# Energy Capacity of Waffle-Flat-Plate Structures with Hysteretic Dampers Subjected to Bidirectional Seismic Loadings

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## Abstract

**:**

## Featured Application

**The results of this study can be applied to design reinforced concrete (RC) waffle-flat-plate structures with hysteretic dampers capable of enduring bidirectional seismic loadings under different seismic performance levels. The levels tested were: fully operational (elastic), operational (damage only in dampers), and near collapse (damage in dampers and in RC structure).**

## Abstract

## 1. Introduction

## 2. Test, Numerical Model, and Validation of Reinforced Concrete (RC) Waffle-Flat-Plate (WFP) Structure

#### 2.1. Shake-Table Tests

#### 2.2. Numerical Model of the WFP Structure

_{p}, which is taken here as the depth of the transverse section, i.e., l

_{p}= 160 mm according to ACI 318M-14 [3]. As for the fiber elements, the transverse section is discretized by a grid of 2x2 mm fibers of concrete, which are replaced by steel when they are occupied by rebars. In addition, the hysteretic behavior of the concrete was defined through the parametric model proposed in [25], adapted to the constitutive model proposed in [26]. For the steel rebars, the Giuffré-Menegotto-Pinto model with isoparametric strain hardening was used. The non-linear behavior of the plastic hinges is characterized by means of the moment-curvature backbone curve, i.e., M-φ, and the corresponding hysteretic law in each main direction. The M-φ curve is defined by yield moment, M

_{y}, and the yield and ultimate curvatures, φ

_{y}and φ

_{u}. M

_{y}was estimated using the empirical expression proposed by Fardis [4] with a reduction of 20% to account for the biaxial cyclic loading-interaction factor [27,28]. First, φ

_{y}and φ

_{u}were calculated as φ

_{y}=θ

_{y}/l

_{p}and φ

_{u}= θ

_{u}/l

_{p}, where θ

_{y}and θ

_{u}are the yielding and ultimate rotation obtained with empirical formulae [4]. Next, these initial values were modified to fit the results obtained from the tests. The stiffness degradation implemented in the hysteretic law was based on the study carried out by Rodriguez et al. [27].

_{cr}, is a simplified version of the model developed by Valipour and Foster [29]. It is defined through three points: cracking torsion (φ

_{1},T

_{cr}); ultimate torsional capacity (φ

_{2},T

_{u}); and torsional failure (φ

_{3},T

_{u}).

#### 2.3. Experimental Validation of the Numerical Model

**C**—classical—is given by

**C**= a

_{0}

**M**, where a

_{0}is a parameter and

**M**is the mass matrix. In turn, a

_{0}is defined through the expression a

_{0}= 2ξ

_{i}ω

_{i}, where ξ

_{i}and ω

_{i}are the damping ratio and the angular frequency—corresponding to the frequency, f

_{i}= ω

_{i}/2π—of the i-th vibration mode. The reference damping ratio and frequency used to calculate a

_{0}were 0.03 and 3.16 Hz, respectively; these values are close to the one obtained experimentally under elastic deformations, 0.024 [7]. This gives a

_{0}= 1.19. When the numerical model collapsed, the frequency of the fundamental mode was 1.3 Hz; using a

_{0}= 1.19 the corresponding damping ratio at failure is ξ

_{1}= a

_{0}/(2ω

_{1}) = a

_{0}/(2f

_{1}2π) = 0.072, a value close to the one measured at the end of the tests, 0.092 [7].

_{I}. The numerical model is seen to predict the input energy until the onset of simulation C200i very well. From this point on, the numerical model underestimates the input energy measured experimentally only for C200i and part of C200. It is important to note that the response history in terms of input energy provides a more convenient and accurate criterion to assess the goodness-of-fit between model and test, preferable to displacement or force histories. Displacements or forces are vectors defined at specific parts of the structure; in contrast, energy is a scalar quantity that synthesizes the overall response of the entire structure [23]. Moreover, variations in the input energy history are better distinguished by using a logarithmic scale.

## 3. Test and Numerical Model of Dampers

_{s}N-

_{s}Δu, obtained from one of the tests before the specimen failed. The test results were used to calibrate a numerical model (to characterize the behavior of the SPD), namely, the Giuffré-Menegotto-Pinto model with isotropic strain hardening. It is implemented in Opensess [24] as Uniaxial Material Steel 02. Table 1 indicates the values of the parameters that control this constitutive model.

_{s}N

_{y}and

_{s}k correspond to the yield-axial force and the stiffness, respectively, while the remaining parameters control the hysteretic behavior of the steel device; their meaning is explained in reference [24]. Figure 7 compares the

_{s}N-

_{s}Δu curves obtained experimentally with those predicted by the numerical model (dash lines). A very good agreement is seen.

## 4. Numerical Analyses of a WFP Structure Upgraded with Slit-Plate Dampers (SPDs)

#### 4.1. Design of the Hysteretic Dampers

_{f}Q

_{yX,1}and

_{f}Q

_{yY,1}, were 57.5 kN and 62.5 kN, respectively [7]. These results were corroborated against those provided by the numerical model through a pushover analysis. Therefore, the base shear of the dampers in X and Y directions,

_{s}Q

_{yX,1}and

_{s}Q

_{yY,1}, was limited to 28.75 kN (=57.5/2) and 31.25 kN (=62.5/2), respectively.

_{s}Q

_{yX,1}and

_{s}Q

_{yY,1}, the axial yield strength of the dampers at the first story for X and Y directions,

_{s}N

_{yX,1}and

_{s}N

_{yY,1}, were obtained as follows. For X direction, ${}_{s}{}^{}Q{}_{yX,1}=2{}_{s}{}^{}N{}_{yX,1}\mathrm{cos}\alpha \mathrm{cos}\gamma $, from which

_{s}N

_{yX,1}= 17.72 kN. For Y direction, ${}_{s}{}^{}Q{}_{yY,1}={}_{s}{}^{}N{}_{yY,1}\left(\mathrm{cos}\theta +2\mathrm{cos}\beta \mathrm{cos}\gamma \right)$, thus N

_{yY,1}= 20.93 kN. The meaning of angles α, β, γ, θ is shown in Figure 8. Finally, a common value for the axial yield strength of the dampers installed in the first-story,

_{s}N

_{y,1}, was determined as

_{s}N

_{y,1}= min {

_{s}N

_{yX,1},

_{s}N

_{yY,1}} = 17.72 kN, in order to fulfill the criteria explained above. Furthermore, the yield interstory drift for the dampers,

_{s}δ

_{y,1}, was limited to 0.40

_{f}δ

_{y,1}on the basis of the results obtained by Oviedo et al. [9], where

_{f}δ

_{y,1}is the yield interstory drift of the WFP system (without dampers) at the first floor. The results of the test [7] and the pushover analyses conducted with the numerical model indicate that

_{f}δ

_{y,1}≈ 0.01h

_{p,1}= 14 mm for both X and Y directions, where h

_{p,1}is the height of the first story; hence, the value adopted for

_{s}δ

_{y,1}is

_{s}δ

_{y,1}= 0.40 × 0.01h

_{p,1}= 5.6 mm. Then, the axial yield relative displacement for the dampers,

_{s}Δu

_{y,1}, is obtained from

_{s}δ

_{y,1}in X direction, for which

_{s}N

_{y,1}was selected as ${}_{s}{}^{}\mathsf{\Delta}{u}_{y,1}={}_{s}{}^{}\delta {}_{y,1}\mathrm{cos}\gamma /\mathrm{cos}\alpha $, giving that

_{s}Δu

_{y,1}is 5.35 mm. Finally, the stiffness of the dampers is ${}_{s}{}^{}k{}_{y,1}={}_{s}{}^{}N{}_{y,1}/{}_{s}{}^{}\mathsf{\Delta}{u}_{y,1}=17.72/5.35=3.31$ kN/mm.

_{y,i}for i > 1, was determined for the purpose of preventing damage concentration. Here, damage is characterized through the parameter ${\eta}_{i}={W}_{p,i}/\left({Q}_{y,i}{\delta}_{y,i}\right)$, where W

_{p,i}is the energy dissipated by plastic deformations at the i story [15]. Q

_{y,i}, can be expressed as dimensionless by the yield-force coefficient, ${\alpha}_{i}$, defined as ${\alpha}_{i}={Q}_{y,i}/{\displaystyle \sum}_{k=i}^{N}{m}_{k}g$, where m

_{k}is the mass of the k story, N is the number of stories and g is the gravity acceleration. The distribution ${\overline{\alpha}}_{i}={\alpha}_{i}/{\alpha}_{1}$ that makes ${\eta}_{i}$ approximately equal in all stories is called optimum yield-shear strength coefficient distribution [15]. Different expressions for ${\overline{\alpha}}_{i}$ have been proposed in the literature [15,16,17,18]. In this study, the approach proposed by Japanese Building Code [18], ${\overline{\alpha}}_{JBC,i}$ is used, which is expressed as follows:

_{1}is the fundamental period of the structure without dampers. The eigenvalue analysis carried out in the numerical model of the WFP system showed that the fundamental periods along X and Y directions are T

_{X,1}= 0.32 s and T

_{Y,1}= 0.38 s, respectively. Therefore, by applying T

_{1,X}= 0.32 s and ${\overline{m}}_{2}=0.527$ in Equation (1) for the X direction, the value obtained for ${\overline{\alpha}}_{JBC,2}$ is equal to 1.278. Further, as the total shear strength at the first story along X direction, Q

_{yX,1}, is obtained as ${Q}_{yX,1}={}_{f}{}^{}Q{}_{yX,1}+{}_{s}{}^{}Q{}_{yX,1}=57.5+28.75=86.25$ kN, this implies that ${\alpha}_{1}$ = 86.25 · 10

^{3}N/[(6450 kg + 5780 kg)9.8 m/s

^{2}] = 0.7196. Then, the optimum shear strength coefficient at the second story in X direction is ${\overline{\alpha}}_{JBC,2}{\alpha}_{1}=0.92$ and the required shear strength is ${\overline{\alpha}}_{JBC,2}{\alpha}_{1}{\displaystyle \sum}_{k=2}^{2}{m}_{k}g=58.15$ kN. Following a similar procedure for Y direction, ${Q}_{yY,1}={}_{f}{}^{}Q{}_{yY,1}+{}_{s}{}^{}Q{}_{yY,1}=62.5+26.5=89$ kN, ${\alpha}_{1}=0.743$, ${\overline{\alpha}}_{JBC,2}=1.302$ (for T

_{Y,1}= 0.38 s), ${\overline{\alpha}}_{JBC,2}{\alpha}_{1}=0.97$ and finally ${\overline{\alpha}}_{JBC,2}{\alpha}_{1}{\displaystyle \sum}_{k=2}^{2}{m}_{k}g=61.3$ kN. In addition, the yield-shear strength at the second story of the WFP specimen along X and Y direction was obtained through a pushover analysis, obtaining that ${}_{f}{}^{}Q{}_{yX,2}\approx {}_{f}{}^{}Q{}_{yY,2}=138$ kN. Therefore, no dampers were considered at the second story, because the strength of the WFP system along X and Y directions (138 kN) exceeded the required strength for the story (i.e., 58.15 kN in the X direction and 61.3 kN in the Y direction). Using dampers only in specific stories (not in all stories) is a practice successfully employed in design [10].

#### 4.2. Numerical Model of WFP System with Dampers

_{s}N

_{y}= 17.72 kN and

_{s}k

_{y}= 3.31 kN/mm; the values of the other dimensionless parameters of the model are indicated in Table 1.

_{1}= 3.08 Hz and f

_{2}= 3.95 Hz. In order to keep the damping ratio of 3% in the second mode established in Section 2.3 for the WFP structure without dampers, the parameter a

_{0}was updated for the new frequency ω

_{2}= 2πf

_{2}= 2π3.95 = 24.82 rad/s, giving a

_{0}= 1.49. Figure 9 shows the mass-damping model of the WFP system both with and without dampers. According to the new damping model, the value of ξ

_{1}corresponding to the fundamental frequency f

_{1}= 3.08 Hz is ξ

_{1}= 3.8%. As the WFP system with dampers enters in plastic range, the fundamental frequency decreases. It is expected that its lowest value be higher than the obtained at the collapse in the tested structure without dampers, i.e., f

_{1}= 1.3 Hz. The value of ξ for f

_{1}= 1.3 Hz in the damping model for the WFP system with dampers corresponds to 9% (Figure 9), which is considered an upper bound for the damping in the numerical model.

#### 4.3. Numerical Analyses

_{70}to be applied to the accelerograms in order to achieve a total input energy—in terms of pseudo velocity V

_{E}= (2E

_{I}/M)

^{0.5}of V

_{E}= 70 cm/s—was between 1/3 and 3. V

_{E}= 70 cm/s is the input energy measured in the WFP specimen tested on the shake-table (Section 2.1) when it was on the brim of yield under uni- and bidirectional loadings [6,7]; it will be referred to as V

_{E,70}(=70 cm/s) herein. It was expected that by adding the hysteretic dampers, the input energy to produce plastic deformations in WFP structure would be higher than V

_{E,70}. The energy input in the X and Y directions at the onset of yielding of the WFP structure without dampers, i.e., when V

_{E,70}(=70 cm/s) is attained, will be respectively referred to as E

_{IX,70}and E

_{IY,70}, and the corresponding equivalent velocities are V

_{EX,70}= (2E

_{IX,70}/M)

^{0.5}and V

_{EY,70}= (2E

_{IY,70}/M)

^{0.5}. In general, V

_{EX,70}and V

_{EY,70}are different for each ground motion. The second criterion is that the PGA of the ground motion after scaling by SF

_{70}, be 0.3 g at most. This condition was imposed to avoid excessively large (i.e., unrealistic) values of PGA when the numerical model collapses. The records were classified in five sets, Set 1 to Set 5, according to the angle θ formed by the components X and Y of V

_{E}, $\theta =\mathrm{atan}\left({V}_{EY,70}/{V}_{EX,70}\right)$, as follows: 22.91° < θ < 32.65° for Set 1; 32.65° < θ < 42.40° for Set 2; 42.40° < θ < 52.14° for Set 3; 52.14° < θ < 61.88° for Set 4; and 61.88° < θ < 71.62° for Set 5. Seven records corresponding to different earthquakes were chosen within each set. The records are identified by the earthquake name followed by the record sequence number in the database (see Table A1, Table A2, Table A3, Table A4 and Table A5 in Appendix A for details).

_{70}were applied simultaneously to the numerical model in successive seismic simulations using a sequence of scaling factors, 100%, 200%, 300%, etc., until the seismic performance level (SPL) of Near Collapse (NC) was achieved. That is, 100% meant that the scaling factor applied to the original ground motion was SF

_{70}, for 200% the scaling factor was 2×SF

_{70}, and so on. In addition, within each set, one ground motion was selected, and the X and Y components were applied separately to the numerical model until failure, so as to investigate the ultimate capacity under unidirectional seismic loads. In total, 45 NLTHAs were launched in a parallel scheme (OpenseesMP.exe) using a Dell Precision Tower 5810 with 12 cores at 3.60 GHz in order to minimize the computational time. As a reference, the mean computational time required was 40 h per core and record.

_{i}, defined as the ratio between the interstory drift and the story height, is used here as the engineering parameter to establish the upper limit of IDI

_{i}for the different SPLs defined above, denoted as IDI

_{FO,i}, IDI

_{OP,i}and IDI

_{NC,i}, respectively. Therefore, for SPL FO, IDI

_{FO,1}= 0.4% (

_{s}δ

_{y,1}= 0.4

_{f}δ

_{y,1}and

_{f}δ

_{y,1}= 0.01h

_{p}) and IDI

_{FO,2}= 1% for the second story, not having dampers. For SPL OP, the reference was the yield-displacement of the WFP system, thus IDI

_{OP,i}= 1% for both stories. Finally, for SPL NC and according to the results obtained in the literature [6,7], the limit was established at IDI

_{NC,i}= 2.6% for both stories.

## 5. Results: Seismic Capacity of a WFP Structure Upgraded with Hysteretic Dampers

#### 5.1. Input Energy and Dissipated Energy

_{E}can be simply calculated from the energy input in the X, E

_{IX}, and in the Y, E

_{IY}, directions by V

_{E}= (V

_{EX}

^{2}+V

_{EY}

^{2})

^{0.5}, where V

_{EX}= (2E

_{IX}/M)

^{0.5}, V

_{EY}= (2E

_{IY}/M)

^{0.5}. V

_{E}can be interpreted as the modulus of a vector in the V

_{EX}-V

_{EY}plane. The same can be applied to the total energy that contributes to damage, E

_{D}, defined [15] as E

_{I}minus the energy dissipated by inherent damping E

_{ξ}, i.e., E

_{D}= E

_{I}− E

_{ξ}. Given the energy balance of the structure [15], it follows that E

_{D}equals the sum of the elastic vibrational energy E

_{e}and the energy dissipated through plastic deformations (hysteretic energy) E

_{h}, i.e., E

_{D}=E

_{e}+E

_{h}. For high levels of plastic deformations E

_{e}becomes negligible in comparison with E

_{h}[15], which leads to E

_{D}≈ E

_{h}. The energy that contributes to damage can be calculated independently in the X and Y directions, E

_{DX}and E

_{DY}, and is nearly equal to the energy dissipated in the X and Y directions, E

_{hX}and E

_{hY}, i.e., E

_{DX}≈ E

_{hX}and E

_{DY}≈ E

_{hY}. The corresponding equivalent velocities are V

_{DX}= (2E

_{DX}/M)

^{0.5}, V

_{DY}= (2E

_{DY}/M)

^{0.5}, and V

_{D}= (V

_{DX}

^{2}+V

_{DY}

^{2})

^{0.5}. Based on the above considerations, the values of V

_{D}, V

_{DX}, and V

_{DY}when the structure is near collapse can be interpreted as the ultimate energy dissipation capacity of the structure in the form of plastic deformations, expressed in terms of equivalent velocities.

_{E}(defined by its components V

_{EX}and V

_{EY}) obtained through numerical simulations using the model described in Section 4.2, subjected to the five sets of bidirectional ground motion records explained in Section 4.3. The results of each set (Set 1 to Set 5) are identified in the figure with different colors. Additionally plotted are the V

_{E}’s obtained with the numerical model subjected to unidirectional ground motions (referred to as Set X and Set Y); they are identified in the legend with the letter X or Y added to the record name. The mean $\overline{x}$ (red dash line) and mean plus/minus one standard deviation σ (blue dash line) are also depicted and identified with V

_{E,FO}, V

_{E,OP}and V

_{E,NC}in Figure 10.

_{I}), the increase is more than sevenfold, i.e., 7.29 (= 2.70

^{2}).

_{E,FO}, which are greater for V

_{E,OP}and especially for V

_{E,NC.}, with COVs around 0.30. Furthermore, the differences between the input energy achieved for the different SPLs under each set of ground motions were analyzed by means of the ratios V

_{E,OP}/V

_{E,FO}and V

_{E,NC}/V

_{E,FO}. Moderate differences could be seen for V

_{E,OP}/V

_{E,FO}, with a mean of 3.22. For V

_{E,NC}/V

_{E,FO}the mean increases to 8.82. The highest values for the latter are found under Set X, Set 1 and especially Set 2 (V

_{E,NC}/V

_{E,FO}= 12.23). The structure under Sets 4, 5, and Y showed the highest values for the elastic input energy V

_{E,FO}, while the trend was reversed for V

_{E,NC}(i.e., when the structure undergoes plastic deformations), giving the highest values under Sets X, 1, and 2. Therefore, a reduction of the seismic capacity of the structure under plastic deformations is observed when the Y component of the seismic action is the strongest. It is worth noting that the structure analyzed in this study presents eccentricities of the center of mass with respect to the center of stiffness in the Y direction.

_{D,OP}and V

_{D,NC}, respectively. The same representation criteria as in Figure 10a were used. The average values shown in the Figures with red dashed lines are ${\overline{V}}_{D,OP}=77$ cm/s and ${\overline{V}}_{D,NC}=241$ cm/s, with standard deviations equal to 22 cm/s and 69 cm/s, respectively. A more detailed analysis of results regarding the energy that contributed to damage is given in Table 3. As seen, under the sets of ground motions X, 1 and 2, the structure shows the highest values for the capacity to dissipate energy by plastic deformation, as with input energy. Moreover, this capacity is on average more than threefold, in terms of equivalent velocity, for SPL NC than for SPL OP, exhibiting the highest differences under unidirectional seismic loadings (Sets X and Y) and the bidirectional seismic loadings with Sets 1 and 2 for which the X component of the seismic action is the strongest.

_{D,OP}/V

_{E,OP}and V

_{D,NC}/V

_{E,NC}obtained for the different set of records for SPL OP and NC, respectively. The mean of V

_{D,OP}/V

_{E,OP}under each set of records was in a range between 0.66 and 0.72, the absolute mean—taking into account all the records—being equal to 0.70. For V

_{D,NC}/V

_{E,NC}, a range between 0.76 and 0.82 was obtained and the absolute mean was 0.80. Two aspects of these results deserve special attention. The first one is the low dispersion achieved for both cases, with COV equal to 0.11 at the most for V

_{D,OP}/V

_{E,OP}and 0.04 for V

_{D,NC}/V

_{E,NC}. The second noteworthy finding is: the stronger the X component of the seismic action applied along a symmetrical direction, the higher the aforementioned relationships, and thus the capacity to dissipate the energy through plastic deformations.

_{D,OP}/V

_{E,OP}was 0.70 (COV = 0.09) and the mean of V

_{D,NC}/V

_{E,NC}was 0.80 (COV = 0.04). Numerous expressions are proposed in the literature to estimate V

_{D}/V

_{E}. Among them, the one proposed by Akiyama [15] is expressed as follows:

_{D}/V

_{E}= 0.74, which is very close to the value obtained, on average, in the analyses for V

_{D,OP}/V

_{E,OP}(=0.70). Nevertheless, for SPL NC both the WFP system and the dampers undergo plastic deformations. This means that higher values are expected for the modified frequency and damping ratio. Substituting the damping ratio 0.09 considered in Section 4.2 for the WFP system with dampers when it is near collapse, Equation (2) gives V

_{D}/V

_{E}= 0.61, far from the value obtained from the analysis, (V

_{D,NC}/V

_{E,NC}= 0.80). The well-known reason [15] is that, for the sake of simplicity, Equation (2) ignores the fact that for a large level of plastic deformation (as is the case of SPL NC), the amount of energy dissipated by damping tends to decrease, and this enlarges V

_{D}/V

_{E}. Akiyama’s Equation (2) therefore provides a reasonable approximation for SPL OP—damage only in dampers—but underestimates the V

_{D}/V

_{E}for SPL NC damage to both the WFP system and the dampers.

#### 5.2. Differences for the Seismic Capacity Under Unidirectional and Bidirectional Loadings

_{E}and 70% to 80% for V

_{D}. The same trend is observed for SPL NC, but with higher values—86% to 89% for V

_{E}, and 78% to 88% for V

_{D}. Similar results were obtained by Rodrigues H. et al. in columns subjected to biaxial cyclic loadings and variable axial loads [27]. Yet, there are exceptions, such as the response under the Y component of Kobe-1115 or the X component of Landers-888 in SPL NC, which show values of V

_{E}and V

_{D}that are higher than those under bidirectional seismic action (about 50% higher for Kobe-115 and about 20% greater for Landers-888). The limited number of records used in the present study impedes the formulation of a general statement about the higher capacity of structures under bidirectional loading versus unidirectional loads. It is worth noting that the columns of the WFP system are prone to undergoing variable axial loads under both unidirectional and bidirectional cyclic loadings due to the scheme of the structure itself—three columns. This could explain the results obtained.

_{E,SPL}and V

_{D,SPL}. For SPL OP the differences, on average, are about 18% higher with COV = 0.07. For SPL NC the differences are greater, being about 27% higher with COV = 0.20.

#### 5.3. Ductility Level for Seismic Performance Levels (SPL) Near Collapse (NC)

_{f}μ

_{X,i}and

_{f}μ

_{Y,i}, and for the dampers (the stiff part),

_{s}μ

_{X,i}and

_{s}μ

_{Y,i}, respectively. For the flexible part (WFP system),

_{f}μ

_{X,i}is defined as ${}_{f}{}^{}\mu {}_{X,i}=\left({\delta}_{maxX,i}-{}_{f}{}^{}\delta {}_{yX,i}\right)/{}_{f}{}^{}\delta {}_{yX,i}$, where δ

_{maxX,i}and

_{f}δ

_{yX,i}are the maximum interstory drift—measured at the center of stiffness of the story—and the yield interstory drift of the main structure for the i story, respectively, along the X direction, and

_{f}μ

_{Y,i}it is as ${\mu}_{Y,i}=\left({\delta}_{maxY,i}-{}_{f}{}^{}\delta {}_{yY,i}\right)/{}_{f}{}^{}\delta {}_{yY,i}$, where δ

_{maxY,i}and

_{f}δ

_{yY,i}are the counterpart variables defined above but along the Y direction. For the stiff part (dampers), the same definition is used for

_{s}μ

_{X,i}and

_{s}μ

_{Y,i}, changing the subindex ‘f’ to ‘s’, and using

_{s}δ

_{yX,i}and

_{s}δ

_{yY,i}for the yield interstory drift of the dampers along X and Y directions instead of

_{f}δ

_{yX,i}and

_{f}δ

_{yY,i}.

_{1}exceeded IDI

_{max}= 2.6% in any direction. In contrast, for the second story, the low values achieved for δ

_{maxX,2}and δ

_{maxY,2}led to values about zero for both

_{f}μ

_{X,2}and

_{f}μ

_{Y,2}. Taking into account that

_{f}δ

_{yX,1}=

_{f}δ

_{yY,1}= 14 mm,

_{s}δ

_{yX,1}=

_{s}δ

_{yY,1}= 5.6 mm and IDI

_{max}= IDI

_{max,NC}= 2.6%, the maximum apparent plastic deformation for the WFP system and for hysteretic dampers would be

_{f}μ

_{max,1}= 1.6 and

_{s}μ

_{max,1}= 5.5, respectively. This means that for the structure investigated in this study, the ductility level in dampers is about triple the one considered for the WFP system. Figure 12 offers the mean of the maximum apparent plastic deformation achieved in the WFP system (Figure 12a) and in hysteretic dampers (Figure 12b) under the different sets of ground motion records for each direction. It is seen that under Sets X, 1, and 2, SPL NC is achieved under the X direction for which the input energy of the X component of the seismic action is the highest, because the mean of

_{f}μ

_{X,1}and

_{s}μ

_{X,1}correspond to

_{f}μ

_{max,1}and

_{s}μ

_{max,1}, respectively. The opposite occurs under Sets 4, 5, and 7. Nevertheless, under Set 3 there is no specific direction for which SPL NC is attained.

#### 5.4. Energy Dissipated by Plastic Deformations for SPL NC

_{p,i}, is obtained by adding the damage in the flexible part (WFP system),

_{f}W

_{p,i}, and that of the stiff part (hysteretic dampers),

_{s}W

_{p,i}, i.e., ${W}_{p,i}={}_{f}{}^{}W{}_{p,i}+{}_{s}{}^{}W{}_{p,i}$. Further, W

_{p,i},

_{f}W

_{p,i}and

_{s}W

_{p,i}can be expressed from their X and Y components as ${W}_{p,i}={W}_{pX,i}+{W}_{pY,i}$, ${}_{f}{}^{}W{}_{p,i}={}_{f}{}^{}W{}_{pX,i}+{}_{f}{}^{}W{}_{pY,i}$ and ${}_{s}{}^{}W{}_{p,i}={}_{s}{}^{}W{}_{pX,i}+{}_{s}{}^{}W{}_{pY,i}$. Then, the following relationships are deduced: ${W}_{pX,i}={}_{f}{}^{}W{}_{pX,i}+{}_{s}{}^{}W{}_{pX,i}$ and ${W}_{pY,i}={}_{f}{}^{}W{}_{pY,i}+{}_{s}{}^{}W{}_{pY,i}$. The dissipated energy of the main structure at the i story along X and Y directions under a given seismic record is obtained as follows: (i) first, the contribution of the j column to the dissipated energy along the X and Y directions is obtained through the expressions ${}_{f}{}^{}W{}_{pX,ij}={\displaystyle \int}{}_{f}{}^{}Q{}_{X,ij}d{\delta}_{X,ij}$ and ${}_{f}{}^{}W{}_{pY,ij}={\displaystyle \int}{}_{f}{}^{}Q{}_{Y,ij}d{\delta}_{Y,ij}$, respectively, where

_{f}Q

_{X,ij}and

_{f}Q

_{Y,ij}are the shear force histories of the j column along X and Y directions, and δ

_{X,ij}and δ

_{Y,ij}are the counterpart interstory drift histories; (ii) next,

_{f}W

_{pX,i}and

_{f}W

_{pY,i}are obtained by adding the contribution of the dissipated energy of all the columns of the story along X and Y directions, i.e., ${}_{f}{}^{}W{}_{pX,i}={\displaystyle \sum}_{j=1}^{Nc}{}_{f}{}^{}W{}_{pX,ij}$ and ${}_{f}{}^{}W{}_{pY,i}={\displaystyle \sum}_{j=1}^{Nc}{}_{f}{}^{}W{}_{pY,ij}$, where N

_{c}is the number of columns at the i story. The dissipated energy of the dampers at the i story is obtained by adding the contribution of all of them along the X and Y directions. The WFP system considered in this research was upgraded with dampers located only at the first story, arranged as indicated in Figure 8. The energy dissipated by a k damper under a seismic record,

_{s}W

_{p,ik}, is obtained through the expression ${}_{s}{}^{}W{}_{p,ik}={\displaystyle \int}{}_{s}{}^{}N{}_{k}d({}_{s}{}^{}\mathsf{\Delta}{u}_{k})$, where

_{s}N

_{k}and

_{s}Δu

_{k}are respectively the axial force history and the relative axial displacement history of the damper. Damper 3 is aligned in Y direction, hence its contribution to the dissipated energy for the first story is entirely accounted for in

_{s}W

_{pY,1}. Dampers 1 and 2 contribute to dissipating energy in both X and Y directions (Figure 8); their contributions

_{s}W

_{pX,1}and

_{s}W

_{pY,1}are ${}_{s}{}^{}W{}_{p,1k}{\left(\mathrm{cos}\left(\alpha \right)\right)}^{2}$ and ${}_{s}{}^{}W{}_{p,1k}{\left(\mathrm{sin}\left(\alpha \right)\right)}^{2}$, respectively.

_{p,1}; this stands as a moderate contribution and means that the dampers are effectively controlling (i.e., limiting) damage to the main structure.

_{,i}defined in Section 4.1. Accordingly, for the WFP system, the damage along X and Y directions is respectively expressed as ${}_{f}{}^{}\eta {}_{X,i}={}_{f}{}^{}W{}_{pX,i}/\left({}_{f}{}^{}Q{}_{yX,1}{}_{f}{}^{}\delta {}_{yX,i}\right)$ and ${}_{f}{}^{}\eta {}_{Y,i}={}_{f}{}^{}W{}_{pY,i}/\left({}_{f}{}^{}Q{}_{yY,1}{}_{f}{}^{}\delta {}_{yY,i}\right)$. For the hysteretic dampers, the counterpart expressions are ${}_{s}{}^{}\eta {}_{X,i}={}_{s}{}^{}W{}_{pX,i}/\left({}_{s}{}^{}Q{}_{yX,1}{}_{s}{}^{}\delta {}_{yX,i}\right)$ and ${}_{s}{}^{}\eta {}_{Y,i}={}_{s}{}^{}W{}_{pY,i}/\left({}_{s}{}^{}Q{}_{yY,1}{}_{s}{}^{}\delta {}_{yY,i}\right)$. Moreover, according to De Stefano and Faella [38], the damage to the main part and the stiff part at the i story can be expressed as ${}_{f}{}^{}\eta {}_{i}={}_{f}{}^{}\eta {}_{X,i}+{}_{f}{}^{}\eta {}_{Y,i}$ and ${}_{s}{}^{}\eta {}_{i}={}_{s}{}^{}\eta {}_{X,i}+{}_{s}{}^{}\eta {}_{Y,i}$, respectively. Figure 14a,b show

_{f}η

_{X,i},

_{f}η

_{Y,i}and

_{f}η

_{i}for the first (i = 1) and second story (i = 2), respectively. Higher values are seen for

_{f}η

_{1}than for

_{f}η

_{2}. This result is a consequence of the greater strength of the WFP system in the second story with respect to the value provided by an optimum distribution (Section 4.1).

_{f}η

_{X,1}or

_{f}η

_{Y,1}, respectively. The same response is observed for the hysteretic dampers in Figure 14c. Nevertheless, the efficiency exhibited by dampers in the X direction under ground motion records of large intensity for the X component is higher than that of the Y direction under seismic actions having large intensity for the Y component; this is reflected by the comparatively higher values observed for

_{s}η

_{X,1}under Sets X, 1, and 2 as opposed to those for

_{s}η

_{Y,1}under Sets 4, 5 and Y.

_{k}(k varies from 1 to 3), achieved under each set of records, as well as the maximum damage index achieved for them under each set of records, ID

_{max}. It is seen that ID

_{max}ranges from 0.07 (Set 3) up to 0.25 (Set Y), indicating that SPL NC is achieved in conjunction with the failure of the WFP system, but not by failure of the dampers. The mean ID

_{k}values obtained for the different dampers result from their distribution at the first story: (i) along X direction, dampers 1 and 2 are in symmetric layout, from the base of columns 2 and 3 to the outer column-plate connection; (ii) along Y direction, a non-symmetric layout has damper 3 located between columns 2 and 3, and aligned in this direction. Therefore, ID

_{1}and ID

_{2}show, on average, similar values under the different record sets, whereas ID

_{3}shows higher values under the sets with large intensity in the Y component. This means that the arrangement of dampers in the story is a key issue for preventing an uneven distribution of damage among them.

#### 5.5. Equivalent Number of Cycles for SPL NC

_{i}; the other is the hysteretic energy dissipated through cumulative cyclic reversals of each story i expressed through the dimensionless variable η

_{i}. The ratio η

_{i}/µ

_{i}is another key aspect of the seismic response of the structure, characterizing the efficiency of the structure in dissipating energy [15]. The ratio η

_{i}/µ

_{i}is referred in the literature as the equivalent number of cycles ${n}_{eq,i}\left(={\eta}_{i}/{\mu}_{i}\right)$ and is influenced by the characteristics of the ground motion and the type of structure [43]. In flexible-stiff mixed structures, this ratio can be defined for the main structure, ${}_{f}{}^{}n{}_{eq,i}={}_{f}{}^{}\eta {}_{i}/{}_{f}{}^{}\mu {}_{i}$, and for the stiff part, ${}_{s}{}^{}n{}_{eq,i}={}_{s}{}^{}\eta {}_{i}/{}_{s}{}^{}\mu {}_{i}$.

_{f}n

_{eq,1}and

_{s}n

_{eq,1}obtained in the analyses for SPL NC at the first story along the X and Y directions, i.e., ${}_{f}{}^{}n{}_{eqX,i}={}_{f}{}^{}\eta {}_{X,i}/{}_{f}{}^{}\mu {}_{X,i}$ and ${}_{f}{}^{}n{}_{eqY,i}={}_{f}{}^{}\eta {}_{Y,i}/{}_{f}{}^{}\mu {}_{Y,i}$ for the WFP system (Figure 15a), and ${}_{s}{}^{}n{}_{eqX,i}={}_{s}{}^{}\eta {}_{X,i}/{}_{s}{}^{}\mu {}_{X,i}$ and ${}_{s}{}^{}n{}_{eqY,i}={}_{s}{}^{}\eta {}_{Y,i}/{}_{s}{}^{}\mu {}_{Y,i}$ for the hysteretic dampers (Figure 15b). It is important to stress that the components X and Y used to calculate

_{f}n

_{eq,1}and

_{s}n

_{eq,1}were those for which the maximum ductility levels, ${}_{f}{}^{}\mu {}_{max,1}$ and ${}_{s}{}^{}\mu {}_{max,1}$, were achieved under each seismic record. Figure 15 also shows

_{f}n

_{eq,min1}and

_{s}n

_{eq,min1}, respectively, the minimum values achieved for

_{f}n

_{eq,1}and

_{s}n

_{eq,1}in the analysis under each set of records.

_{f}n

_{eqX,1}ranges on average between 1 and 1.5, with a mean of 1.39 and COV = 0.24; in turn

_{f}n

_{eqY,1}ranges between 1.5 and 2.5, with a mean of 2.0 and COV = 0.28. Further, it can be observed that

_{f}n

_{eq,min1}ranges between 0.53 (set X) and 1.66 (set Y), with a mean of 1.20 and COV = 0.29. These values obtained for

_{f}n

_{eq,1}are similar to the proposal by Akiyama,

_{f}n

_{eq}= 2 for flexible-stiff mixed systems whose flexible part exhibits pinching (as is the case of WFP systems) [15].

_{s}n

_{eqX,1}ranges on average between 7 and 35, with a mean about 24 and COV = 0.60.

_{s}n

_{eqY,1}ranges between 10 and 20, with a mean about 15 and COV = 0.59. The higher values for both

_{s}n

_{eqX,1}and

_{s}n

_{eqY,1}are concentrated under the sets of records for which the correspondent seismic component (X or Y) is large, i.e., Sets X, 1, and 2 for the former, and Sets 5 and Y for the latter. Furthermore,

_{s}n

_{eq,min1}is found to range between 3.1 (Set 4) and 10.72 (Set X), with a mean of 6.36 and COV = 0.39. For flexible-stiff systems whose stiff parts exhibit an elastic-perfectly plastic behavior, Akiyama [15] proposed

_{s}n

_{eq}= 8. This value is slightly higher than

_{s}n

_{eq,min1}, but close to the lower limit of the range of both

_{s}n

_{eqX,1}and

_{s}n

_{eqY,}indicated above.

_{f}n

_{eq,1}and

_{s}n

_{eq,1}come from the successively scaled seismic simulations until achieving SPL NC. Therefore, they can be used to obtain the cumulative damage in WFP structures with hysteretic dampers subjected to shocks and aftershocks.

## 6. Conclusions

- The total input energy or the total hysteretic energy—expressed in the form of equivalent velocities V
_{E}or V_{D}—required to attain SPLs FO, OP, and NC remains basically constant, irrespective of the ground motion considered when the two horizontal components of the ground motion are simultaneously applied. - The V
_{E}that the structure can endure until SPL NC was, on average, about three times the value obtained SPL OP, and eight times the value for OP and FO. Meanwhile, the V_{D}when the structure reaches SPL NC was, on average, about three times that obtained for SPL OP. - Past studies on conventional structures pointed out that torsion effects tend to redistribute the damage in the structure, which in terms of energy means to balance the energy input by the X and Y components of the ground motion. This was not found for the structure under study, most probably due to the fact that the dampers controlled the torsional movements.
- The capacity of the structure under bidirectional loadings is in most cases slightly higher than under unidirectional loadings. Nevertheless, for some ground motions the response is opposite.
- The values of V
_{E}and V_{D}estimated from the energies obtained independently for the X and Y components of the ground motion (unidirectional analysis) are always larger than the actual value obtained applying the two components simultaneously (bidirectional analysis). The values are 18% larger for SPL OP and 27% larger for SPL NC. - The relationship V
_{D}/V_{E}obtained for SPL OP (0.70 with COV = 0.09) was very similar to that predicted with Akiyama’s Equation (2) (0.74). Nevertheless, for SPL NC, the value obtained (0.80) was markedly higher than that provided by Akiyama’s equation (0.61). - The maximum ductility level, estimated through the maximum apparent plastic deformation, was 1.6 in the WFP system and 5.5 in SPDs.
- Most of the energy (80%) input and dissipated by the structure was absorbed by the dampers.
- The equivalent number of cycles obtained for the flexible and stiff parts would rely on the characteristics of the main structure and the layout of dampers in X and Y directions. The average and minimum values obtained are
_{f}n_{eq,1}= 2 and_{f}n_{eq,min1}= 1.20, respectively, for the flexible part;_{s}n_{eq,1}ranged between 15 and 24, with a minimum value of_{s}n_{eq,min1}= 6.36. The values found here are in accordance with those proposed by Akiyama [15].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

Earthquake Name | Year | Sta. Name (Record Seq. No.) | M_{w} | SF_{70} | PGA_{X}(g) | PGA_{Y}(g) | V_{EX}(cm/s) | V_{EY}(cm/s) | θ (°) |
---|---|---|---|---|---|---|---|---|---|

Manjil (Irán) | 1990 | Qazvin (1636) | 7.37 | 1.05 | 0.18 | 0.13 | 57.94 | 32.84 | 29.54 |

Cape Mendocino (USA) | 1992 | Eur.Myrtle&West (826) | 7.01 | 1.07 | 0.15 | 0.18 | 57.15 | 32.41 | 29.56 |

San Fernando (USA) | 1971 | Hollywood Stor FF (68) | 6.61 | 1.12 | 0.21 | 0.17 | 52.63 | 33.57 | 32.53 |

Sierra Madre (USA) | 1991 | LA-City Terrace (1643) | 5.61 | 1.70 | 0.11 | 0.09 | 35.24 | 21.37 | 31.23 |

Chi-Chi (Taiwan) | 1999 | TAP051 (1435) | 7.62 | 1.74 | 0.11 | 0.06 | 35.08 | 19.89 | 29.55 |

Imperial Valley (USA) | 1979 | Niland Fire Sta. (186) | 6.53 | 1.77 | 0.11 | 0.07 | 33.29 | 21.19 | 32.48 |

Landers (USA) | 1992 | Anaheim-WB Rd (833) | 7.28 | 2.69 | 0.05 | 0.04 | 22.08 | 13.76 | 31.94 |

Earthquake Name | Year | Sta. Name (Record Seq. No.) | M_{w} | SF_{70} | PGA_{X}(g) | PGA_{Y}(g) | V_{EX}(cm/s) | V_{EY}(cm/s) | θ (°) |
---|---|---|---|---|---|---|---|---|---|

Chi-Chi (Taiwan) | 1999 | TCU050 (1490) | 7.62 | 1.00 | 0.15 | 0.13 | 53.39 | 45.78 | 40.62 |

Northridge (USA) | 1994 | Elizabeth Lake (971) | 6.69 | 0.98 | 0.15 | 0.11 | 57.76 | 41.97 | 36.00 |

Hector Mine (USA) | 1999 | Amboy (1762) | 7.13 | 0.95 | 0.18 | 0.15 | 59.38 | 43.37 | 36.15 |

Duzce (Turkey) | 1999 | Mudumu (1619) | 7.14 | 1.06 | 0.12 | 0.06 | 53.85 | 38.40 | 35.50 |

Morgan Hill (USA) | 1984 | Gilroy Array 7 (460) | 6.19 | 0.89 | 0.19 | 0.11 | 60.11 | 50.94 | 40.28 |

Imperial Valley (USA) | 1979 | EL Centro 12 (175) | 6.53 | 1.18 | 0.14 | 0.12 | 46.25 | 37.23 | 38.83 |

Loma Prieta (USA) | 1989 | Intern. Airport (799) | 6.93 | 0.59 | 0.24 | 0.33 | 91.34 | 77.28 | 40.24 |

Earthquake Name | Year | Sta. Name (Record Seq. No.) | M_{w} | SF_{70} | PGA_{X}(g) | PGA_{Y}(g) | V_{EX}(cm/s) | V_{EY}(cm/s) | θ (°) |
---|---|---|---|---|---|---|---|---|---|

Chi-Chi (Taiwan) | 1999 | TCU107 (1534) | 7.62 | 1.01 | 0.12 | 0.16 | 45.00 | 53.05 | 49.69 |

Kobe (Japan) | 1995 | Sakai (1115) | 6.90 | 0.99 | 0.16 | 0.12 | 50.13 | 50.13 | 45.00 |

Northridge (USA) | 1994 | Brentwood-VAHosp. (986) | 6.69 | 1.03 | 0.19 | 0.16 | 43.79 | 51.58 | 49.67 |

Landers (USA) | 1992 | Desert Hot Springs (850) | 7.28 | 0.94 | 0.17 | 0.15 | 51.91 | 53.78 | 46.02 |

Hector Mine (USA) | 1999 | Baker Fire Sta. (1766) | 7.13 | 1.25 | 0.13 | 0.09 | 36.32 | 42.60 | 49.55 |

Coalinga (USA) | 1983 | Parkfield VinC2W (362) | 6.36 | 1.59 | 0.07 | 0.08 | 27.53 | 34.44 | 51.37 |

Whittier Narrows (USA) | 1987 | Panorama City-Roscoe (673) | 5.99 | 1.65 | 0.10 | 0.11 | 29.51 | 30.61 | 46.04 |

Earthquake Name | Year | Sta. Name (Record Seq. No.) | M_{w} | SF_{70} | PGA_{X}(g) | PGA_{Y}(g) | V_{EX}(cm/s) | V_{EY}(cm/s) | θ (°) |
---|---|---|---|---|---|---|---|---|---|

Superstition Hills (USA) | 1987 | Wild Life Liq. Ar (729) | 6.54 | 0.84 | 0.18 | 0.21 | 43.34 | 71.56 | 58.80 |

Irpinia (Italy) | 1980 | Calitri (289) | 6.90 | 0.82 | 0.13 | 0.18 | 44.95 | 72.46 | 58.18 |

Kocaeli (Turkey) | 1999 | Goynuk (1162) | 7.51 | 1.26 | 0.13 | 0.12 | 34.03 | 44.19 | 52.40 |

Hector Mine (USA) | 1999 | San Bernardino – Mont.M (1829) | 7.13 | 1.28 | 0.09 | 0.13 | 29.25 | 46.39 | 57.77 |

Coalinga (USA) | 1983 | Parkfield – Gold Hill 3W (352) | 6.36 | 1.55 | 0.14 | 0.12 | 24.03 | 38.40 | 57.96 |

Landers (USA) | 1992 | San Bernardino – E&Hosp. (888) | 7.28 | 1.48 | 0.08 | 0.09 | 27.36 | 38.49 | 54.59 |

Friuli (Italy) | 1976 | Tolmezzo (125) | 6.50 | 0.56 | 0.35 | 0.31 | 64.73 | 107.60 | 58.97 |

Earthquake Name | Year | Sta. Name (Record Seq. No.) | M_{w} | SF_{70} | PGA_{X}(g) | PGA_{Y}(g) | V_{EX}(cm/s) | V_{EY}(cm/s) | θ (°) |
---|---|---|---|---|---|---|---|---|---|

Hector Mine (USA) | 1999 | Joshua Tree (1794) | 7.13 | 0.79 | 0.15 | 0.19 | 36.78 | 80.52 | 65.45 |

Chi-Chi (Taiwan) | 1999 | ILA066 (1349) | 7.62 | 1.28 | 0.08 | 0.10 | 19.92 | 51.12 | 68.71 |

Chalfant Valley (USA) | 1986 | Bishop – LADWP (549) | 6.19 | 0.75 | 0.25 | 0.17 | 33.97 | 86.80 | 68.62 |

Landers (USA) | 1992 | Featherly Park –Maint (854) | 7.28 | 1.68 | 0.05 | 0.05 | 18.72 | 37.18 | 63.28 |

Northridge (USA) | 1994 | Seal Beach – Off Bldg. (1079) | 6.69 | 1.84 | 0.06 | 0.08 | 17.16 | 33.99 | 63.21 |

Chi-Chi (Taiwan) | 1999 | ILA036 (1328) | 7.62 | 2.54 | 0.06 | 0.07 | 9.82 | 25.75 | 69.12 |

Loma Prieta (USA) | 1989 | Gilroy Array 1 (765) | 6.93 | 0.36 | 0.41 | 0.47 | 79.93 | 179.16 | 65.96 |

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**Figure 4.**Top displacement history: (

**a**) for the successive simulations in X and Y directions; (

**b**) for C100 in X direction; (

**c**) for C100 in Y direction.

**Figure 8.**Layout of dampers at the first story: (

**a**) elevation of the installation scheme of damper 3; (

**b**) plan view; (

**c**) elevation of the installation scheme of dampers 1 and 2.

**Figure 10.**Energy capacity of the structure: input energy for seismic performance levels (SPL) FO (

**a**), SPL OP (

**b**) and SPL NC (

**d**); hysteretic energy for SPL OP(

**c**) and SPL NC (

**e**).

**Figure 12.**Mean and error bars for

_{f}μ along X and Y directions: (

**a**) main structure; (

**b**) hysteretic dampers.

**Figure 13.**Ratio of the dissipated energy in the hysteretic dampers with respect to the total value at the first story, overall (${}_{s}{}^{}W{}_{p,1}/{W}_{p,1}$), and by components (${}_{s}{}^{}W{}_{pX,1}/{W}_{pX,1}$ and ${}_{s}{}^{}W{}_{pY,1}/{W}_{pY,1}$ ).

**Figure 14.**Dimensionless representation of the damage: for WFP system at the first story (

**a**) and at second story (

**b**); for hysteretic dampers, damage level, (

**c**) and damage index (

**d**).

**Figure 15.**Equivalent number of cycles obtained in the analyses at the first story for WFP system (

**a**) and for hysteretic dampers (

**b**).

**Table 1.**Parameters of the constitutive model of a slit-plate damper (SPD) calibrated with an experimental test.

Uniaxial Material | _{s}N_{y}[kN] | _{s}k[kN/mm] | b | R0 | cR1 | cR2 | a1 | a2 | a3 | a4 | sigInit |
---|---|---|---|---|---|---|---|---|---|---|---|

Steel02 | 16 | 100 | 0.005 | 22.0 | 0.925 | 0.15 | 0.13 | 1.0 | 0.13 | 1.0 | 0.0 |

Set | mean V_{E,FO}(cm/s) | COV V_{E,FO} | mean V_{E,OP}(cm/s) | COV V_{E,OP} | mean V_{E,NC}(cm/s) | COV V_{E,NC} | mean $\frac{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}}{{\mathit{V}}_{\mathit{E},\mathit{F}\mathit{O}}}$ | COV $\frac{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}}{{\mathit{V}}_{\mathit{E},\mathit{F}\mathit{O}}}$ | mean $\frac{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}}{{\mathit{V}}_{\mathit{E},\mathit{F}\mathit{O}}}$ | COV $\frac{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}}{{\mathit{V}}_{\mathit{E},\mathit{F}\mathit{O}}}$ |
---|---|---|---|---|---|---|---|---|---|---|

X | 30.36 | 0.25 | 98.58 | 0.33 | 294.89 | 0.28 | 3.28 | 0.21 | 10.15 | 0.31 |

1 | 32.15 | 0.15 | 104.14 | 0.17 | 289.98 | 0.30 | 3.29 | 0.18 | 9.19 | 0.31 |

2 | 32.94 | 0.29 | 127.21 | 0.29 | 391.20 | 0.24 | 4.00 | 0.29 | 12.23 | 0.23 |

3 | 38.64 | 0.20 | 136.04 | 0.16 | 301.72 | 0.12 | 3.64 | 0.24 | 8.04 | 0.19 |

4 | 40.87 | 0.35 | 113.43 | 0.40 | 255.92 | 0.24 | 2.92 | 0.32 | 6.69 | 0.26 |

5 | 40.85 | 0.36 | 92.30 | 0.26 | 278.61 | 0.16 | 2.44 | 0.24 | 7.55 | 0.30 |

Y | 36.77 | 0.25 | 105.36 | 0.27 | 293.24 | 0.31 | 2.93 | 0.25 | 8.02 | 0.19 |

All records | 36.31 | 0.31 | 111.81 | 0.31 | 301.39 | 0.28 | 3.22 | 0.30 | 8.82 | 0.33 |

Set | mean V_{D,OP}(cm/s) | COV V_{D,OP} | mean $\frac{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}}{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}}$ | COV $\frac{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}}{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}}$ | mean V_{D,NC}(cm/s) | COV V_{D,NC} | mean $\frac{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}}{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}}$ | COV $\frac{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}}{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}}$ | mean $\frac{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}}{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}}$ | COV $\frac{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}}{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}}$ |
---|---|---|---|---|---|---|---|---|---|---|

X | 70.04 | 0.27 | 0.72 | 0.09 | 241.06 | 0.27 | 0.82 | 0.02 | 3.53 | 0.26 |

1 | 73.08 | 0.11 | 0.71 | 0.11 | 235.98 | 0.31 | 0.81 | 0.03 | 3.22 | 0.28 |

2 | 89.77 | 0.33 | 0.70 | 0.06 | 320.27 | 0.25 | 0.82 | 0.02 | 3.77 | 0.29 |

3 | 94.53 | 0.13 | 0.70 | 0.07 | 240.51 | 0.13 | 0.80 | 0.03 | 2.57 | 0.16 |

4 | 78.98 | 0.37 | 0.70 | 0.07 | 202.71 | 0.21 | 0.80 | 0.04 | 2.80 | 0.33 |

5 | 61.92 | 0.19 | 0.68 | 0.09 | 220.27 | 0.15 | 0.79 | 0.03 | 3.60 | 0.10 |

Y | 67.36 | 0.18 | 0.66 | 0.11 | 223.43 | 0.31 | 0.76 | 0.04 | 3.27 | 0.18 |

All records | 77.22 | 0.29 | 0.70 | 0.09 | 241.35 | 0.29 | 0.80 | 0.04 | 3.24 | 0.28 |

Rec. Name No. | $\frac{{\mathit{V}}_{\mathit{E}\mathit{X},\mathit{O}\mathit{P}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}}$ | $\frac{{\mathit{V}}_{\mathit{E}\mathit{Y},\mathit{O}\mathit{P}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}}$ | $\frac{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}^{*}}{{\mathit{V}}_{\mathit{E},\mathit{O}\mathit{P}}}$ | $\frac{{\mathit{V}}_{\mathit{E}\mathit{X},\mathit{N}\mathit{C}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}}$ | $\frac{{\mathit{V}}_{\mathit{E}\mathit{Y},\mathit{N}\mathit{C}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}}$ | $\frac{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}^{*}}{{\mathit{V}}_{\mathit{E},\mathit{N}\mathit{C}}}$ | $\frac{{\mathit{V}}_{\mathit{D}\mathit{X},\mathit{O}\mathit{P}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}}$ | $\frac{{\mathit{V}}_{\mathit{D}\mathit{Y},\mathit{O}\mathit{P}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}}$ | $\frac{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}^{*}}{{\mathit{V}}_{\mathit{D},\mathit{O}\mathit{P}}}$ | $\frac{{\mathit{V}}_{\mathit{D}\mathit{X},\mathit{N}\mathit{C}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}}$ | $\frac{{\mathit{V}}_{\mathit{D}\mathit{Y},\mathit{N}\mathit{C}}^{\mathit{u}\mathit{n}\mathit{i}}}{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}}$ | $\frac{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}^{*}}{{\mathit{V}}_{\mathit{D},\mathit{N}\mathit{C}}}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|

SMADRE 1643 | 0.86 | 0.80 | 1.18 | 0.85 | 0.56 | 1.02 | 0.88 | 0.80 | 1.19 | 0.85 | 0.54 | 1.01 |

CHICHI 1490 | 0.63 | 0.92 | 1.12 | 0.78 | 0.65 | 1.02 | 0.63 | 1.00 | 1.19 | 0.80 | 0.62 | 1.01 |

KOBE 1115 | 0.78 | 1.06 | 1.31 | 0.71 | 1.46 | 1.62 | 0.81 | 1.07 | 1.34 | 0.72 | 1.49 | 1.66 |

LANDERS 888 | 0.81 | 0.68 | 1.06 | 1.17 | 1.00 | 1.54 | 0.79 | 0.67 | 1.04 | 1.23 | 1.03 | 1.61 |

CHALFANT 549 | 0.86 | 0.79 | 1.17 | 0.81 | 0.78 | 1.13 | 0.88 | 0.84 | 1.22 | 0.82 | 0.79 | 1.13 |

mean | 0.79 | 0.85 | 1.17 | 0.86 | 0.89 | 1.26 | 0.80 | 0.70 | 1.19 | 0.88 | 0.78 | 1.28 |

COV | 0.10 | 0.14 | 0.07 | 0.17 | 0.33 | 0.19 | 0.10 | 0.17 | 0.07 | 0.19 | 0.36 | 0.21 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Donaire-Ávila, J.; Galé-Lamuela, D. Energy Capacity of Waffle-Flat-Plate Structures with Hysteretic Dampers Subjected to Bidirectional Seismic Loadings. *Appl. Sci.* **2020**, *10*, 3133.
https://doi.org/10.3390/app10093133

**AMA Style**

Donaire-Ávila J, Galé-Lamuela D. Energy Capacity of Waffle-Flat-Plate Structures with Hysteretic Dampers Subjected to Bidirectional Seismic Loadings. *Applied Sciences*. 2020; 10(9):3133.
https://doi.org/10.3390/app10093133

**Chicago/Turabian Style**

Donaire-Ávila, Jesús, and David Galé-Lamuela. 2020. "Energy Capacity of Waffle-Flat-Plate Structures with Hysteretic Dampers Subjected to Bidirectional Seismic Loadings" *Applied Sciences* 10, no. 9: 3133.
https://doi.org/10.3390/app10093133