earthquake records. (a) El Centro. (b) Kobe.
(a)(b)The reduction ratios of the different outrigger systems using stochastic optimization approach in comparison with the BS under typical earthquake records are summarized in Table3, and the corresponding time history curves of structural bottom story drift ratio and top peak acceleration are presented in Figures 16 and 17. From both the table and figures, all four damped outrigger systems could reduce the maximum harmful drift and floor acceleration significantly, with an average reduction of 45%–55% in the former and 32%–41% in the latter in comparison with the BS. Compared to other outrigger systems, the DN system with adequate negative stiffness devices has the best performance both in harmful interstory drift ratio and floor absolute acceleration. Additionally, while the both mixed damped outrigger systems have compatible reduction effect in terms of acceleration, the N&C system performs slightly better than the DN (0.1) and C&N systems with the same amount of negative stiffness devices, where the variation of vibration reduction is due to the variation of frequency content of the specific earthquake records.Table 3
Control results of structural RMS response subjected to different seismic excitation.
Outrigger systems (OS)Reduction ratioλ (%)Bottom interstory drift RMSTop acceleration RMSEl CentroKobeAverageEl CentroKobeAverageC&N49.3940.4844.9429.0236.4132.72N&C55.8443.5449.6931.1235.8733.49DN62.0348.1255.0836.5945.3540.97DN (0.1)52.5542.1647.3630.0634.8132.44λ is the percentage reduction of the RMS response of different outrigger systems compared to the BS, i.e., λ=BSResponse−OSResponse/BSResponse, where OS refers to four outrigger systems in the table.Figure 16
Comparison of time history curves of bottom story drift ratio of different outrigger systems under typical earthquake. (a) El Centro. (b) Kobe.
(a)(b)Figure 17
Comparison of time history curves of structural top acceleration of different outrigger systems under typical earthquake. (a) El Centro. (b) Kobe.
(a)(b)


### 5.2. Seismic Energy Distribution

As the damped outrigger systems are using the dampers as energy dissipation devices, it is natural to analyze the energy distribution of the systems. From the energy balance perspective, the seismic input energyEI to the systems when the main structures remain linear is transformed into kinetic energy EK, strain energy ES, and the dissipated energy both by inherent damping EC and the installed dampers ED, i.e., EK+ES+EC+ED=EI. As opposed to the typical analysis only focusing on the energy dissipated by dampers, the dampers installed affect not only the dissipated energy, but also the input energy, or to put it this way, the purpose of installing dampers is to protect the main structure, i.e., the actual concern of seismic input energy influencing the systems’ behavior is EK+ES. Therefore, the quantity EI−EC−ED is a more meaningful performance index rather than ED itself.Again, the significant seismic performance of four damped outrigger systems is evidenced by Figure18, which shows how the energy quantities EI−EC−ED of different outrigger systems evolve during earthquakes, and by Table 4 that summarizes the corresponding maximum energy that could damage the target main structures. Compared to BS structure, the DN system cut the main structure energy input by around 39% and 27% under El Centro and Kobe records, respectively, as opposed to 36% and 25% for the N&C system, respectively, which has slightly better performance than the DN (0.1) and C&N systems.Figure 18
Energy dissipation with different outrigger systems under typical earthquake records. (a) El Centro. (b) Kobe.
(a)(b)Table 4
Maximum energy dissipation under selected earthquake records.
Outrigger systemsEl CentroKobeEI − ED − EC (e6J)EI − ED − EC (e6J)BS3.59356.6775C&N2.45045.0661N&C2.29624.9841DN2.20114.8466DN (0.1)2.33195.0219


### 5.3. Seismic Response Envelopes

To evaluate the response envelopes of different outrigger systems, 28 near-fault earthquake records are selected [59]. Figure 19 presents the response spectra of these selected records. Comparisons of mean response envelopes by different outrigger systems under selected earthquake records are given in Figure 20. Compared to BS structure, C&N, N&C, DN, and DN (0.1) can reduce the average value of the structural maximum harmful story drift ratio by 19.26%, 20.34%, 25.89%, and 19.63% and absolute acceleration by 20.27%, 22.56%, 30.61%, and 20.48%, respectively. Such result is consistent with the previous analysis, i.e., the D&N system has the best performance, followed by the N&C, DN (0.1), and C&N systems which have similar performance levels.Figure 19
Response spectra of the selected seismic records.Figure 20
Comparisons of mean response envelopes by different outrigger systems under selected earthquake records. (a) Harmful interstory ratio. (b) Peak acceleration.
(a)(b)


## 5.1. Time History Response

In this section, the obtained optimal designs of different outrigger systems are further examined for structures under El Centro and Kobe records. Figure15 shows the time history records of El Centro and Kobe on the left column and the corresponding power spectrum densities on the right with the peak ground acceleration adjusted to 0.35 g.Figure 15
Acceleration and frequency-amplitude relationships for earthquake records. (a) El Centro. (b) Kobe.
(a)(b)The reduction ratios of the different outrigger systems using stochastic optimization approach in comparison with the BS under typical earthquake records are summarized in Table3, and the corresponding time history curves of structural bottom story drift ratio and top peak acceleration are presented in Figures 16 and 17. From both the table and figures, all four damped outrigger systems could reduce the maximum harmful drift and floor acceleration significantly, with an average reduction of 45%–55% in the former and 32%–41% in the latter in comparison with the BS. Compared to other outrigger systems, the DN system with adequate negative stiffness devices has the best performance both in harmful interstory drift ratio and floor absolute acceleration. Additionally, while the both mixed damped outrigger systems have compatible reduction effect in terms of acceleration, the N&C system performs slightly better than the DN (0.1) and C&N systems with the same amount of negative stiffness devices, where the variation of vibration reduction is due to the variation of frequency content of the specific earthquake records.Table 3
Control results of structural RMS response subjected to different seismic excitation.
Outrigger systems (OS)Reduction ratioλ (%)Bottom interstory drift RMSTop acceleration RMSEl CentroKobeAverageEl CentroKobeAverageC&N49.3940.4844.9429.0236.4132.72N&C55.8443.5449.6931.1235.8733.49DN62.0348.1255.0836.5945.3540.97DN (0.1)52.5542.1647.3630.0634.8132.44λ is the percentage reduction of the RMS response of different outrigger systems compared to the BS, i.e., λ=BSResponse−OSResponse/BSResponse, where OS refers to four outrigger systems in the table.Figure 16
Comparison of time history curves of bottom story drift ratio of different outrigger systems under typical earthquake. (a) El Centro. (b) Kobe.
(a)(b)Figure 17
Comparison of time history curves of structural top acceleration of different outrigger systems under typical earthquake. (a) El Centro. (b) Kobe.
(a)(b)


## 5.2. Seismic Energy Distribution

As the damped outrigger systems are using the dampers as energy dissipation devices, it is natural to analyze the energy distribution of the systems. From the energy balance perspective, the seismic input energyEI to the systems when the main structures remain linear is transformed into kinetic energy EK, strain energy ES, and the dissipated energy both by inherent damping EC and the installed dampers ED, i.e., EK+ES+EC+ED=EI. As opposed to the typical analysis only focusing on the energy dissipated by dampers, the dampers installed affect not only the dissipated energy, but also the input energy, or to put it this way, the purpose of installing dampers is to protect the main structure, i.e., the actual concern of seismic input energy influencing the systems’ behavior is EK+ES. Therefore, the quantity EI−EC−ED is a more meaningful performance index rather than ED itself.Again, the significant seismic performance of four damped outrigger systems is evidenced by Figure18, which shows how the energy quantities EI−EC−ED of different outrigger systems evolve during earthquakes, and by Table 4 that summarizes the corresponding maximum energy that could damage the target main structures. Compared to BS structure, the DN system cut the main structure energy input by around 39% and 27% under El Centro and Kobe records, respectively, as opposed to 36% and 25% for the N&C system, respectively, which has slightly better performance than the DN (0.1) and C&N systems.Figure 18
Energy dissipation with different outrigger systems under typical earthquake records. (a) El Centro. (b) Kobe.
(a)(b)Table 4
Maximum energy dissipation under selected earthquake records.
Outrigger systemsEl CentroKobeEI − ED − EC (e6J)EI − ED − EC (e6J)BS3.59356.6775C&N2.45045.0661N&C2.29624.9841DN2.20114.8466DN (0.1)2.33195.0219


## 5.3. Seismic Response Envelopes

To evaluate the response envelopes of different outrigger systems, 28 near-fault earthquake records are selected [59]. Figure 19 presents the response spectra of these selected records. Comparisons of mean response envelopes by different outrigger systems under selected earthquake records are given in Figure 20. Compared to BS structure, C&N, N&C, DN, and DN (0.1) can reduce the average value of the structural maximum harmful story drift ratio by 19.26%, 20.34%, 25.89%, and 19.63% and absolute acceleration by 20.27%, 22.56%, 30.61%, and 20.48%, respectively. Such result is consistent with the previous analysis, i.e., the D&N system has the best performance, followed by the N&C, DN (0.1), and C&N systems which have similar performance levels.Figure 19
Response spectra of the selected seismic records.Figure 20
Comparisons of mean response envelopes by different outrigger systems under selected earthquake records. (a) Harmful interstory ratio. (b) Peak acceleration.
(a)(b)


## 6. Conclusions

This paper proposed a stochastic optimization method to investigate optimal configurations of combined negative stiffness damped outriggers (NSDOs) and conventional damped outriggers (CDOs) subjected to nonstationary stochastic seismic excitation. The simplified analysis model of various combinations of damped outriggers was developed, and the state-space representation of outrigger systems was formulated. The nonstationary seismic excitation was modeled as a uniformly modulated stationary Gaussian process with a time-modulating function following Clough–Penzien spectrum. Combining equations of motions of outrigger systems and seismic excitation gave rise to the augmented state-space representation of structure-damper-excitation and subsequently the differential Lyapunov equation. The optimal designs of damped outriggers were defined via the solution of the differential Lyapunov equation. The multiobjective optimization with Pareto optimal fronts was adopted to deal with conflicting objectives. The sensitivity of negative stiffness ratio on the design objectives of reducing maximum harmful interstory drift and floor absolute acceleration was studied. The efficacy of the proposed method was demonstrated through a tall building designed with various damped outriggers, and the optimal designs were further examined under real earthquake records. These investigations showed the following:(1)
The proposed multiobjective optimization method incorporated uniformly modulated Clough–Penzien filter to model nonstationary stochastic seismic excitation and Pareto optimal fronts to deal with conflicting objectives, whereby the designs of the combined negative stiffness damped outrigger and conventional damped outrigger systems were optimized. The efficacy of the proposed method was verified through an illustrated example.(2)
The proposed method was applied to optimally design the proposed combined NSDO and CDO systems subjected to nonstationary seismic excitation. With this method, the obtained optimal parameters of dampers and outrigger arrangement for such systems could reduce around 36%–50% in maximum RMS values of harmful interstory drift ratio and over 30% in peak RMS values of floor absolute acceleration compared to the bare structure (BS). Provided with sufficient negative devices, the double negative stiffness outrigger (DN) system performed the best, followed by the two combined systems, the N&C and C&N systemswithin which the NSDO is correspondingly placed above and below the CDO.(3)
Stochastic optimizations of three damped outrigger systems considering nonstationary and stationary seismic excitations were compared, and the stationary-based optimization was further evaluated under nonstationary excitation. Although the stationary assumption of seismic excitation could lead to overestimation of structural responses with respect to both harmful interstory drift and floor acceleration, both the optimal designs of three damped outrigger systems between stationary and nonstationary earthquakesand the deviations in optimal objectivevalues using the stationary and nonstatoinary earthquakes-based optimal designs were less than 2%. Such findings justify that the less time-consumingstationary-based seismic optimization is adequate for engineering design practice.(4)
The sensitivity analysis was conducted by analyzing the stochastic optimization results with respect to the maximum allowable negative stiffness ratio. The DN system would be the best design option provided with adequate negative stiffness devices that do not cause the system to become unstable, while the N&C system would be the optimal configuration for its best utilization of devices with limited amount negative stiffness devices. When the available amount of negative stiffness devices lies in between, there might be a trade-offbetween the N&C and DN systems depending on the weights associatedwithharmful interstory drift and floor absolute acceleration. Therefore, the proposed combined NSDO and CDO damped outrigger systems could provide valuable design flexibilities with damped outrigger configurations tailored for various objectives.(5)
The optimal designs of damped outrigger systems using the proposed optimization method were further examined under typical as well as near-fault earthquake records. With emphasis both on harmful drift and floor acceleration, the optimal designs of four damped outrigger systems were evaluated regarding time history dynamic response, energy distribution, and response envelopes. Compared to BS, all damped outrigger systems on average could reduce the harmful interstory drift by 45%–55% and floor absolute acceleration by 32%–41% as well as reduce the energy input to the main structures by 25%–39% under El Centro and Kobe earthquake records; however, such reduction effect in the case of near-fault earthquakes decreased to 19%–26% and 20%–31% in terms of average response envelopes, respectively.



---

*Source: 4024741-2023-02-08.xml*