Numerical Investigation and Design of Reinforced Concrete Shear Wall Equipped with Tuned Liquid Multiple Columns Dampers

The tuned liquid multiple column damper (TLMCD) is a variation of the tuned liquid column damper (TLCD) that includes multiple vertical columns. A new damping system that embeds TLMCDs within reinforced concrete shear wall systems, termed tuned liquid wall damper (TLWD), is proposed, augmenting the traditional structural component with energy dissipation capabilities. The objective of this study is to assess energy mitigation and strength trade-offs in designing TLWDs and demonstrating the promise of TLWD systems in tall buildings through vertically distributed applications. This is done by investigating the performance of the proposed TLWD through the finite element model (FEM) of a simplified representation of a 42-story building equipped with the multifunctional component. A strength model for the TLWD is developed to empower faster performance evaluation on more complex models. Results from the FEM are used to validate the strength model and show that the model could be used conservatively in assessing strength performance. Design considerations are discussed based on the simplified representation. In particular, to improve mitigation performance while maintaining strength, it is found that a single-layer arrangement of the vertical columns is preferred, while distributing the inertia among a higher number of smaller columns. The proposed TLWD is numerically evaluated on a more realistic system consisting of a multi-degrees-of-freedom representation of the 42-story building under stochastic wind excitation. Simulation results demonstrate that the TLWD, used in a vertically distributed configuration through the building, could be used to mitigate vibrations, outperforming a traditional TLCD system with geometric constraints under 20 design wind realization. Results from the numerical simulations also confirmed the design considerations established through the simplified representation. Disciplines Civil Engineering | Geotechnical Engineering Comments This article is published as Wang, Zhe, Liang Cao, Filippo Ubertini, and Simon Laflamme. "Numerical Investigation and Design of Reinforced Concrete Shear Wall Equipped with Tuned Liquid Multiple Columns Dampers." Shock and Vibration 2021 (2021): 6610811. DOI: 10.1155/2021/6610811. Posted with permission. Creative Commons License This work is licensed under a Creative Commons Attribution 4.0 License. Authors Zhe Wang, Liang Cao, Filippo Ubertini, and Simon Laflamme This article is available at Iowa State University Digital Repository: https://lib.dr.iastate.edu/ccee_pubs/286 Research Article Numerical Investigation and Design of Reinforced Concrete Shear Wall Equipped with Tuned Liquid Multiple Columns Dampers Zhe Wang , Liang Cao , Filippo Ubertini , and Simon Laflamme 1,4 Department of Civil, Construction, and Environmental Engineering, Iowa State University, Ames, IA, USA ATLSS Engineering Research Center, Lehigh University, Bethlehem 18015, PA, USA Department of Civil and Environmental Engineering, University of Perugia, Perugia, Italy Department of Electrical and Computer Engineering, Iowa State University, Ames, IA, USA Correspondence should be addressed to Zhe Wang; zhew@iastate.edu Received 13 November 2020; Revised 12 January 2021; Accepted 16 February 2021; Published 28 February 2021 Academic Editor: Salvatore Caddemi Copyright © 2021 Zhe Wang et al. *is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. *e tuned liquid multiple column damper (TLMCD) is a variation of the tuned liquid column damper (TLCD) that includes multiple vertical columns. A new damping system that embeds TLMCDs within reinforced concrete shear wall systems, termed tuned liquid wall damper (TLWD), is proposed, augmenting the traditional structural component with energy dissipation capabilities. *e objective of this study is to assess energy mitigation and strength trade-offs in designing TLWDs and demonstrating the promise of TLWD systems in tall buildings through vertically distributed applications. *is is done by investigating the performance of the proposed TLWD through the finite element model (FEM) of a simplified representation of a 42-story building equipped with the multifunctional component. A strengthmodel for the TLWD is developed to empower faster performance evaluation onmore complex models. Results from the FEM are used to validate the strength model and show that the model could be used conservatively in assessing strength performance. Design considerations are discussed based on the simplified representation. In particular, to improve mitigation performance while maintaining strength, it is found that a single-layer arrangement of the vertical columns is preferred, while distributing the inertia among a higher number of smaller columns. *e proposed TLWD is numerically evaluated on a more realistic system consisting of a multi-degrees-of-freedom representation of the 42-story building under stochastic wind excitation. Simulation results demonstrate that the TLWD, used in a vertically distributed configuration through the building, could be used to mitigate vibrations, outperforming a traditional TLCD system with geometric constraints under 20 design wind realization. Results from the numerical simulations also confirmed the design considerations established through the simplified representation.


Introduction
High-strength and lightweight materials combined with advanced construction techniques have empowered the design of taller and more flexible high-rise structures, yet making them vulnerable to excessive motion that may cause discomfort during daily operations and safety concerns against natural hazards [1]. A solution is the integration of supplemental damping strategies, including passive, semiactive, and active systems [2]. Amongst these strategies, passive systems are now widely accepted by the civil engineering community due to their low cost and known long-term mechanical robustness and reliability.
Of interest to this paper is the use of tuned mass dampers (TMDs), a vibration mitigation system that dissipates energy through inertia [3]. e tuned liquid column damper (TLCD) is a variation of TMD proposed by Sakai et al. [4,5] that consists of a partially water-filled U-shaped tank and an internal orifice to generate hydrodynamic head loss in addition to the internal surface friction while liquid sloshing occurs [6]. e technology is considered as an attractive alternative due to its ease of installation, low maintenance requirements, cost-effectiveness, and high mechanical robustness [7,8]. In the last decades, the concept of TLCD was well investigated in terms of design optimization [9,10], also with applications to multihazard mitigation [11,12]. However, TLCDs require significant space for implementation [12] and are only effective over a limited frequency range. Cao et al. studied a variation of the TLCD termed tuned liquid multiple columns damper (TLMCD), which consists of multiple vertical columns joined with a horizontal tube [13]. e TLMCD is capable of tuning at multiple frequencies and providing enhanced performance in a constrained geometry.
Recently, the authors proposed to implement a TLMCD system within a reinforced concrete (RC) shear wall, termed tuned liquid wall damper (TLWD), with the intent to improve vibration mitigation capabilities of shear wall components [13]. e advantage of the TLWD is in providing a geometrically restricted and distributed location for the integration of TLMCD systems while enabling multifrequency tuning. Others have proposed similar systems. Ye et al. studied cast-in situ hollow floor slabs equipped with an internal tuned liquid damper, augmenting the structural damping ratio by approximately 2% [14]. Matia and Gat investigated an elastic beam embedded with a fluid-filled parallel-channel network, and simulation results demonstrated that the fluid-solid interaction reduced structural motion [15]. To the best knowledge of the authors, none have studied the integration of TLCD systems within RC shear wall components.
Prior work from the authors in [13] was focused on the development and validation of an analytical model for TLMCD systems. In this paper, the investigation extends to the TLWD system, whereas perforated openings within RC shear walls are used to integrate a TLMCD. Increasing the height and length of perforated openings may increase structural damping by increasing the size of the internal TLMCD, and it may also significantly reduce the strength capacity and lateral motion performance of the RC shear wall, yet with less significant adverse effects on ultimate flexural capacity [16][17][18]. Here, both strength and mitigation functionalities are investigated within given geometric constraints to satisfy both strength and motion performance requirements, as well as the trade-off between strength and energy dissipation is studied. e rest of the paper is organized as follows. Section 2 describes the analytical and numerical models used in conducting the study. is includes the proposed simplified strength model. Section 3 conducts parametric studies for a simplified structure equipped with a TLWD. Section 4 validates the proposed strength model and, further, the parametric evaluation by conducting additional simulation using the validated model. Results from both Sections 3 and 4 are used to establish design considerations for implementing a TLWD system. Section 5 verifies the findings by demonstrating the proposed TLWD on a 42-story building. Section 6 concludes the paper.

Models
is section presents the models used in conducting the numerical study. It starts by a presentation of the TLMCD analytical model developed in prior work [13], followed by a simplified analytical model to evaluate strength of a TLWD. After that the finite element modeling methodology is described, followed by that of the numerical simulations in MATLAB.

TLMCD Model.
A TLMCD consists of multiple vertical columns jointed with a horizontal tube equipped with multiple orifices located in between adjacent vertical columns. Figure 1(a) is the schematic of a TLMCD and Figure 1(b) illustrates a TLWD system. In Figure 1(a), x .. g is the acceleration transmitted from the floor, x i is the liquid surface displacement in the ith column, A is the cross-section area of vertical column, h is the initial vertical liquid surface height (equal for every column), l i is the horizontal center-to-center distance between the ith and i + 1th columns, v is cross-section area ratio of vertical column to the horizontal tube, and ψ i is the ith orifice blocking ratio. A nonlinear dynamic model for the TLMCD can be derived using Lagrange equations [13]. Starting with d dt where _ x i is the liquid velocity in the ith column, t is time, and Q oi , Q fi , and Q ei are orifice damping force, friction force between the liquid and inner surface of columns, and liquid inertia force, respectively, the kinematic energy T and potential energy V are taken as where ρ l is the liquid density. e analytical model is based on the following assumptions: (1) all vertical columns are identical in geometry, (2) liquid compression and liquid-air interface diffusion are negligible, resulting in a constant total liquid volume during vibrations, and (3) the characteristic dimensions of the columns' cross sections are considerably smaller than the total length of the tube, resulting in uniform motion of the liquid surface over the cross sections. An Ncolumn TLMCD can be treated as N-1 degree of freedoms (DOFs) by allowing the displacement of the last column be dependent on the other columns' displacements using the assumed kinematic constraint x N � − N−1 i�1 x i . It should be noted that the analytical model becomes that of a conventional TLCD for N � 2. e damping force Q oi is assumed to be proportional to the square of the liquid velocity _ x 2 l [9]. e energy dissipated by the orifice damping force W o can be written as

Shock and Vibration
Q oi on the ith DOF and selected head loss coefficient model η are written as [19] In addition to the head loss effect caused by the orifice blocks, friction resistance between the liquid and inner tube surface provokes a friction head loss effect. For mathematical tractability, the friction force is assumed to be proportional to the liquid velocity _ x 2 l . An expression for the energy dissipated by the friction force W f can be written as where μ is the head loss coefficient due to the friction resistance. e mathematical expression for the friction force Q fi over a small unit of time for the ith DOF is expressed as e inertial force Q ei is induced by the lateral motion of the building due to an external excitation. Since the liquid mass remains constant during vibrations, the inertial force Q ei on the ith DOF is solely given by the contribution from the liquid motion in the horizontal tube. e energy dissipated by inertial force W e can be written as Figure 1: Schematic of (a) TLMCD [13] and (b) TLWD system consisting of a shear wall with embedded TLMCDs.  [20] are used to calculate the strength of TLWD with a given TLMCD embedment profile. e strength model is built upon considering only the void caused by the vertical columns. A special reinforcement scheme would be required to cope with the void caused by the horizontal tube, which is left to future work. Lateral stiffness k, nominal shear strength V n , nominal flexural strength M n , and nominal axial strength P n are considered as strength objectives. e moment of inertia of TLWD along the in-plane direction can be written where b w is the width of the wall section, l w is the wall length, A i is the area of ith vertical column, y i is the distance of the neutral axis to the center of ith circular opening, N is the number of vertical columns, and r is the radius of openings. Stiffness value k for the shear wall with respect to the section installed above is inversely proportional to its deflection when subjected to a unit point load applied at the top. e total deflection of the TLWD is determined from the sum of shear deflection Δ v and moment deflection Δ m , with k defined as [21] k Quantities Δ v and Δ m for a TLWD subjected to a unit horizontal load F can be calculated as , for a fixed wall for a cantilever wall, where h w is height of wall, E c is Young's modulus, G is shear modulus, v is Poisson's ratio, f c ′ is compressive strength, and A w � A t -NA is the cross-section area of the TLWD taken by removing the cross-section area of the vertical columns NA from the cross-section area of the wall geometry A t . e shear strength V n of a TLWD consists of the summation of the shear strength from concrete V c and that from transverse reinforcements V s : Shear strength values are modified based on the simplified method from ACI-318 Section 11.5.4.5 using the reduction factor α: with the expressions for V c and V s written as where d is the effective depth of the cross section, taken as 0.8 l w from the strain compatibility analysis [20]. A v is the crosssectional area of transverse reinforcement, f y is the yield strength of the steel reinforcement, and s is the center-tocenter spacing of transverse reinforcement. Value P n is governed by two failure cases, namely, crushing of concrete and buckling of steel reinforcement. From ACI 318-14, P n is taken as the sum of axial load capacity of concrete and steel reinforcement, with where A s is the cross-sectional area of longitudinal reinforcement. Both the effects of axial load and bending moment are considered in the axial-bending interaction M n illustrated for an 8-column TLWD in Figure 2, assuming an equivalent rectangular compressive stress distribution for the concrete and an elasto-plastic constitutive law for steel reinforcement. In Figure 2, ε cu is the ultimate compressive strain of concrete, ε s is the strain of steel reinforcement, ε c is the strain of concrete, ε y is the yield strain of steel reinforcement, β is the ratio of the depth of the rectangular stress block to the neutral axis, C c is the compression force from concrete, C s is the compression force from steel reinforcement, and T s is the tension force from steel reinforcement. e value for M n given an applied axial load N u can be calculated using equations (17)-(21) based on the strain distribution under force equilibrium: where ε i c and ε i s are, respectively, the compressive and tensile strain of the reinforcement bar at the ith depth with ε s ≤ ε y , 4 Shock and Vibration d i c and d i s are, respectively, the depth of the ith reinforcement bar in compression and tension, E s is the Young modulus of steel, A i s is the cross-section area of reinforcement bar at the ith depth, and A cb is the area of compression block. is TLWD strength model will be validated in Section 4 using data produced by numerical simulations.

Finite Element Model.
e finite element model of a TLWD is constructed to validate the TLWD strength model. e concrete damage plasticity model (CDPM) in ABAQUS is used to simulate the nominal strength of TLWDs. e CDPM considers both tensile cracking and compressive crushing failure mechanisms based on the models proposed by Lubliner et al. [22] and Lee and Fevens [23]. In the CDPM, the five following parameters characterize the stressstrain behavior and damage of inelastic and cracking strain. e dilation angle ψ defines the angle of the inclination surface towards the hydrostatic axis. e eccentricity ε defines the shape of the hyperbola and can be calculated as the ratio of tensile strength to compressive strength [24]. Parameter f b0 /f c0 is the ratio of compressive strength in biaxial direction f b0 to uniaxial direction f c0 , recommended as 1.16 in [25]. Parameter K c defines the shape of the cross section at failure in the deviatoric plane, recommended as 0.67 in [25]. e viscosity parameter μ is used to reduce the time interval to prevent divergence. For nonviscoelastic materials such as concrete, the value should be as small as possible [24]. Parameters used in characterizing the CDPM are listed in Table 1.
e uniaxial compressive stress-strain relationship is characterized following Hsu and Hsu model [26]. e stressstrain relationship, plotted in Figure 3(a), solely relies on f c ′ , with where σ c is the compressive stress, ε l is the compressive strain at the end of linear phase, and ε 0 is the compressive strain at peak stress. e tensile stress-strain relationship of concrete is characterized using Nayal and Rasheed tension stiffening model [27] modified by Wahalathantri et al. [28]. e tensile behavior, plotted in Figure 3(b), is linear until reaching the average splitting strength f ct , corresponding to the cracking strain ε cr . After, the stress decreases at various rates after reaching 0.77 f ct , corresponding to 1.25 ε cr , 0.45 f ct , corresponding to 4 ε cr , and 0.1 f ct , corresponding to 8.7 ε cr , the cracking strain ε cr and average splitting strength f ct are written as e constitutive model of steel reinforcement bars is assumed to be bilinear elastic-perfectly plastic with identical stress-strain behavior in tension and compression, as shown in Figure 3(c). f y , ε y , and E s are the yield strength, yield strain, and Young's modulus of steel reinforcement accordingly.
e bond between reinforcement steel and concrete is assumed to be perfect: Concrete and steel reinforcement are modeled using linear 3D 8-node solid and linear 2D 2-node truss elements. e embedded region constraint method is employed, where concrete and steel reinforcement is modeled as host and embedded regions, respectively. e embedded TLMCDs are modeled as vertical voids occupied by liquid. A mesh sensitivity study is conducted to achieve convergence. e element size of concrete is determined as 50 × 80 × 110 mm 3 (width × length × height) and the meshing length of steel as 150 mm, with a total of approximately 85,000 elements.

Numerical Simulation Model.
After the TLWD validation using the finite element model, the TLWD will be numerically simulated in the MATLAB environment. e equation of motion for a structure equipped with a TLWD can be written as where M, C, and K are the mass, damping, and stiffness system matrices, respectively, x is the displacement vector, the dot denotes a time derivative, F d is the damping forces generated by TLWD, F e is the excitation force vector, and E d and E e are the associated force location matrices. e statespace formulation of equation (25) is written as 6 Shock and Vibration where X is the state vector, A is the state matrix, and B d and B e are the input vector. From [13], an N-column symmetric TLWD can be simplified into a linear system having N/2 equal pairwise natural frequencies and can be tuned through adjusting column spacings. e system stiffness matrix K ∈ R ((N/2)+1)×((N/2)+1) , mass matrix M ∈ R ((N/2)+1)×((N/2)+1) , and damping matrix C ∈ R ((N/2)+1)×((N/2)+1) of the coupled SDOF structure can be written as where m d is the liquid mass of the TLMCD and c s is the structural damping coefficients. It should be noted that the linear damping coefficients of TLMCDs are taken equal to 0 in the linear damping matrix C because the TLMCD damping force is proportional to the square of velocity (equations (4) and (7)). e state vector X � x _ x T ∈ Shock and Vibration R (N+2) is defined for the coupled SDOF system. An explicit discrete-time formulation is used to solve the state-space equation: where e is the exponential function and I the identity matrix.

Performance Investigation: Single Wall Section
In this section, the motion and strength performance of the TLWD are preliminarily investigated on a simplified SDOF system and a single RC wall geometry, respectively. e investigation starts by a parametric study on the TLMCD configuration for motion engineering. It is followed by an investigation of embedment strategies to optimize performance versus both motion and strength by comparing the numerical and FEM results. After that, results from the finite element analysis are used to validate the simplified analytical model for strength. en, the validated model is used to further the study on trade-offs between motion and strength performance.
e SDOF system is based on a representation of a 42story residential building (Building 1A) [29] equipped with a TLWD. Note that the behavior of the SDOF system does not fully represent that of the multi-degree-of-freedom system (MDOF). e structural mass m s and fundamental frequency ω s are obtained from the literature [29]. e structural damping coefficients are obtained assuming 5% natural damping ratio. e SDOF system is schematized in Figure 4 and model parameters are summarized in Table 2.
e vertical liquid columns in the attached TLWD are equally spaced, and the first vibration mode is tuned to the fundamental frequency of the structure. e TLWD represents the assembly of numerous TLMCDs occupying an equivalent cross-section A eq . e strength evaluation assumes the TLMCDs are embedded in an interstory wall at the ground level from Building 1A [29]. e wall, illustrated in Figure 5, has a height h w � 4,167 mm, length l w � 5,334 mm, and width b w � 610 mm. It is reinforced using 1.39% longitudinal and 0.58% transverse reinforcement ratio. e concrete compressive strength f c ′ and steel reinforcement yield strength f y are taken as 55.16 MPa and 413.69 MPa, respectively.
Transfer functions H 1 (ω) and H 2 (ω) are selected as the main motion performance objectives. ey correspond to the amplification of displacement and acceleration responses in the frequency domain, respectively. ey are evaluated numerically from the steady-state response using Nominal shear V n , flexural M n , and axial strength P n are selected as the main strength objectives, simulated through the displacement-controlled pushover analysis at the ultimate stage. For the same wall cross section, slender wall and short wall geometries are used to reach the flexural and shear failure modes separately. Among these, M n is determined based on balanced axial-moment interaction from strain compatibility analysis with approximated axial load ratio ρ axial � 0.36.

Motion Performance.
e performance of the TLMCD is first investigated on single-layered configurations (i.e., all columns organized along the same horizontal line in Figure 5), with the overall geometry constrained to the shear wall's geometry. e parametric studies on the number of columns N and their diameter D are conducted with D expressed as a function of wall thickness b w . A harmonic excitation of frequency ranging from 0.8ω s to 1.2ω s is used.
Tuning frequency ratio f and head loss coefficient η are adjusted to obtain the optimal transfer functions H 1 (ω)| opt and H 2 (ω)| opt . Both H 1 (ω)| opt and H 2 (ω)| opt over various values for D and N are plotted in Figures 6 and 7, respectively. Tables 3 and 4 list the corresponding mass ratio m of the equivalent TLMCD to the structural mass, the optimal tuning frequency f| opt , the optimal head loss coefficient η| opt     Results show that increasing the columns' diameters or numbers has a positive effect on mitigation capabilities, as it would be expected due to the increase in liquid mass. e lower column diameter cases reduce the maximum total damping forces but result in a larger liquid displacement x l | max in vertical columns, as shown in Table 3. Similarly, the amplification of energy dissipation is observed from results listed in Table 4 when the column number increases from 2 to 8.
is is mainly because the increased mass in the fundamental mode contributes a higher total damping force.
Overall, results confirm that the mitigation performance of the TLWD can be ameliorated by adding liquid mass, but there is a physical limit in doing so because the increased liquid mass yields a decrease in the concrete cross-section area, which will yield a loss in strength. Section 3.2 studies TLMCD-embedded strategies to optimize performance versus both motion and strength.

Embedment Strategies.
A strength performance investigation is conducted under different embedded TLMCD configurations designed to yield similar mitigation performance under various optimal TLMCD configurations. First, single-layer TLMCD configurations are studied. ese configurations, designed to arbitrarily maintain H 1 | max and H 2 | max constant at 6.7, along with the shear strength V n , axial-bending strength M n , axial strength P n , and stiffness k of the wall obtained from the finite element analysis, are listed in Table 5. Results reveal that achieving mitigation performance through a distributed column strategy increases the shear strength and the axial-bending strength, but decreases the axial strength and lateral stiffness, yet with a marginally decreasing change in variations. Overall, distributing the TLMCD over 12 columns instead of 6 provoked a decrease in the concrete area A c of 1.5%, with V n and M n increasing approximately by 9.7% and 7.8%, P n and k decreasing about 3.1% and 1%.
Next, the effect of distributing inertia through a different number of layers of tubes on the TLWD strength is investigated. is study is done on the layered configurations illustrated in Figure 8 (one-, two-, and three-layer configurations), with the TLMCD configurations designed to maintain a constant total cross-section area across each configuration and by optimizing their design to arbitrarily maintain H 1 | max and H 2 | max constant at 7.9. Table 6 lists the strength values V n , M n , P n , and k under each configuration. Results reveal that while the axial strength and lateral stiffness remains unchanged due to the constant concrete cross-section area, distributing liquid mass amongst additional layers negatively affect both shear and axial-bending strengths, with the axial-bending strength significantly decreasing (27.46%) between the one-and three-layer configurations.
is can be attributed to the larger area occupied by the tubes over the width of the wall, causing stress concentration at the thinnest concrete cross section around middle section of the wall. is is shown by the FEM investigation in Figure 9. In the numerical analysis, concrete cracks were first observed over the middle sections between layers and then propagated to the outer surface. e failure stage was reached right after the formation of vertical sliding cracks. A similar failure pattern was also observed in the one-layer configuration, but, at higher thresholds, it was consistent with Table 6.
Lastly, the effect of distributing individual TLMCDs over the in-plane direction on mitigation performance is examined. e investigated embedded geometries have a constant number of columns, but the arrangements are divided into n TLMCDs as illustrated in Figure 10, where the concrete strength is approximated as constant across all configurations (i.e., ignoring changes in the horizontal tube) and properties are optimized to minimize the maximum value of the transfer functions. Table 7 lists the results. e examination shows that the utilization of a single-TLMCD arrangement yields significantly higher mitigation capability, attributed to the longer horizontal distance between the opposite columns. Because the length of the TLMCD is constrained by the size of the shear wall, the tuning frequency is only dependent on the ratio of column area to horizontal tube area v. Disjointed TLMCDs have a larger v| opt , resulting in a less combined liquid mass.

Design of TLWD
In this section, results from the performance investigation presented in the previous section are used to validate the analytical TLWD strength model. After, the analytical TLWD strength model is used to further investigate tradeoffs between motion and strength performance when designing a TLWD. Lastly, possible design strategies based on findings are discussed.

TLWD Strength Model Validation.
e analytical model developed in Section 2.2 is validated using FEM data from

Motion-Strength Trade-offs.
e performance of the TLWD versus both motion and strength is investigated under a wide range of configurations using the validated analytical strength model. Under each configuration, the embedded TLMCD is tuned to minimize the maximum transfer function (i.e., keeping both H 1 | max and H 2 | max approximately equal), here assembled under the performance metric H 1,2 | max . From the findings in Section 4.1, the study is limited to single-layer and single-TLMCD arrangements. e axial-bending strength M n is obtained at the balanced axial-bending interaction ρ axial � 0.36. Figure 11 plots the results of the multiconfiguration investigation, where a reduction in H 1,2 | max in Figure 11(a) is a desired feature, while that of the strength in Figures 11(b)-11(e) is not. One can observe an obvious trade-off between motion and strength performance. However, the gain in motion reduction is marginally decreasing while the loss in strength is marginally increasing with N and Db w increasing, showing that motion performance can be substantially increased through design of the TLWD system while minimizing the adverse effects on strength. Shear strength V n is the most sensitive to variations in configurations, with a total variation of 47% in strength amongst all configurations, while other strength components M n , P n , and k only vary maximally by 12%, 24%, and 26% respectively.

Design Considerations.
Results from the parametric investigation presented in Section 4.2, along with the extended investigation conducted using the validated strength model, point to the following considerations in designing a TLWD system. First, mitigation performance is generally achieved by adding liquid mass in the TLWD and implementing TLMCD designs with longer horizontal distances between opposite columns yields better performance. Second, in order to satisfy strength requirements, the TLMCD system should be arranged in a single layer within the wall if possible because distributing liquid mass along the width of the wall has a significant adverse effect on shear and axialbending strength. ird, the use of many smaller columns is advantageous over fewer, larger columns in the constrained TLWD environment with respect to strength, except for axial strength due to the provoked decrease in the concrete cross-section area over similar mitigation performance. Fourth, the proposed analytical strength model could be used to quickly investigate various design configurations, whereas the model will generally yield conservative estimates with respect to strength. Fifth, there is a trivial trade-off between motion and strength performance in designing a TLWD, but altering the geometry of the TLMCD does not affect both motion and strength performance proportionally, therefore pointing towards an equilibrium point depending on the design requirements. Sixth, when tuning the geometry of the TLMCD system, one must pay particular attention to the shear strength, as it is highly sensitive to the  selected configuration and constitute a key design property in the design of a shear wall.

High-Rise Building Example
In this section, the performance of the TLWD is numerically validated on a realistic high-rise building subject to stochastic wind excitations. Motion and strength performance is assessed, with strength analyzed using the proposed analytical model.

Numerical Model.
A 42-story RC residential building (Building 1A) [29] is used for the simulations. It consists of six centrally located core walls along east-west direction surrounded on the perimeter by concrete columns. e walls have constant lengths, but their width varies along the height of the structure. Geometries and material properties and reinforcement details can be found in literature [29]. Here, the east-west direction of the building is simulated as a lumped-mass shear system. e fundamental damping ratio of the building is taken as 2% with stiffness proportional damping. e dynamic properties of the building used in the simulation are tabulated in Table 9. e simulated wind speed consists of two components, the steady mean speed V m and the fluctuating speed V f , due to aerodynamic turbulence: e steady mean speed is determined based on the wind profile power law [30] as follows: where V 0 is the 3-second gust speed obtained from hazard maps in ASCE 7-16 [31], V terrain is the mean velocity for a given terrain, v * and v 0 are the shear velocities of the building site and open terrain, and z * and z 0 are the surface roughness for the building site and open terrain, respectively. A multivariate stochastic Gaussian process with crossspectral density function is used for modeling turbulence [32] as follows: where S k is the Kaimal power spectral density function in along-wind direction, O is the excitation frequency, and z is the height of floors. e coherence function between ith and jth locations in equation (32) is defined as e cross-spectral density matrix, S(Ω), containing cross-spectral density functions at different heights, is decomposed using Cholesky's decomposition: where H(Ω) is a lower triangular matrix with generally complex off-diagonal elements. Once the matrix is decomposed, the stochastic fluctuating wind speed V f (z i , t) can be obtained by the following series: where Φ kl is the random phase ranged from 0 to 2π; the phase θ ik (Ω kl ) and the double-indexing frequency Ω kl is given by where Ω c is an upper cutoff frequency and N Ω is the total number of random frequency points. 14 Shock and Vibration e wind load is converted from wind speed to concentrated forces applied on the building floors [30] using where ρ is the air density, A p is the projected area exposed to the wind pressure, and C D is the drag coefficient of the structure. Across-wind forces and vortex shedding are neglected for simplicity, and it is assumed that the alongwind response of the structure dominates motion. A total of 20 wind realizations are produced taking a suburban region based on occupancy category II (residential building) [31] with 7% probability of exceedance in 50 years. Figure 12 plots a typical wind realization speed V (Figure 12(a)) and associated load F w (Figure 12(b)) time histories occurring at the top floor of the building using the input parameters listed in Table 10.

TLWD Design.
In the following numerical investigations, three optimally tuned TLWD configurations are considered: TLWD1 has a 2-column TLMCD of column cross-section diameter D � 0.5b w , TLWD2 has a 6-column TLMCD of column cross-section diameter D � 0.5b w , and TLWD3 has a 12-column TLMCD of column cross-section diameter D � 0.37b w . TLWD1 is selected to mimic a distributed TLCDs system, TLWD2 is selected to improve on TLWD1 through added inertia, and TLWD3 is selected from Table 5 and to perform similar to TLWD2 for motion mitigation, yielding minimal strength reduction. e simplified strength model is used to calculate the strength reduction in parameters V n , M n , P n , and k caused by the integration of the TLMCDs. In this numerical example, TLWDs are vertically distributed within the top 20 floors (from the 23rd to the 42nd floor inclusively). e shear wall parameters including the wall width b w , longitudinal reinforcing ratio ρ w , and transverse reinforcing ratio ρ t are listed in Table 11 along with the resulting reductions in strength.
A comparison of strength reductions shows that TLWD1 yields a significantly stronger shear wall, except for the axial-bending strength compared with TLWD3. However, the effect of embedding a TLMCD on the axial-bending strength is not substantial, with a maximum reported reduction of 5.1%. Configuration TLWD3 does yield an improvement in strength relative to TLWD2. Nevertheless, the differences in strength reductions between both configurations are all within 3.6%. Overall, adding more column cross-section areas within the shear wall (TLWD1 versus TLWD2) does have an adverse effect on strength, while distributing the cross-section areas with the wall (TLWD2 versus TLWD3) improves on strength, consistent with findings in Section 5.1.

Motion Performance.
For the numerical investigation on motion performance, it is assumed that all TLWDs have identical geometries and dynamic properties, and they are modeled as a single equivalent TLWD at each floor, with a maximum of three TLMCDs per interstory wall. e placement of TLWD is first investigated using TLWD2 configuration and selected because it adopts the geometry of a TLCD (TLWD1) but with added inertia. Four vertical distributions are considered, with the TLWDs installed over the top 5, top 10, top 15, and top 20 floors. Figure 13 plots the fitted normal distributions of the maximum structural responses under the 20 wind realizations. Table 12 lists the average drift μ d and acceleration responses μ a , the standard deviation of the drift σ d and acceleration responses σ a , and the probability of exceeding the acceleration threshold P a . Results show that, for this particular building, increasing the

Parameters
Value number of TLWDs over more floors yields marginally decreasing gains in acceleration mitigation and the utilization of TLWDs does not significantly affect interstory drift. In the case of acceleration control, the use of TLWDs over at least the last ten floors is necessary in ensuring that the structure does not significantly exceed the acceleration threshold of 40 mg, where 40 mg corresponds to an acceptable motion performance for essential structures under rare winds [33]. It can also be noted that performance under drift is always satisfied, even in the uncontrolled case. From these results, arrangement of TLWDs over the top 15 floors is selected to continue the numerical investigation. Mitigation performance for TLWDs1-3 is now assessed against that of a conventional TLCD installed at the top of the building. To enable fair comparison, the TLCD's capacity is equal to that of TLWD1 by constraining its geometry, where its horizontal length is equals to that of one embedded TLMCD, and the area of its vertical columns is equal to the sum of all of the single column areas of each TLMCD. Note that such TLCD geometry may not be practical and is simply used to benchmark performance. Figure 14 plots the fitted normal distributions of the maximum structural responses under the 20 wind realizations, and Figure 15 plots the response profiles using the averaged maximum structural responses. Table 13 lists the average drift μ d and acceleration responses μ a , the standard deviation of the drift σ d and acceleration responses σ a , and the probability of exceeding the acceleration threshold P a .
Results show that configurations TLWD2 and TLWD3 successfully reduce the mean maximum acceleration of the uncontrolled case by 18% and slightly outperform the TLCD case by approximate 3%, yet not necessitating large space at top of the building. A cross-comparison amongst TLWDs reveals that TLWD2 and TLWD3 slightly outperform TLWD1 by approximate 4.2% in terms of acceleration mitigation. However, this improvement in mitigation keeps the structure's maximum acceleration motion under the performance threshold at least 99% of the time. An      inspection of the floor drift and acceleration profiles also shows that TLWD2 and TLWD3 outperform TLWD1 and the TLCD. is enhancement in performance, in particular, with respect to TLWD1, is consistent with the findings in Section 5.2.

Conclusion
In this study, a multifunctional shear wall, termed tuned liquid wall damper (TLWD), capable of vibration mitigation was investigated. e TLWD consists of a tuned liquid multiple column damper (TLMCD) system integrated within a reinforced concrete shear wall component. e objective of the paper was to evaluate trade-offs between motion and strength performance in designing a TLWD. To empower fast evaluation of strength properties on multidegrees-of-freedom representations, a simplified analytical strength model was proposed based on ACI 318-14 and validated against finite element model simulations conducted on the simplified representation of a 42-story building.
e finite element analysis combined with numerical simulations of the simplified representation was used to establish design considerations. In particular, it was found that distributing liquid mass among a higher number of single columns over a single-layer configuration was preferable in minimizing the reduction in strength while providing enhanced mitigation.
After that, numerical simulations were conducted on a more realistic representation of the 42-story building subjected to 20 stochastic wind loads. First, TLWDs were sequentially added starting from the top floor of the structure to address the mitigation performance on the vertical distribution of TLWDs. e results showed a decreasing marginal gain in mitigation performance. Further simulations were conducted comparing three different TLMCD configurations installed on the last 15 floors. Results showed that adding liquid inertia through additional columns (TLWD1 versus TLWD2) decreased strength while increasing mitigation capabilities and distributing the liquid inertia through additional, smaller columns (TLWD2 versus TLWD3) increased strength while also increasing mitigation capabilities, thus confirming design considerations established through the simplified representation. e use of TLWDs showed important gain in mitigation relative to an uncontrolled case, in particular, for acceleration mitigation. TLWDs also outperformed, yet not significantly, a geometrically constrained TLCD installed at the top of the structure occupying a notably larger space. e use of a distributed liquid inertia (TLWD3) exhibited the best mitigation performance.
Overall, this study demonstrated the potential of the TLWD system as a passive energy mitigation system. While a distributed TLWD system utilizes a higher total liquid mass compared to a conventional TLCD installed at the top floor, its advantage resides in constraining the energy dissipation system with an existing structural component, therefore eliminating the need to utilize a dedicated area in the build as long as structural strength can be ensured. e proposed TLWD is practically feasible due to the low maintenance requirements, cost-effectiveness, and high mechanical robustness characteristics of TLMCDs. e performance of the TLWD system could be potentially improved by optimizing designs at each floor or by integrating semiactive capabilities by actively controlling the orifice block ratios. ese investigations are left to future work.
Data Availability e data that support the findings of this study are available from the corresponding author upon request.

Disclosure
Any opinions, findings, and conclusions or recommendations expressed in this study are those of the authors and do not necessarily reflect the views of the National Science Foundation.

Conflicts of Interest
e authors declare no conflicts of interest.