Optimum Strength Distribution for Structures with Metallic Dampers Subjected to Seismic Loading

A key aspect of the seismic design of structures is the distribution of the lateral strength, because it governs the distribution of the cumulative plastic strain energy (i.e., the damage) among the stories. The lateral shear strength of a story i is commonly normalized by the upward weight of the building and expressed by a shear force coefficient αi. The cumulative plastic strain energy in a given story i can be normalized by the product of its lateral strength and yield displacement, and expressed by a plastic deformation ratio ηi. The distribution αi/α1 that makes ηi equal in all stories is called the optimum yield-shear force distribution. It constitutes a major aim of design; a second aim is to achieve similar ductility demand in all stories. This paper proposes a new approach for deriving the optimum yield-shear force coefficient distribution of structures without underground stories and equipped with metallic dampers. It is shown, both numerically and experimentally, that structures designed with the proposed distribution fulfil the expected response in terms of both damage distribution and inter-story drift demand. Moreover, a comparison with other distributions described in the literature serves to underscore the advantages of the proposed approach.


Introduction
Conventional seismic design allows for damage to a structure under rare or very rare seismic events for economic reasons. The philosophy is to design the structural elements (e.g., beams and columns in frames) so that they can dissipate most of the energy introduced by the earthquake through plastic deformations, without leading to the collapse of the structure. To the present day, the seismic design of structures is made with the so-called force-based approach, which characterizes the loading effect of earthquakes on structures in terms of forces. The force-based approach implements the above philosophy through the use of ductility factors and the assumption of the "equal-displacement" rule. The origin of this approach can be found in the studies carried out for single degree of freedom (SDOF) systems by Veletsos and Newmark [1]. The equal displacement rule assumes that the maximum displacement of a SDOF elastoplastic system and that of the counterpart elastic system under a given ground motion are the same. Based on this assumption, the design lateral strength of the structure is reduced by a ductility-based factor µ (called q in Eurocode 8 [2]) with respect to the value that would be required for the structure to remain elastic. However, many difficulties arise when this approach is extended to multi-story structures. One resides in how to predict the distribution of the cumulative plastic energy dissipation demand E hi among stories. Controlling this distribution is a key aspect of earthquake resistant design [3]. Force-based methods resolve the inability to deal explicitly with E hi by prescribing rules aimed at avoiding/alleviating the concentration of plastic strain energy strut. The equation proposed in [12] approximated the "exact" curve that is obtained by solving the partial differential equation of the elastic undamped distributed parameter system. The new optimum distribution proposed in this paper is obtained representing the building by a discrete model with the mass lumped at several points and the lateral stiffness represented by discrete springs, under the two assumptions motioned above. The new optimum distribution applies modal analysis formulation without resorting to approximations. The application of the proposed optimum distribution to buildings with underground stories, for which the soil-structure interaction could be significant, is beyond the scope of this study. The proposed optimum distribution can be used in buildings with uneven height of the story.

Idealization of a Structure with Metallic Dampers
Structures with metallic dampers consist of two parts working in parallel. The "main structure" is entrusted with sustaining the gravity loads while the structure moves laterally, remaining basically elastic. The other part is the "energy dissipation system", entrusted with dissipating most of the energy input by the earthquake. The latter comprises the dampers and the auxiliary elements needed to connect them with the "main structure". Figure 1 shows in bold lines the idealized curve that represents the ith story shear force versus inter-story drift, Qi-δi, of a structure with metallic dampers subjected to monotonic ( Figure 1a) and cyclic (Figure 1b) loading. This curve is the sum of two curves. One represents the main structure, fQi-δi, and is characterized by the elastic stiffness fki. The other represents the energy dissipation system, sQi-δi, and is defined by the initial stiffness ski, the yield strength, sQy,i, and the yield displacement sδy,i. The stiffness ratio Ki is defined by Ki = ski/fki. The "energy dissipation system" is typically made stiffer laterally than the "main structure". Accordingly, hereafter, the "energy dissipation system" will be designated with the subindex "s" (stiff part) and the "main structure" with the subindex "f" (flexible part). The flexible part is assumed elastic and the stiff part is assumed to exhibit elastic-perfectly plastic restoring force characteristics. The main structure must have a large enough elastic deformation capacity to achieve the maximum lateral displacement expected under seismic loads. As for the strength, past research [7] recommended the following value for the ratio rq,i between the averaged maximum shear force attained by the main structure in the positive and negative domains, � , (=[ , + , ]/2), and the yield-shear force of the energy dissipation system, sQy,i at each story level: The main structure must have a large enough elastic deformation capacity to achieve the maximum lateral displacement expected under seismic loads. As for the strength, past research [7] recommended the following value for the ratio r q,i between the averaged maximum shear force attained by the main structure in the positive and negative domains, f Q max,i (=[ f Q max,i + f Q min,i ]/2), and the yield-shear force of the energy dissipation system, s Q y,i at each story level:

Energy-Based Design Framework
The dynamic response of an N-degree of freedom system subjected to a seismic ground motion is governed by the following force-equilibrium equation:  u g (t) (2) where M and C are the mass and damping matrices, F(t) is the restoring force vector, r is the directivity ground motion vector, ü g (t) is the ground acceleration, u is the relative displacement vector respect to the foundation ground and t the time. Pre-multiplying Equation (2) by du(t) T = . u(t) T dt and integrating over the duration of the earthquake, t = t d , the energy balance equation is obtained: where E k is the kinetic energy, E ξ is the damping energy, E a is the absorbed energy and E I is the relative input energy, (hereafter input energy). Further, E I can also be expressed as an equivalent velocity, V E , defined by V E = √ 2E I /M, where M is the total mass of the structure, and Equation (3) can be rewritten as: where E a is the sum of the elastic strain energy, E s , and the energy dissipated by plastic deformations or hysteretic energy, E h , i.e., E a = E s + E h . By definition, the sum of E k and E s is the so-called vibrational elastic energy, E e (=E k + E s ). Housner [6] defined the sum of E e and E h as the energy that contributes to the damage E D (=E e + E h ). E D can also be expressed in the form of an equivalent velocity, V D , by V D = √ 2E D /M and thus Equation (4) can be rearranged as follows: Furthermore, several empirical expressions have been proposed in the literature to estimate V D from V E [7,[13][14][15][16][17][18]. In this study, the one proposed by Akiyama [7] is used: where ξ is the inherent fraction of damping of the structure. For elastic MDOF systems, F(t) can be expressed by F(t) = Ku(t), where K is the elastic stiffness matrix. The damping matrix C can be in general non-proportional; yet, even in this case it can be approximated by a damping matrix that can be diagonalized by the natural vibration modes. This approximated damping matrix is obtained ignoring the off-diagonal coupling coefficients of the modal coordinate damping matrix. Then, the response vector u can be obtained by superimposing the contribution of each vibrational mode, i.e., u = N n=1 ϕ n x n (t), where x n (t) is the normal coordinate corresponding to mode n and ϕ n the corresponding mode vector. Using a damping matrix C that can be diagonalized by the undamped vibration modes, taking into account the orthogonality properties, and pre-multiplying Equation (2) by ϕ n T , gives the following set of N-uncoupled equations in normal coordinates: ..
x n (t) + k n x n (t) = −m n Γ n ..

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where m n = ϕ n T Mϕ n , c n = ϕ n T Cϕ n and k n = ϕ n T Kϕ n are the n-th generalized mass, damping and stiffness, respectively, and Γ n = ϕ n T Mr/m n is the n-th modal participation factor. The displacements x n (t) obtained from Equation (7) are Γ n times larger than those of an equivalent SDOF system, x SDOF,n (t), with mass, damping and stiffness given by m n , c n and k n , respectively, subjected to the ground acceleration ü g . That is, x n (t) = Γ n x SDOF,n (t). Then, multiplying both sides of the Equation (7) by dx n = .
x n dt and integrating between t = 0 and t ≤ t d , the energy balance equation corresponding to the uncoupled n-th vibration mode is obtained as follows: The first three terms of the left-hand side are the kinetic energy E k,n , the damping energy E ξ,n and the absorbed energy E a,n corresponding to vibration mode n, respectively. It is worth noting that, given the linear nature of Equation (8), the third term on the left-hand side should be, strictly speaking, denoted as the elastic strain energy, which coincides with the absorbed energy in this case. The term on the right-hand side is the input energy corresponding to vibration mode n, E I,n . Each term of Equation (8) is Γ n 2 times the corresponding value in the n-th equivalent SDOF system; in particular, is the absorbed energy of the n-th equivalent SDOF system. Moreover, substituting u = N n=1 ϕ n x n (t) in Equation (3) and taking into account that M, K and C are orthogonal, the energy balance equation of the elastic MDOF systems at t ≤ t d is: From Equations (3), (8), and (9), it can be observed that E k , E ξ , E a and E I in the elastic MDOF systems are obtained summing up the contribution of all n-th vibration mode energies, E kn , E ξn , E an and E In . In particular, E a = N n=1 E an can also be expressed as follows: Similar expressions can also be obtained for E k , E ξ and E I . Equation (10) shows that the absorbed energy of an elastic MDOF system can be obtained as the superposition of the absorbed energy of N equivalent SDOF systems, each one associated with a vibration mode n, subjected to the ground acceleration ü g and scaled by Γ 2 n . Chou and Uang [19] proposed applying the superposition of E a,n likewise for structures that undergo plastic deformations to estimate the total absorbed energy at t = t d . Further, Akiyama [7] showed that as the structure enters the nonlinear range, E e (=E k + E s ) becomes negligible compared with E h at t ≥ t d and can be disregarded, i.e., E h = E a . Therefore, the following Equation (10), E h can be estimated by: where E SDOF hn is the hysteretic energy dissipated by the n-th equivalent SDOF system. If the structure is classically damped, it suffices to add the contribution of the first r vibrational modes that mobilize 90% of the total mass of the structure. In the more general case that the structure is non-classically damped, it has been shown in recent research [20] that the modal mass ratio alone is not sufficient to ascertain the accuracy of a truncated model. A measure that is related to the damping matrix, called modal dissipation ratio of each mode in a complex modal analysis framework, should necessary be considered when constructing a reduced-order model.
In a general MDOF system subjected to a given ground motion characterized by its hysteretic energy spectrum E h SDOF -T, the hysteretic energy accumulated in the i-th story E h,i can be estimated through Equation (13), first, calculating ψ mn,i from a pushover analysis with the n-th vibrational mode loading pattern and, second, taking as E hn SDOF the ordinate of the E h SDOF -T spectrum at period T n .

Proposal of New Optimum Yield-Shear Force Coefficient Distribution
The main purpose of installing dampers is to avoid damage in the main structure and concentrate the plastic deformations in the energy dissipation system. Accordingly, in the following discussion the main structure is assumed to remain elastic and the cumulative plastic strain energy ratioη i for this type of structure is defined as η i = s E h,i /( s Q y,i × s δ y,i ), where s E h,i denotes the hysteretic energy dissipated by the energy dissipation devices (dampers) installed at the i-th story. For convenience, noting that s Q y,i = N j=i m j g s α i , s δ y,i = s Q y,i / s k i = N j=i m j g s α i / K i f k i and using Equation (13), the plastic strain energy ration η i can be rewritten as follows: where the sum in the numerator is extended to r vibration modes. For a given ground motion, the s α i 's that make η i equal in all stories, i.e., η i = η, are defined here as the optimum yield-shear force coefficients of the energy dissipation system, s α opt,i . The corresponding distribution, s α opt,i / s α 1 , is referred to hereafter as optimum yield shear force distribution s α i (= s α opt,i / s α 1 ), where s α 1 is the yield-shear force coefficient of the energy dissipation system at the first (ground) story. Using Equation (14) and noting that η i = η, then s α i is given by: Taking into account that s δ y,i = s Q y,i / s k i , recalling Figure 1 and noting that Q y,i = s Q y,i + f k i s δ y,i = s Q y,i + ( s k i /K i ) s δ y,i = s Q y,i + s Q y,i (1/K i ) = s Q y,i (K i + 1)/K i , the yield shear force coefficient of the whole structure (including main structure and energy dissipation system) at the i-th story α i is related to s α i by: Therefore, the distribution of the optimum total yield shear force coefficient, If K i is the same in all stories, it is obvious that α i coincides with s α i . Using Equation (15) α i gives, then: Past research [7] shows that in an elastic-perfectly plastic MDOF system designed with the optimum distribution associated with a given ground motion, damage spreads out evenly among the stories essentially regardless of the level of plastic deformation. This allows to assume that if the ground motion is scaled down by a factor s y so that the level of plastic deformation in the dampers is zero, i.e., η i = η ≈ 0, the maximum total elastic force developed by the dampers at each story i in each mode n, s Q maxn,i , coincides with the lateral strength required on the dampers in each mode n, s Q yn,i , to carry all stories to the brink of yielding, that is s Q maxn,i = s Q yn,i . These forces can be expressed in terms of shear force coefficients, s α max,i , and their distribution, i.e., s α max,i = s α max,i / s α max,1 , would coincide with the optimum distribution s α i , i.e., s α max,i = s α i . The vector of maximum elastic forces on the whole structure (i.e., main structure and energy dissipation system) at the n-th vibration mode, F n , of an elastic MDOF system subjected to a given ground motion characterized by its elastic response spectrum S a -T scaled by factor s y can be easily obtained from modal analysis as follows: F n = Mϕ n Γ n s y S an (18) where S an is the ordinate of the S a -T spectrum at the period of vibration mode n. The corresponding maximum total shear force on the i-th story in mode n, Q maxn,i , is simply obtained by summing up the components F n , i of vector F n above the i-th story as follows: where ϕ n,j are the components of mode vector ϕ n . The maximum force in the energy dissipation system at the i-th story in mode n, s Q maxn,i , can be related with Q maxn,i by Q maxn,i = s Q maxn,i + ( s Q maxn,i / s k i ) f k i = s Q maxn,i (K i + 1)/K i . Solving for s Q maxn,i in this expression and recalling that is has been assumed that s Q maxn,i = s Q yn,i , the following expression is obtained: In turn, for a design earthquake that causes plastic deformations in the energy dissipation system (i.e., a ground motion not scaled down by s y ), the maximum and minimum normalized plastic deformations experienced by the energy dissipation system of each story i in each vibration mode n are defined as follows: µ maxn,i = (δ maxn,i − s δ yn,i )/ s δ yn,i and µ minn,i = |(δ minn,i − s δ yn,i )|/ s δ yn,i . Here δ maxn,i and δ minn,i are the maximum and minimum inter-story drifts attained at the i-th story in mode n, and s δ yn,i is defined as s δ yn,i = s Q yn,i / s k i . The mean value is µ n,i = (µ maxn,i + µ minn,i )/2. It has been shown [7] that, in mixed systems combining in each story an elastic part with an elastic-perfectly plastic part, as shown in Figure 1, the maximum deformations are roughly the same in the positive and negative domains and thus it can be assumed δ maxn,i ≈ |δ minn,i |, or µ maxn,i ≈ µ minn,i . Moreover, it has been shown [7] that if the structure is designed with the optimum distribution, then µ n,i is roughly the same in all stories and thus it can be assumed that µ n,i = s µ n . Consequently (δ maxn,i − s δ y,i ) = s µ n s δ y,i . The hysteretic energy accumulated at the i-th story, if pushed monotonically beyond the elastic range following the distribution of lateral forces of the n-th mode given by Equation (18) up to δ maxn,i , is: Recalling that s δ yn,i = s Q yn,i / s k i and K i = s k i / f k i , using Equations (12), (20), and (21) ψ mn,i can estimated with the following expression: Equation (22) indicates that when the distribution of lateral strength of the structure, expressed in terms of the yield-shear force coefficient, follows the optimum distribution, then ψ mn,i can be calculated from the mass and the elastic properties of the structures, with no need to perform a pushover analysis. In this study, Equation (15) with ψ mn,i calculated using Equation (22) is proposed as the optimum yield-shear force coefficient distribution for the energy dissipation system and will be denoted as s α prop,i herein. Similarly, Equation (17) with ψ mn,i calculated using Equation (22) is proposed as the optimum yield-shear force coefficient distribution for the whole structure and will be denoted as α prop,i hereafter.

Prototype Structures, Modelization and Ground Motions
In order to validate the optimum yield shear-force coefficient distribution α prop,i proposed in this study, several prototype reinforced concrete (RC) structures consisting of waffle-flat plates supported on RC columns having 3, 6, and 9-stories (main structure) equipped with metallic dampers (energy dissipation system) were investigated. To model the prototype structures, it was assumed that the mass of each plate is concentrated in one point of mass m i located at the centre of mass of the plate (lumped-mass model). Torsional and rocking effects are not considered and therefore a single degree of freedom is assigned to each mass: the horizontal displacement in the direction of the ground motion. The rest of parameters that define the numerical model are the i-th story lateral stiffness of the main structure f k i ; and the lateral stiffness s k i and lateral strength s Q yi provided by the dampers. f k i is defined as the quotient between the shear force endured by the main structure at the i-th story and the corresponding interstory drift in elastic conditions. s k i is defined as the quotient between the shear force endured by the metallic dampers installed at the i-th story and the corresponding interstory drift in elastic conditions. s Q yi is the maximum shear force that can be endured by the metallic dampers installed at the i-th story. The main structure is assumed to remain elastic and the metallic dampers exhibit an elastic-perfectly plastic hysteretic behaviour, as shown in Figure 1. The inherent damping fraction ξ = 0.05. The main structures are designed to sustain only the gravity loads specified under Spanish standards [21,22]. Therefore, f k i and m i for the prototypes with the same number of stories are identical. f k i was determined from a static analysis in which the main structure was subjected to a distribution of lateral forces that followed the pattern of the first vibration mode. Table 1 shows f k i , m i , the height of the story i h i and the fundamental period of the main structure f T 1 . To validate the proposed optimum distribution in a large number of different cases, a variety of levels of plastic deformation for the dampers-expressed in terms of coefficient s µ-, and a variety of ground motions-characterized by the seismological parameters V D , T G , T NH , I d and the proximity to the source (i.e., near-fault or far-field earthquakes)-were considered. T G is the predominant period of the ground motion, T NH is the corner period of the pseudo-velocity Newmark-Hall spectrum, and I D is a seismological parameter proposed by Cosenza and Manfredi [23] defined by I D = 2gI A /(π PGA PGV) where I A is the Arias intensity, PGA is the peak ground acceleration and PGV the peak ground velocity. Appendix A presents the original (unscaled) ground motions selected, together with T G , T NH , I D , PGA and PGV. The lateral stiffness provided by the dampers at each story i, s k i , was made proportional to the lateral stiffness of the main frame f k i , i.e., K i = f k i / s k i = K. Each ground motion was scaled in acceleration by applying a factor SF. The values of SF were kept in the range 0.3 ≤ SF ≤ 3 to distort as little as possible the characteristics of the original signal, and were determined to keep the maximum allowed inter-story drift IDI ,i below 0.75%, while providing a K smaller than 20. The limit IDI ,i ≤ 0.75% was fixed on the basis of code recommendations [8,24] and experimental results obtained by the authors in previous studies [25][26][27]. Also, values of K larger than 20 are not realistic because they lead to very small values of s δ yi that can hardly be applied in practice. For a given set of values of parameters s µ, SF, V D , T G , T NH , I d and the proximity to the source, the required K, s Q y1 -expressed in terms of the base shear force coefficient s α 1 (= s Q y1 /Mg)-, and the normalized energy dissipation demand of the dampers η were calculated applying the method developed by Benavent-Climent [12], but using the optimum yield shear force distribution proposed in this study, with Equations (15) and (22) instead of the optimum distribution proposed in [12]. Tables 2-4 summarize the combinations of values for SF, K, V D , s α 1 , η, s µ investigated, together with the resulting fundamental period of the whole structure in the elastic range T 1 . In applying Equation (15), the input energy E SDOF hn corresponding to each mode n was determined from the inelastic hysteretic energy spectrum E h -T of the scaled ground motion.
The inelastic hysteretic energy spectrum E h -T was calculated by following Akiyama's approach [7], as explained next. First, the elastic input energy spectrum E I -T of the ground motion was obtained with the expression of the right-hand side of Equation (3). Second, the inelastic input energy spectrum, E I − T, was determined by averaging the elastic spectrum E I -T in the range T ≤ T ≤ T max ; that is, the E I corresponding to a given period T of the spectrum was obtained by E I = T max T E I dT/(T max − T). Here, T max is a period longer than T, influenced by the level of plastic deformations experienced by the system characterized by s µ. The calculation of T max is explained in Appendix B as an extension of the formulation proposed by Akiyama [7] for conventional structures to systems with metallic dampers. Third, recalling that the elastic vibrational energy E e can be neglected as explained in Section 2.2. (i.e., Table 1. Lateral stiffness, mass and height of the main structure.

3-Story 6-Story 9-Story
Story  Table 2. Properties of the designed metallic in three-story buildings.  Table 3. Properties of the designed metallic in six-story buildings.  Table 4. Properties of the designed metallic in nine-story buildings.

SF
Records  (17) and (22)-obtained for the near-fault ( Figure 2) and far-field ( Figure 3) ground motions. In all cases the grow is approximately linear in the lower two thirds of the building, and increases exponentially in the upper stories, especially for taller buildings. This is due to the influence of the higher modes of vibration. It is also worth noting that in the lower two thirds part of the building α prop,i is basically the same for all records, whereas clear differences are observed in the upper one third depending on the ground motion considered. These tendencies and features have been identified in previous studies [7,12].

Comparison between Proposed and "Exact" Distributions
Similarly to the optimum yield shear force coefficient distributions proposed in previous studies [7,11,12] the new distribution proposed in this study provides an approximation to the "exact" distribution, referred to as � , hereafter, that would make ηi exactly the same in all stories.

�
, is different for each ground motion and can be obtained only through an iterative trial-anderror procedure of nonlinear time history analyses. A number of multi-purpose algorithms existing in the literature may be used to efficiently conduct such an iterative process [28,29]. In this study, � , was calculated using the pattern search method (PSM), which belongs to the direct search

Comparison between Proposed and "Exact" Distributions
Similarly to the optimum yield shear force coefficient distributions proposed in previous studies [7,11,12] the new distribution proposed in this study provides an approximation to the "exact" distribution, referred to as � , hereafter, that would make ηi exactly the same in all stories.

�
, is different for each ground motion and can be obtained only through an iterative trial-anderror procedure of nonlinear time history analyses. A number of multi-purpose algorithms existing in the literature may be used to efficiently conduct such an iterative process [28,29]. In this study, � , was calculated using the pattern search method (PSM), which belongs to the direct search

Comparison between Proposed and "Exact" Distributions
Similarly to the optimum yield shear force coefficient distributions proposed in previous studies [7,11,12] the new distribution proposed in this study provides an approximation to the "exact" distribution, referred to as α exact,i hereafter, that would make η i exactly the same in all stories. α exact,i is different for each ground motion and can be obtained only through an iterative trial-and-error procedure of nonlinear time history analyses. A number of multi-purpose algorithms existing in the literature may be used to efficiently conduct such an iterative process [28,29]. In this study, α exact,i was calculated using the pattern search method (PSM), which belongs to the direct search methodology for optimization of n-dimensional functions [30]. PSM has been used in previous studies on optimization of buildings with dampers [31] and on damage control of buildings [32,33]. There are other approaches such as the gradient-based method or genetic algorithms used successfully in structural optimization problems [28,29] that could potentially be more efficient to reach the "exact" optimum distribution. Nevertheless, it is out of the scope of this study to determine the most efficient method. In short, PSM is an iterative procedure to minimize an objective function. Some advantages of the PSM are: (i) it is a derivative-free method and (ii) it is simple. In this case, the objective function is the standard deviation of η i , while the independent n-dimensional variable is the lateral strength of the structure. Through non-linear time history analyses with a given accelerogram, PSM varies the lateral strength distribution of the structure until the standard deviation of η i is zero. The main drawback of using PSM in this study has been the significant increase of the calculation time with the increase of the number of stories of the buildings. PSM took about fifteen minutes to reach the optimum distribution for a three-story building subjected to a single ground motion. This time increased up to four hours for a six-story building and up to ten hours for a nine-story building.
Using the PSM, the exact distribution α exact,i was calculated for each prototype structure and for each ground motion, and then was compared with the counterpart approximate distribution proposed in this study, α prop,i in "exact" optimum distribution. Nevertheless, it is out of the scope of this study to determine the most efficient method. In short, PSM is an iterative procedure to minimize an objective function. Some advantages of the PSM are: (i) it is a derivative-free method and (ii) it is simple. In this case, the objective function is the standard deviation of ηi, while the independent n-dimensional variable is the lateral strength of the structure. Through non-linear time history analyses with a given accelerogram, PSM varies the lateral strength distribution of the structure until the standard deviation of ηi is zero. The main drawback of using PSM in this study has been the significant increase of the calculation time with the increase of the number of stories of the buildings. PSM took about fifteen minutes to reach the optimum distribution for a three-story building subjected to a single ground motion. This time increased up to four hours for a six-story building and up to ten hours for a nine-story building.
Using the PSM, the exact distribution � , was calculated for each prototype structure and for each ground motion, and then was compared with the counterpart approximate distribution proposed in this study, �   Under near-field ground motions, the coefficients of variation (COV) of � , / � , calculated for all stories are 0.05, 0.07 and 0.10 for the three-, six-, and nine-story prototype structures, respectively. The counterpart COVs for far-field ground motions are 0.05, 0.06 and 0.08, respectively. Noticeably, there are not significant differences between the two types of earthquakes, the COV of � , / � , being less than or equal than 0.10, independently the height of the building. The optimum distribution proposed by Benavent-Climent [12] provides a good approximation to the "exact" distribution in the lower part of the building but tends to over-estimate the required strength in the upper stories. The COV of � , / � , under near-field records are 0.14, 0.14, 0.17 for three, six and nine-story prototype structures respectively, and 0.13, 0.13, 0.17 under far-field records. Therefore, the proposed optimum distribution � , gives a better approximation to the "exact" one than to that obtained with � , . It is worth noting that, from a practical point of view, the main differences between � , and � , , are: (i) � , is based only on properties of the main structure while � , requires also to know the ratio between the lateral stiffness of the dampers and the lateral stiffness of the main frame at each story; (ii) � , depends on the predominant period of the ground motion TG while � , requires to know the inelastic input energy spectrum. Therefore, the time required to obtain the optimum distribution with � , is larger than with � , , and the analytical expression of the former is more complex than the later. However, from a practical point of view, since these calculations are made usually with a computer, the differences in time and in programming complexity are negligible.

Damage Distribution among Stories
As explained in previous sections, the damage in a given story i can be characterized by the ratio ηi. It has likewise been explained in previous sections that the optimum distribution of the shear force coefficients is the one that makes ηi equal in all stories. This strictly occurs only when the distribution � , is used, and the common value of ηi in this case will be named ηexact (=ηi) hereafter. When the approximate distributions � , or � , are used, in general the ηi's do not have exactly the same value in all stories. In order to evaluate such deviations, the ηi obtained at each story through nonlinear time story analyses conducted for each individual ground motion and for each prototype structure designed with � , , referred to as ηprop,i hereafter, are plotted in Figures 6 and 7 normalized by ηexact. Also shown in the figures with bold lines are the mean curves of ηprop,i/ηexact obtained by averaging the response at each story level throughout the seven ground motions applied to each prototype structure. As can be seen, the mean curve ηprop,i/ηexact is close to the vertical line passing Under near-field ground motions, the coefficients of variation (COV) of α prop,i /α exact,i calculated for all stories are 0.05, 0.07 and 0.10 for the three-, six-, and nine-story prototype structures, respectively. The counterpart COVs for far-field ground motions are 0.05, 0.06 and 0.08, respectively. Noticeably, there are not significant differences between the two types of earthquakes, the COV of α prop,i /α exact,i being less than or equal than 0.10, independently the height of the building. The optimum distribution proposed by Benavent-Climent [12] provides a good approximation to the "exact" distribution in the lower part of the building but tends to over-estimate the required strength in the upper stories. The COV of α Benavent,i /α exact,i under near-field records are 0.14, 0.14, 0.17 for three, six and nine-story prototype structures respectively, and 0.13, 0.13, 0.17 under far-field records. Therefore, the proposed optimum distribution α prop,i gives a better approximation to the "exact" one than to that obtained with α Benavent,i . It is worth noting that, from a practical point of view, the main differences between α Benavent,i and α prop,i , are: (i) α Benavent,i is based only on properties of the main structure while α prop,i requires also to know the ratio between the lateral stiffness of the dampers and the lateral stiffness of the main frame at each story; (ii) α Benavent,i depends on the predominant period of the ground motion T G while α prop,i requires to know the inelastic input energy spectrum. Therefore, the time required to obtain the optimum distribution with α prop,i is larger than with α Benavent,i , and the analytical expression of the former is more complex than the later. However, from a practical point of view, since these calculations are made usually with a computer, the differences in time and in programming complexity are negligible.

Damage Distribution among Stories
As explained in previous sections, the damage in a given story i can be characterized by the ratio η i . It has likewise been explained in previous sections that the optimum distribution of the shear force coefficients is the one that makes η i equal in all stories. This strictly occurs only when the distribution α exact,i is used, and the common value of η i in this case will be named η exact (=η i ) hereafter. When the approximate distributions α prop,i or α Benavent,i are used, in general the η i 's do not have exactly the same value in all stories. In order to evaluate such deviations, the η i obtained at each story through nonlinear time story analyses conducted for each individual ground motion and for each prototype structure designed with α prop,i , referred to as η prop,i hereafter, are plotted in Figures 6 and 7 normalized by η exact . Also shown in the figures with bold lines are the mean curves of η prop,i /η exact obtained by averaging the response at each story level throughout the seven ground motions applied to each prototype structure. As can be seen, the mean curve η prop,i /η exact is close to the vertical line passing through the abscissa 1.0. This indicates that the proposed distribution provides a reasonably uniform distribution of damage among stories. More precisely, the COV of η prop,i /η exact calculated for all stories and all the near-field earthquakes are 0.40, 0.33 and 0.42, for the three-, six-, and nine-story prototype structures, respectively; the counterpart values for far-field earthquakes are 0.33, 0.33, and 0.37. For comparison, the ηi obtained at each story through nonlinear time story analyses conducted for each individual ground motion and for each prototype structure designed with � , was also calculated and it is referred to as ηBenavent,i hereafter. Figures 6 and 7 show in bold lines the mean curves of ηBenavent,i/ηexact obtained by averaging the response at each story level through the seven ground motions applied to each prototype structure. When � , is used the COV of ηBenavent,i/ηexact are 0.81, 0.61, and 0.63 for near-field earthquakes and 0.80, 0.59, and 0.52 for far-field earthquakes, respectively. It follows from these results that the optimum yield shear force distribution proposed in this study For comparison, the ηi obtained at each story through nonlinear time story analyses conducted for each individual ground motion and for each prototype structure designed with � , was also calculated and it is referred to as ηBenavent,i hereafter. Figures 6 and 7 show in bold lines the mean curves of ηBenavent,i/ηexact obtained by averaging the response at each story level through the seven ground motions applied to each prototype structure. When � , is used the COV of ηBenavent,i/ηexact are 0.81, 0.61, and 0.63 for near-field earthquakes and 0.80, 0.59, and 0.52 for far-field earthquakes, respectively. It follows from these results that the optimum yield shear force distribution proposed in this study For comparison, the η i obtained at each story through nonlinear time story analyses conducted for each individual ground motion and for each prototype structure designed with α Benavent,i was also calculated and it is referred to as η Benavent,i hereafter. Figures 6 and 7 show in bold lines the mean curves of η Benavent,i /η exact obtained by averaging the response at each story level through the seven ground motions applied to each prototype structure. When α Benavent,i is used the COV of η Benavent,i /η exact are 0.81, 0.61, and 0.63 for near-field earthquakes and 0.80, 0.59, and 0.52 for far-field earthquakes, respectively. It follows from these results that the optimum yield shear force distribution proposed in this study provides a much more uniform distribution of damage among the stories than the one proposed by Benavent-Climent [12]. The reason lies in the bias of Benavent-Climent's equation, overestimating the required strength at the higher stories, which leads to lower values of η Benavent,i for the high stories, and higher values for the low stories.
Furthermore, Figure 8 compares the histories of energy E h dissipated by the 6-story building designed with α prop,i and with α Benavent,i for two ground motions: Calitri ( Figure 8a) and Montebello (Figure 8b). It can be seen that there are not significant differences between them. provides a much more uniform distribution of damage among the stories than the one proposed by Benavent-Climent [12]. The reason lies in the bias of Benavent-Climent's equation, overestimating the required strength at the higher stories, which leads to lower values of ηBenavent,i for the high stories, and higher values for the low stories. Furthermore, Figure 8 compares the histories of energy Eh dissipated by the 6-story building designed with � , and with � , for two ground motions: Calitri ( Figure 8a) and Montebello (Figure 8b). It can be seen that there are not significant differences between them.

Maximum Interstory Drifts
Figures 9 and 10 show the maximum inter-story drifts normalized by the story height, IDIi, obtained from the time history analyses conducted on the prototype structures designed with � , , denoted by IDIprop,i (record-to-record, mean and the envelope of the maximum values "maxenv"), and for those designed with � , , denoted by IDIexact,i (the mean and the envelope of the maximum values "maxenv"). Overall, comparing the mean curves it is seen that the IDIi's obtained with the proposed distribution (i.e., "mean IDIprop,I" in the figures) are very close, almost coinciding with the exact values (i.e., "mean IDIexact,i" in the figures). Similar observations can be made with the maximum envelope curves (i.e., "maxenv IDIprop,I" versus "maxenv IDIexact,I"). Moreover, the target IDIi (=0.75%) is not exceeded in any case. It is also noteworthy that despite the differences observed in the distribution of ηi for the structures designed with � , and with � , (Figures 6 and 7), the response in terms of displacement are in very good agreement. In other words, the small deviations of ηi among stories scarcely affect the maximum lateral inter-story drifts. Deviations of the yield shear force coefficient distribution with respect to the exact value � , inevitably cause deviations from the aspired even distribution of damage (i.e., ηi equal in all stories). Nonetheless, when these deviations are limited (to about 20% as occurs with the proposed distribution � , according to the results of this study), then the damage concentration (i.e., values of ηi in specific stories markedly larger than in the rest of stories) is prevented and the response of the structure in terms of displacements IDIprop,i scarcely deviates from that obtained for a structure designed � , , i.e from IDIexact,i. These results emphasize the key importance of designing structures with a strength distribution close to that prescribed by the optimum yield-shear force distribution so as to prevent damage concentration and to control the response of the structure in terms of displacement. The results of this study also underline the improvements of the proposed optimum distribution with respect to the one proposed by Benavent-Climent [12] in past studies.  Figures 9 and 10 show the maximum inter-story drifts normalized by the story height, IDI i , obtained from the time history analyses conducted on the prototype structures designed with α prop,i , denoted by IDI prop,i (record-to-record, mean and the envelope of the maximum values "maxenv"), and for those designed with α exact,i , denoted by IDI exact,i (the mean and the envelope of the maximum values "maxenv"). Overall, comparing the mean curves it is seen that the IDI i 's obtained with the proposed distribution (i.e., "mean IDI prop,I " in the figures) are very close, almost coinciding with the exact values (i.e., "mean IDI exact,i " in the figures). Similar observations can be made with the maximum envelope curves (i.e., "maxenv IDI prop,I " versus "maxenv IDI exact,I "). Moreover, the target IDI i (=0.75%) is not exceeded in any case. It is also noteworthy that despite the differences observed in the distribution of η i for the structures designed with α prop,i and with α exact,i (Figures 6 and 7), the response in terms of displacement are in very good agreement. In other words, the small deviations of η i among stories scarcely affect the maximum lateral inter-story drifts.

Maximum Interstory Drifts
Deviations of the yield shear force coefficient distribution with respect to the exact value α exact,i inevitably cause deviations from the aspired even distribution of damage (i.e., η i equal in all stories). Nonetheless, when these deviations are limited (to about 20% as occurs with the proposed distribution α prop,i according to the results of this study), then the damage concentration (i.e., values of η i in specific stories markedly larger than in the rest of stories) is prevented and the response of the structure in terms of displacements IDI prop,i scarcely deviates from that obtained for a structure designed α exact,i , i.e from IDI exact,i . These results emphasize the key importance of designing structures with a strength distribution close to that prescribed by the optimum yield-shear force distribution so as to prevent damage concentration and to control the response of the structure in terms of displacement. The results of this study also underline the improvements of the proposed optimum distribution with respect to the one proposed by Benavent-Climent [12] in past studies. Finally, a comparison was carried out between the IDIi's of the prototype structures designed using � , or with � , , respectively IDIprop,i and IDIBenavent,i. The goodness of the agreement of IDIi or IDIBenavent,i with respect to IDIexact,i, is quantified in terms of the normalized square mean error, NMSE, defined as follows: NMSE ranges between −∞ (very bad fit) and 1 (perfect fit). Values above 0.70 are considered a reasonably good fit. The results are shown in Figure 11. It is seen that the structures designed using � , lead to NMSE values greater than 0.70 ( Figure 11a) and 0.50 (Figure 11b) in the case of near- Finally, a comparison was carried out between the IDIi's of the prototype structures designed using � , or with � , , respectively IDIprop,i and IDIBenavent,i. The goodness of the agreement of IDIi or IDIBenavent,i with respect to IDIexact,i, is quantified in terms of the normalized square mean error, NMSE, defined as follows: NMSE ranges between −∞ (very bad fit) and 1 (perfect fit). Values above 0.70 are considered a reasonably good fit. The results are shown in Figure 11. It is seen that the structures designed using � , lead to NMSE values greater than 0.70 ( Figure 11a) and 0.50 (Figure 11b) in the case of near- Figure 10. Interstory drift ratios IDI prop,i (record-to-record, mean and maximum envelope) and IDI exact,i (mean and maximum envelope) obtained for 3 (a), 6 (b) and 9 (c) story buildings subjected to far-field earthquakes.
Finally, a comparison was carried out between the IDI i 's of the prototype structures designed using α prop,i or with α Benavent,i , respectively IDI prop,i and IDI Benavent,i . The goodness of the agreement of IDI i or IDI Benavent,i with respect to IDI exact,i , is quantified in terms of the normalized square mean error, NMSE, defined as follows: NMSE ranges between −∞ (very bad fit) and 1 (perfect fit). Values above 0.70 are considered a reasonably good fit. The results are shown in Figure 11. It is seen that the structures designed using α prop,i lead to NMSE values greater than 0.70 (Figure 11a) and 0.50 (Figure 11b) in the case of near-field and far-field records, respectively. Nonetheless, for the structures designed with α Benavent,i , even though the target IDI i is not exceeded and most analyses show NSME greater than 0.50, there are results giving poor values for NMSE, especially with near-field records (Figure 11a). field and far-field records, respectively. Nonetheless, for the structures designed with � , , even though the target IDIi is not exceeded and most analyses show NSME greater than 0.50, there are results giving poor values for NMSE, especially with near-field records (Figure 11a).

Ductility Distribution among Stories
Following past studies [7], it was assumed in Section 3 that when the structure is designed with the optimum yield-force coefficient distribution the ductility demand ̅ is rougly the same in all stories (i.e., ̅ = ). This assumption is further examined in this subsection. Using the IDIprop,i's of Section 4.5, the corresponding ductility factor ̅ at each story i, referred to as ̅ , herein, was calculated for the structures designed with the proposed optimum distribution � , by: where hi is given in Table 1. The yield displacement of the dampers is obtained with: where mi, are given in Table 1 and Tables 2-4. ̅ , was compared with the ductility factor assumed for designing the prototype structures shown in Tables 2-4. The quotient ̅ , / calculated for each ground motion and the mean curve "mean ̅ , / " are plotted in Figures 12 and 13. Also plotted in the figures are the mean curves "mean ̅ , / " and "mean ̅ , / " obtained using IDIBenavent,i, � , and IDIexact,i, � , instead of IDIprop,I and � , in Equations (24) and (25). The curves "mean ̅ , / " , "mean ̅ , / " and "mean ̅ , / " are found to lie approximately in a vertical line. Noting that the denominator is the same in all cases and has the same value for all stories, this means that the ductility demand ̅ is approximately the same in all stories. The variation of "mean ̅ , / ", "mean ̅ , / " or "mean ̅ , / " throughout the stories was further quantified by calculating the corresponding COV. The results are shown in Table 5. It can be seen that the structures designed with � , so that η is exactly the same in all stories also exhibit a very uniform distribution of ̅ among stories, the COVs ranging from 0.05 to 0.12. This confirms Akiyama's [7] results and further supports one of the assumptions made in this study. For structures designed with the proposed approximate optimum distribution � , , the COV slightly increases with respect to � , , in particular for the three Figure 11. Normalized mean square error of IDI prop,i and IDI Benavent,i with respect to IDI exact,i for (a) near-field earthquakes and (b) far-field earthquakes.

Ductility Distribution among Stories
Following past studies [7], it was assumed in Section 3 that when the structure is designed with the optimum yield-force coefficient distribution the ductility demand µ i is rougly the same in all stories (i.e., µ i = s µ). This assumption is further examined in this subsection. Using the IDI prop,i 's of Section 4.5, the corresponding ductility factor µ i at each story i, referred to as µ prop,i herein, was calculated for the structures designed with the proposed optimum distribution α prop,i by: where h i is given in Table 1. The yield displacement of the dampers is obtained with: where m i , f k i are given in Table 1 and s α 1 , Tables 2-4. µ prop,i was compared with the ductility factor s µ assumed for designing the prototype structures shown in Tables 2-4. The quotient µ prop,i / s µ calculated for each ground motion and the mean curve "mean µ prop,i / s µ" are plotted in Figures 12  and 13. Also plotted in the figures are the mean curves "mean µ Benavent,i / s µ" and "mean µ exact,i / s µ" obtained using IDI Benavent,i , α Benavent,i and IDI exact,i , α exact,i instead of IDI prop,I and α prop,i in Equations (24) and (25). The curves "mean µ prop,i / s µ", "mean µ Benavent,i / s µ" and "mean µ exact,i / s µ" are found to lie approximately in a vertical line. Noting that the denominator s µ is the same in all cases and has the same value for all stories, this means that the ductility demand µ i is approximately the same in all stories. The variation of "mean µ prop,i / s µ", "mean µ Benavent,i / s µ" or "mean µ exact,i / s µ" throughout the stories was further quantified by calculating the corresponding COV. The results are shown in Table 5. It can be seen that the structures designed with α exact,i so that η is exactly the same in all stories also exhibit a very uniform distribution of µ i among stories, the COVs ranging from 0.05 to 0.12. This confirms Akiyama's [7] results and further supports one of the assumptions made in this study. For structures designed with the proposed approximate optimum distribution α prop,i , the COV slightly increases with respect to α exact,i , in particular for the three story structure, but is still small. The distribution proposed by Benavent-Climent [12] presents the largest COVs.
Metals 2019, 9, x FOR PEER REVIEW 18 of 27 story structure, but is still small. The distribution proposed by Benavent-Climent [12] presents the largest COVs. Finally, it is likewise observed in Figures 12 and 13 that "mean ̅ , / ", "mean ̅ , / " and "mean ̅ , / " are in general below 1. The reason is that the expression used in reference [12] to relate ̅ with , developed by Cosenza and Manfredi [23], is somewhat conservative and provides in general a safe-side estimation of the maximum displacements.
Metals 2019, 9, x FOR PEER REVIEW 18 of 27 story structure, but is still small. The distribution proposed by Benavent-Climent [12] presents the largest COVs. Finally, it is likewise observed in Figures 12 and 13 that "mean ̅ , / ", "mean ̅ , / " and "mean ̅ , / " are in general below 1. The reason is that the expression used in reference [12] to relate ̅ with , developed by Cosenza and Manfredi [23], is somewhat conservative and provides in general a safe-side estimation of the maximum displacements.

Test Model
A waffle-flat slab RC structure supported on RC columns and equipped with metallic dampers (test model) was built in the Laboratory of Structures of the University of Granada (Spain), and subjected to unidirectional dynamic loadings with a shaking table. The test model is representative of a prototype two story building at a scale of two fifths. The overall dimensions and characteristics of the test model are shown in Figure 14. Due to space limitations in the laboratory, the test model does not represent the whole buildings but rather a portion of it. More precisely, to reduce its height it was assumed that under lateral loadings the sections of zero bending moment on the columns of the second story are located at mid-height. With this assumption, the columns of the second story were replaced by half columns with a pin connection at the top. The weight that the prototype building supports above mid-height of these columns was replaced by steel blocks put on the top (indicated as "added weight" in Figure 14). Similarly, to reduce the horizontal dimensions of the test model it was assumed that under lateral loading the mid-span section of the RC slab moves horizontally (i.e., there is no vertical movement) and that the bending moment is zero. To simulate these boundary conditions, the mid-span section of the RC slab was connected to the steel blocks put on the top of the half-columns of the second story with pin-ended steel bars. The very large flexural stiffness of these steel blocks prevented any possible vertical displacement. Steel blocks were also attached to the slab to represent the gravity loads acting on the floor. The mass m 1 of the RC floor slab with the added weight was m 1 = 4079 kg. The mass m 2 of the added weight put on top of the half columns of the second story was m 2 = 7058 kg. The total mass of the test model was 11137 kg.
One metallic damper was installed in each story along the direction of shaking. The type of metallic damper used was developed by the authors in past studies [34] and is named web plastification damper (WPD). The details of the WPDs are shown in Figure 15. The WPD is constructed by assembling several short-length segments of I-shaped steel sections and two U-shaped steel bars. The WPD is installed in the structure as a diagonal bar connecting two consecutive floor levels. When the damper is subjected to axial deformations, the webs of the I-sections are forced to undergo out of plane plastic deformations. The two U-shaped steel bars function as auxiliary elements aimed to transfer the horizontal forces from the slabs to the I-shaped steel sections and are designed to remain elastic.

Test Set-Up and Instrumentation
The RC elements of the test model were instrumented with strain gages attached to the longitudinal reinforcement, uniaxial accelerometers and displacement transducers (linear variable differential transformers, LVDT). The ends of the WPDs were also instrumented with strain gages that measured the axial forces acting on them. The accelerometers and the LVDT's provided the horizontal accelerations and displacements of the masses. Additional LVDTs installed in the WPDs measured their axial displacements. Data were acquired continuously with a scan frequency of 200 Hz. Figure 16 offers a general view of the experimental set up.

Test Set-Up and Instrumentation
The RC elements of the test model were instrumented with strain gages attached to the longitudinal reinforcement, uniaxial accelerometers and displacement transducers (linear variable differential transformers, LVDT). The ends of the WPDs were also instrumented with strain gages that measured the axial forces acting on them. The accelerometers and the LVDT's provided the horizontal accelerations and displacements of the masses. Additional LVDTs installed in the WPDs measured their axial displacements. Data were acquired continuously with a scan frequency of 200 Hz. Figure 16 offers a general view of the experimental set up.

Test Set-Up and Instrumentation
The RC elements of the test model were instrumented with strain gages attached to the longitudinal reinforcement, uniaxial accelerometers and displacement transducers (linear variable differential transformers, LVDT). The ends of the WPDs were also instrumented with strain gages that measured the axial forces acting on them. The accelerometers and the LVDT's provided the horizontal accelerations and displacements of the masses. Additional LVDTs installed in the WPDs measured their axial displacements. Data were acquired continuously with a scan frequency of 200 Hz. Figure 16 offers a general view of the experimental set up.

Seismic Simulations
The test model was subjected to three consecutive seismic simulations. In each simulation, the shaking table reproduced the accelerogram recorded at Calitri during the Campano-Lucano earthquake (Italy, 1980) scaled in time by the factor 0.63. In the first seismic simulation, C100, the original accelerogram was applied without scaling in amplitude. In the second simulation, C200, the amplitude of the original accelerogram was multiplied by two, and in the third simulation C300 by three. The resulting peak accelerations applied to the shaking table were 0.16, 0.31, and 0.47 g, respectively (here g is the acceleration of gravity). Figure 17a shows the accelerogram (scaled only in time by 0.63) and Figure   17b the corresponding elastic spectrum EI-T corresponding to a damping fraction of ξ = 1.8% (i.e., the damping fraction measured during the test as discussed later).

Test Results
The readings provided by the gauges attached to the steel reinforcement of the RC elements indicate that the strains remained below or very close to the yield strain, meaning that the RC main structure remained elastic during the tests. This is corroborated by the fact that the damping fraction ξ obtained experimentally from free vibration tests conducted at the end of each seismic simulation remained constant (ξ = 1.8%). The maximum inter-story drifts IDI were 0.74% in the first story and

Seismic Simulations
The test model was subjected to three consecutive seismic simulations. In each simulation, the shaking table reproduced the accelerogram recorded at Calitri during the Campano-Lucano earthquake (Italy, 1980) scaled in time by the factor 0.63. In the first seismic simulation, C100, the original accelerogram was applied without scaling in amplitude. In the second simulation, C200, the amplitude of the original accelerogram was multiplied by two, and in the third simulation C300 by three. The resulting peak accelerations applied to the shaking table were 0.16, 0.31, and 0.47 g, respectively (here g is the acceleration of gravity). Figure 17a shows the accelerogram (scaled only in time by 0.63) and Figure 17b the corresponding elastic spectrum E I -T corresponding to a damping fraction of ξ = 1.8% (i.e., the damping fraction measured during the test as discussed later).

Seismic Simulations
The test model was subjected to three consecutive seismic simulations. In each simulation, the shaking table reproduced the accelerogram recorded at Calitri during the Campano-Lucano earthquake (Italy, 1980) scaled in time by the factor 0.63. In the first seismic simulation, C100, the original accelerogram was applied without scaling in amplitude. In the second simulation, C200, the amplitude of the original accelerogram was multiplied by two, and in the third simulation C300 by three. The resulting peak accelerations applied to the shaking table were 0.16, 0.31, and 0.47 g, respectively (here g is the acceleration of gravity). Figure 17a shows the accelerogram (scaled only in time by 0.63) and Figure   17b the corresponding elastic spectrum EI-T corresponding to a damping fraction of ξ = 1.8% (i.e., the damping fraction measured during the test as discussed later).

Test Results
The readings provided by the gauges attached to the steel reinforcement of the RC elements indicate that the strains remained below or very close to the yield strain, meaning that the RC main structure remained elastic during the tests. This is corroborated by the fact that the damping fraction ξ obtained experimentally from free vibration tests conducted at the end of each seismic simulation remained constant (ξ = 1.8%). The maximum inter-story drifts IDI were 0.74% in the first story and

Test Results
The readings provided by the gauges attached to the steel reinforcement of the RC elements indicate that the strains remained below or very close to the yield strain, meaning that the RC main structure remained elastic during the tests. This is corroborated by the fact that the damping fraction ξ obtained experimentally from free vibration tests conducted at the end of each seismic simulation remained constant (ξ = 1.8%). The maximum inter-story drifts IDI were 0.74% in the first story and 0.86% in the second one. These IDIs are slightly below and above, respectively, the value assumed in codes for the elastic limit of RC waffle-flat slab structures (0.75%). The corresponding ductility factors s µ i were s µ 1 = 3.9 and s µ 2 = 4.0 for the first and second stories respectively.
In contrast to the main structure, the WPDs underwent plastic deformations. Figures 18 and 19 show the axial force versus axial displacement curves, N-d, measured on the metallic dampers installed in the first ( Figure 18) and second story ( Figure 19) in each seismic simulation. The energy dissipated by each damper at the end of the tests is obtained by integrating these N-d curves. Specifically, at the end of the tests the hysteretic energy dissipated by the damper of the first story was E h1 = 4885 kNmm and that of the damper of the second story E h2 = 3073 kNmm. The corresponding normalized ratios η i = E hi /( s Q yi × s δ yi ) are η 1 = 33.4 and η 2 = 27.2 for the first and second stories. The values of s Q yi and s δ yi used for obtaining η i were calculated with the equations developed in [34], as discussed in the next subsection. 0.86% in the second one. These IDIs are slightly below and above, respectively, the value assumed in codes for the elastic limit of RC waffle-flat slab structures (0.75%). The corresponding ductility factors were 1 = 3.9 and 2 = 4.0 for the first and second stories respectively. In contrast to the main structure, the WPDs underwent plastic deformations. Figures 18 and 19 show the axial force versus axial displacement curves, N-d, measured on the metallic dampers installed in the first ( Figure 18) and second story ( Figure 19) in each seismic simulation. The energy dissipated by each damper at the end of the tests is obtained by integrating these N-d curves. Specifically, at the end of the tests the hysteretic energy dissipated by the damper of the first story was Eh1 = 4885 kNmm and that of the damper of the second story Eh2 = 3073 kNmm. The corresponding normalized ratios ηi = Ehi/(sQyi × sδyi) are η1 = 33.4 and η2 = 27.2 for the first and second stories. The values of sQyi and sδyi used for obtaining ηi were calculated with the equations developed in [34], as discussed in the next subsection. (a) (b) (c) Figure 19. Axial force vs. axial displacement in the metallic damper of the second story for the seismic simulations C100 (a), C200 (b) and C300 (c).

Discussion
A nonlinear finite element model representing the test model without the dampers was developed using Engineer's Studio software (V.1.07.02, Forum-8, London, UK) [35]. The numerical model was subjected to nonlinear static pushover analysis by applying an inverted triangle distribution of lateral forces. From this analysis, the lateral stiffness fki and yield strength fQyi of the main RC structure (without dampers) was obtained, giving fk1 = 3.6 kN/mm and fQy1 = 20.6 kN in the first story, and fk2 = 4.1 kN/mm and fQy2 = 22.4 kN in the second story. The lateral stiffness ski and yield strength sQyi provided by the dampers was calculated with the equations developed in [34] giving sk1 = 31.3 kN/mm and sQy1 = 68.7 kN in the first story, and sk2 = 24.8 kN/mm and sQy2 = 53 kN in the second story. Given these sQyi's, the actual yield shear force coefficients of the energy dissipation system (= / ∑ = ) corresponding to the lateral strengths sQyi indicated above are sα1 = 0.63 in the first story and sα2 = 0.77 in the second 0.86% in the second one. These IDIs are slightly below and above, respectively, the value assumed in codes for the elastic limit of RC waffle-flat slab structures (0.75%). The corresponding ductility factors were 1 = 3.9 and 2 = 4.0 for the first and second stories respectively. In contrast to the main structure, the WPDs underwent plastic deformations. Figures 18 and 19 show the axial force versus axial displacement curves, N-d, measured on the metallic dampers installed in the first ( Figure 18) and second story ( Figure 19) in each seismic simulation. The energy dissipated by each damper at the end of the tests is obtained by integrating these N-d curves. Specifically, at the end of the tests the hysteretic energy dissipated by the damper of the first story was Eh1 = 4885 kNmm and that of the damper of the second story Eh2 = 3073 kNmm. The corresponding normalized ratios ηi = Ehi/(sQyi × sδyi) are η1 = 33.4 and η2 = 27.2 for the first and second stories. The values of sQyi and sδyi used for obtaining ηi were calculated with the equations developed in [34], as discussed in the next subsection.
(a) (b) (c) Figure 18. Axial force vs. axial displacement in the metallic damper of the first story for the seismic simulations C100 (a), C200 (b) and C300 (c).
(a) (b) (c) Figure 19. Axial force vs. axial displacement in the metallic damper of the second story for the seismic simulations C100 (a), C200 (b) and C300 (c).

Discussion
A nonlinear finite element model representing the test model without the dampers was developed using Engineer's Studio software (V.1.07.02, Forum-8, London, UK) [35]. The numerical model was subjected to nonlinear static pushover analysis by applying an inverted triangle distribution of lateral forces. From this analysis, the lateral stiffness fki and yield strength fQyi of the main RC structure (without dampers) was obtained, giving fk1 = 3.6 kN/mm and fQy1 = 20.6 kN in the first story, and fk2 = 4.1 kN/mm and fQy2 = 22.4 kN in the second story. The lateral stiffness ski and yield strength sQyi provided by the dampers was calculated with the equations developed in [34] giving sk1 = 31.3 kN/mm and sQy1 = 68.7 kN in the first story, and sk2 = 24.8 kN/mm and sQy2 = 53 kN in the second story. Given these sQyi's, the actual yield shear force coefficients of the energy dissipation system (= / ∑ = ) corresponding to the lateral strengths sQyi indicated above are sα1 = 0.63 in the first story and sα2 = 0.77 in the second  Funding: This research was funded by Spanish Ministry of Economy and Competitivity, research project reference MEC BIA2017 88814 R and received funds from the European Union (Fonds Européen de Dévelopment Régional).

Conflicts of Interest:
The authors declare no conflict of interest.
Appendix A Table A1. Near-field ground motion records used in the analyses.

Appendix B
Akiyama [7] proposed a simple method to obtain the lengthened period, T max , as opposed to the elastic one T, of a SDOF with elastic-perfectly plastic restoring force characteristics that undergoes plastic deformations characterized by the maximum deformation ratio, µ. T max is obtained by calculating the time it takes the system to complete a cycle of deformation in free vibration (path O-A-B-C-D-E-O in Figure A1a). The procedure proposed by Akiyama is extended here to structures with metallic dampers that are characterized by a post elastic stiffness (A-B and DE, Figure A1b) exerted by the flexible part.
The differential equation that governs the movement of the SDOF system along paths O-A and B-C in free vibration ( Figure A1b Akiyama [7] proposed a simple method to obtain the lengthened period, Tmax, as opposed to the elastic one T, of a SDOF with elastic-perfectly plastic restoring force characteristics that undergoes plastic deformations characterized by the maximum deformation ratio, µ. Tmax is obtained by calculating the time it takes the system to complete a cycle of deformation in free vibration (path O-A-B-C-D-E-O in Figure A1a). The procedure proposed by Akiyama is extended here to structures with metallic dampers that are characterized by a post elastic stiffness (A-B and DE, Figure A1b) exerted by the flexible part.
(a) (b) Figure A1. Deformation cycle of a SDOF system corresponding to µ: (a) Elastic-Perfectly plastic structure; (b) structure with metallic dampers.