Wall‐diaphragm interactions in seismic response of building systems ii: Inelastic response and design

An elastic parametric study of simplified single‐ and multi‐story building models presented in a companion paper shows that seismic building response depends on the period(s) of the vertical lateral force resisting system (vLFRS), the period of the diaphragm (i.e. the horizontal LFRS, or hLFRS) and the mass distribution between the walls and floors. This paper extends the study to include the potential for inelasticity (ductility) in both the v‐ and hLFRS and examines the impact of this inelasticity on the building response. Increased ductility in the vLFRS reduces vLFRS forces, but to what extent are hLFRS (diaphragm) demands reduced? Similarly, ductile hLFRS presumably reduce hLFRS forces, but what about vLFRS forces? Past work has shown that hLFRS forces are difficult to predict. The building model responses indicate wall‐diaphragm interactions may occur between higher modes of the vertical system and the rigid diaphragms causing increased force demands. Ductility demands are observed to be driven by the element (wall or diaphragm) with the larger inelasticity. Significant ductility demands are observed for short period buildings. A simplified formula for an effective ductility seismic force response modification coefficient for the horizontal LFRS is proposed that considers force reductions from both the inelastic vertical and horizontal LFRS.


INTRODUCTION
Seismic design of buildings is largely based on the assumption that diaphragms behave rigidly. Most seismic design codes, such as ASCE 7-16, 1 provide an estimation of the fundamental building period considering only the type of vertical lateral force resisting system (vLFRS) and the height of the building. In turn, the estimated approximated building period is used to estimate the seismic base shear and thereby the forces in the vLFRS and horizontal lateral force resisting system (hLFRS, i.e., diaphragm). The building period is not dependent on the stiffness of the hLFRS or the diaphragm span.

NOVELTY
• This paper uses reduced order multi-story mass-spring models to study the inelastic time-history responses of the vertical lateral force resisting system (LFRS) and the horizontal LFRS (diaphragm) when both systems are ductile. • Through a parametric study, where both the stiffness of the vertical and horizontal LFRS are varied alongside varying the level of ductility in the two systems, the inelastic response of the two LFRS are analyzed. • Ductility demands are found to go with the component with the larger inelasticity.
• Additionally, the component with the larger inelasticity shields the other component from elevated force demands.
However, according to numerous studies, the diaphragm can affect the fundamental period of the building, [2][3][4] and in some cases, the building response may be influenced by or even be dominated by the response of the hLFRS. 5,6 New provisions for rigid wall flexible diaphragm (RWFD) buildings in ASCE 7-22 (Section 12.10.4), 7 determine the seismic diaphragm demands based on the fundamental period of the diaphragm. 8 The alternative design method introduced in ASCE 7-16 (Section 12.10.3) allows for inelasticity in the vertical and horizontal LFRS through two response modification coefficients: and . The diaphragm demands are directly reduced by of the horizontal system and indirectly reduced by the ductility portion of the vLFRS' response modification coefficient, that is, ∕Ω 0 . However, it is not clear if this assumption is correct in practice. In this paper, the parametric study of the simplified building model presented in Fischer and Schafer 9 is extended to include different levels of ductility in both the vertical and horizontal LFRS. Vertical and horizontal LFRS force demands developed in the model are compared to the current seismic design forces prescribed in ASCE 7-16. The simplified building models may represent any building type, but in this paper, the models are based on archetype building models developed by Foroughi et al. 47 and Wei et al. 35 The models are steel frame buildings with buckling restrained-braces or concentric braced frames as the vertical LFRS and the horizontal LFRS are concrete-filled steel deck floors and bare steel deck roof diaphragms.

Model setup
The mass-spring model illustrated in Figure 1 consists of three degrees of freedom (DOF) per floor that are aligned with the lumped masses. Two DOF and masses for the vLFRS (i.e., the walls, subscript ) and one DOF/mass for the hLFRS (i.e., the floor/roof diaphragm, subscript ) per story. The lumped masses are connected with shear springs, as depicted in the figure.
The masses in the diaphragm(s) and walls are determined based on the total mass of the floor, , and the mass distribution factor, . The mass of a typical floor is constant across the height = for ∈ [1, − 1], and the roof weight is half the weight of a typical floor: = ∕2. The number of floors in the building is denoted .
The stiffness of the vertical and horizontal LFRS (diaphragm and walls) are based on the isolated wall and diaphragm periods: and . The hLFRS stiffness is directly connected to the isolated period and mass of the diaphragm(s): The vLFRS stiffness is assumed linear decreasing with height per 9 to match the stiffness variation in archetype building models developed by Foroughi et al. 47 Where is the wall stiffness at the top story and is a function of the first isolated wall period, , the typical floor mass and the number of stories in the model through the determination of the first eigenfrequency in the isolated wall system: Wherêand̂are unitless matrices representing the variation of stiffness and mass in the model. The vertical and horizontal LFRS springs assume an elastic perfectly plastic response, essentially the simplest hysteretic model. The capacities are determined based on the geometric mean of the elastic forces developed in the models with ground motions scaled to MCE level, divided by the ductility portion of the response modification coefficient at MCE level: 1.2 ASCE 7 seismic design demands, ground motions and scaling ASCE 7-16 chapter 12 prescribes one method for determining the seismic design demands in the vLFRS (walls) and two methods for the hLFRS (diaphragms). All three methods rely on a base excitation level, type of vLFRS, and the height of the building. For simplification, the same base excitation, the maximum considered earthquake (MCE) defined in ASCE 7-10, is used to scale the ground motion records in the time-history analysis and to determine the design demands for a direct comparison between the demands. In ASCE 7, the design demands for the vLFRS are determined with the equivalent lateral force (ELF) method, which uses equivalent static forces to distribute the base shear across the height of the building. The diaphragm force at story from the traditional diaphragm method (ASCE 7 12.10.1) is strictly bounded: 0.2 ≤ ≤ 0.4 . While the newer alternative diaphragm design method (ASCE 7-16 12.10.3) 1 uses only a lower bound of 0.2 ≤ with no upper bound. The seismic design demands according to ASCE 7 are summarized in Fischer and Schafer 9 (Section 2).
In Fischer and Schafer, 9 it was found that the alternative diaphragm design method can accurately predict the elastic hLFRS forces developed in the models when both walls and diaphragms are stiff, < 0.5 and ∕ < 0.5 and in some other instances. However, in the majority of cases, the alternative method is unconservative in its estimate of elastic hLFRS forces.
The 22 far-field earthquake record sets from FEMA P695 10 are used for the time-history analyses of the mass-spring models. The ground motion records are scaled according to the method used in FEMA P695 and as described in the companion paper. 9

Seismic response modification factors
The required seismic force demand according to the ELF method is: Where the elastic design force at the design earthquake (DE) level is denoted and is the seismic response modification coefficient. Uang 11 separates into a reduction in the system's ductility, and the system's overstrength, 0 (nearly equivalent to Ω 0 in ASCE 7): The yield force in the walls is set as the design force divided by the ductility coefficient, . = Due to the fixed relation between the maximum considered earthquake (MCE) level and DE level in ASCE 7, the elastic design force can be expressed with the design force at MCE level: = 2∕3 . Further, a ductility coefficient at MCE level is defined for use in the following results and discussion: Figure 2 illustrates the response of a wall spring and how the design and yield forces relates to the response modification coefficient and ductility coeffficients. The wall ductility demand is defined by the peak story drift of the vLFRS, , divided by the yield displacement, : ASCE 7-16 Section 12.10.3, also known as the alternative diaphragm design procedure, introduces a seismic response modification coefficient, , explicit to diaphragms (i.e., hLFRS). Accordingly, a similar set of expressions between design forces, response modification coefficients, and ductility can be established for the hLFRS using as the hLFRS response modification coefficient: The MCE level ductility coefficient for the walls and diaphragm, and , are introduced to study the inelastic behavior of both the vertical and horizontal LFRS, in the same manner as , and are varied to represent different building types. The parametric study in this paper includes values from 1 to 5 and values from 1 to 2.5, that result in a variety of different levels for depending on the overstrength factor 0 , see Table 1, and for depending on  Table 12.10-7 provides response modification coefficients for the horizontal LFRS, , in the range 0.7 to 3.0. Currently, ASCE 7-16 does not provide an overstrength factor for , however, due to the diaphragm's multiple design considerations, they may have significant overstrength, which are considered in Table 2.

RESPONSE
In the following, nonlinear time-history analyses of the 1, 2, 4, 8, and 12-story mass-spring building models are performed for variation in , ∕ , , , and across the 22 scaled earthquake record sets, in total 1,645,600 time-history analyses are conducted. Forces and displacements in both the walls and diaphragm(s) (v-and hLFRS) are recorded doing the time-history analysis, with the peak values being of interest. The force levels in both the vertical and horizontal LFRS for a subset of models are compared to the code predictions, followed by an examination of the ductility demands from the time-history analyses. Special attention will be directed at a few selected models that parallel the SDII archetype building models, as introduced in Fischer and Schafer, 9 with characteristic parameters listed in Table 3. The archetype building models are steel frame buildings with buckling restrained braces (BRB) and concentric braced frames (CBF) as the vLFRS and concrete-filled steel deck floors and bare steel deck roof diaphragms as the hLFRS. These characteristic parameters represent buildings with realistic stiffness properties of both walls and diaphragm within the larger set of models in the parametric study. Figure 3 illustrates the geometric mean of the peak elastic forces in v-and hLFRS for the subset of building models listed in Table 3 where the heavy floor (α_m = 0.9) models are indicated with solid lines, while the circles indicate the same models with light floors (α_m = 0.2). One standard deviation above and below the geometric mean is indicated with a shaded area for the heavy floor models. In Figure 3A, the vLFRS forces are illustrated across the height in the models along with the predicted ELF design demands according to ASCE 7, as indicated with black lines. The vLFRS forces developed in the models are in agreement with the design demands, though the higher building models show signs of higher mode effects in the response. Horizontal LFRS forces are provided in Figure 3B along with code-based bounds and predictions. As the figure shows the observed elastic hLFRS forces are greater than the code predictions and bounds consistently. However, the vertical distribution of forces predicted by the alternative diaphragm design method is reasonable.

Inelastic response of a 12-story model
The response of the 12-story heavy floor ( = 0.9) model to each of the ground motions for the various cases of elastic and inelastic vertical and horizontal LFRS (i.e., varied and ) is shown in Figure 4. The black lines indicate the yield capacity and the blue line is the geometric mean of the response. The case with = = 1 in Figure 4A-D is the "elastic" case; however, because the yield level is set to the geometric mean of the elastic force demand in the walls and diaphragms, minor inelasticity may still occur under some ground motion records.
Increasing the wall ductility with = 2 while the hLFRS (diaphragm) remains near elastic at = 1 results in the responses illustrated in Figure 4E-H. Observe that nearly all vLFRS forces are equal to the vLFRS capacity ( Figure 4E), F I G U R E 3 Elastic geometric mean of peak forces across the height of the building models (colored lines) and one standard deviation above and below the geometric mean (colored shaded area) for a subset of building models. (A) Wall forces distributions with ELF predicted forces (black lines) and (B) diaphragm force distribution with ASCE 7-16 alternative force predictions (black line) and ELF bounds (hatched area). resulting in shielding of the diaphragms, as the diaphragms experience decreased demands across the height of the model ( Figure 4F). The hLFRS in the top two stories are responding linearly, in that they do not reach the yield capacity. Ductility demands of the walls fluctuate around ≈ = 2 ( Figure 4G), while the diaphragm ductility demand stay below ≈ = 1 ( Figure 4H). Figure 4I-L provide the response for a ductile diaphragm (hLFRS) = 2 with near elastic = 1 walls. In the preceding case where yielding initiates in the walls, the walls shield the hLFRS; in this case, where yielding initiates in the hLFRS, wall forces are only modestly reduced from the elastic levels. Figure 4J indicates that the hLFRSreach their yield level in nearly all ground motion records. The ductility demand in the walls ( Figure 4K) indicates that the walls above the eighth story respond linearly, while below the eighth story, some instances are inelastic. A surprising observation from Figure 4L is the larger than expected ductility demand of ≥ = 2 for the hLFRS. The final case in Figure 4 M-P is when both diaphragms and walls are ductile with = = 2. Both the walls and diaphragm reach their yield capacity in most records. The ductility demands in the walls, , are around = 2, while the ductility demand in the diaphragm decreases with increasing height, to levels below the ductility level ≤ = 2.

Wall and diaphragm forces
Normalized v-and hLFRS forces for the 1, 2, 4, 8, and 12-story archetype building models across all levels of studied ductility ( and  per Tables 1 and 2) are compared to the ASCE 7 design predictions in Figure 5. The subplots allow for a systematic study of the impact of wall and diaphragm ductility, with results in the upper left subfigure representing the elastic case with = = 1. The top row of subfigures is for models with elastic hLFRS = 1 with increasing inelasticity in the vLFRS going left to right and the leftmost column of subfigures signifies models with elastic vLFRS = 1 with increasing levels of inelasticity in the hLFRS going down through the subfigures. To connect a particular case of or one may use Tables 1 and 2.
The vLFRS force demands are shown in Figure 5A and are found to agree with the ELF force predictions. With the increase in the vLFRS demands according to the models and the ELF predictions are decreased with the same amount. For the case with elastic vLFRS and inelastic hLFRS, the vLFRS demands are decreasing with increasing . This was F I G U R E 4 Peak force demands and ductility demand in the walls and diaphragm across the height for each of the 44 ground motion records in the 12-story model in Table 3  described above, as the inelastic diaphragms shield the walls. The decrease in vLFRS demands caused by is not considered in the ELF method, however, the reduction is small for this subset of models and on the conservative side.
Similar to the vLFRS forces, the hLFRS forces across the height for the 1, 2, 4, 8, and 12-story building models are illustrated in Figure 5B, where the geometric mean forces and the standard deviation of the 22 ground motion sets are depicted with the alternative diaphragm design predictions and the upper and lower bounds for the traditional diaphragm F I G U R E 5 Inelastic mean peak forces across the height of the building models (colored lines), one standard deviation above and below the mean (colored shaded area) for building models with heavy floors = 0.9. (A) Wall forces distributions with ELF predicted forces (black lines) and (B) diaphragm force distribution with ASCE 7-16 alternative force predictions (shaded grey area) and ELF bounds (hatched area).
design. As discussed above on the elastic diaphragm forces, the alternative method does not predict the elastic or inelastic hLFRS forces for any of the selected building models accurately, see Figure 5B. Across the 16 different ductility levels (for both walls and diaphragm) the alternative method consistently predicts design forces below the recorded forces in the models. The upper bound in the traditional design manages to become effective in capturing the hLFRS forces in the heavy floor building models for ≥ 2.5, and for ≥ 5.

F I G U R E 6
Geometric mean of the ductility demands across the height of the building models (colored lines), one standard deviation above and below the mean (colored shaded area) for a subset of building models listed in Table 3. (A) Wall ductility demand distribution and (B) diaphragm ductility demand distribution.

Ductility demand
With forces reduced with inelasticity, ductility demands arise. In this section, the ductility demands in the mass-spring models are presented as the geometric mean of the ductility demand across the 22 earthquake record sets for the different values of and . Initially, attention is directed at the selection of building models with characteristic values listed in Table 3 to see the variation of ductility demands across typical buildings with various ductility levels. In Figure 6, the geometric mean of the ductility demands is illustrated with colored lines and the variation of the ductility demand across the 22 ground motion sets is given by a shaded colored area for the heavy floor models only. The building models illustrated in Figure 6 are a representative selection of the general behavior of all the models across the parametric space. Though, at short periods (be that or ) the ductility demand increases significantly, which cannot be observed in this figure. The upper left subfigure in both Figure 6A and B is the near elastic case with = = 1. Ductility demands in the vLFRS are shown in Figure 6A. In the cases with = 1 (first column of Figure 6A), the vLFRS ductility demands are smaller than 1, as the diaphragms are partially shielding the walls. In the cases of inelastic vLFRS ≥ 2, the ductility demands are increasing with values near . The lower half of the stories experience ductility demand above , while the second story and the upper stories see a ductility demand below . Ductility demands in the hLFRS are illustrated in Figure 6B. They are generally as expected with < . When the walls yield first ( > ) ductility demands may be even less. However, when both vertical and horizontal LFRS are inelastic, the hLFRS ductility demands are significant for the lower stories with values above . In addition, if the only real source of ductility is the diaphragms, and thus the diaphragms yield first, then the ductility demands in the hLFRS may be much greater than . This indicates that building systems that aim to use the hLFRS as the only or primary source of yielding may require the hLFRS to undergo substantial drifts and deformations.
One way to ensure reduced ductility demands is for the wall and diaphragm periods to be "long." This generally results in a longer building period, as discussed in Fischer and Schafer. 9 Figure 7 shows the ductility demands in the v-and hLFRS in the models with long periods, that is, ≥ 0.5 for the walls and ≥ 0.4 for the diaphragm. The minimum, mean and maximum ductility demands are indicated in Figure 7, where it can be observed that the variance in the v-and hLFRS ductility demands are decreased and they are essentially always less than the assumed level of ductility, and . In Figure 8, the mean ductility demands for both walls and diaphragm are illustrated for short periods: < 0.5 and < 0.4 , along with the minimum and maximum ductility demands for the set of models. The ductility demands in both v-and hLFRS are observed to increase significantly for short periods. This is not surprising as other researchers have found significant ductility demands in short-period models 12,13 and these demands are in reference to very small elastic drifts (associated with short periods).
In the ductility examples in Figure 6, the ductility demands were compared to or , however, in the case of short period models, researchers suggest that the ductility demand may be estimated with 12 : Both estimates of the ductility demand, ∼ , ∼ and Equations 17 and 18, are indicated in Figure 8 with black solid and dashed lines. The vLFRS ductility demands with short periods, < 0.5 , in Figure 8A, develops small ductility demands in the cases with elastic walls = 1 and inelastic diaphragms ≥ 1. Additionally, the ductility demand levels in the vLFRS decrease with increasing diaphragm inelasticity. In cases with > 1, the vLFRS experience significant ductility demands. Generally, the mean ductility demands are above the value of and the alternative ductility estimate in Equation 17 is a better estimate of the vLFRS ductility demand for short period models. However, first story vLFRS ductility demands are significantly larger than either estimate.
Diaphragm ductility demands increase significantly for short diaphragm periods, < 0.4 , as can be observed in Figure 8B. The ductility estimate for short periods in Equation 18 does not provide a good approximation of the hLFRS ductility demand. The ductility demands are consistently above both expected values when the hLFRS inelasticity is larger than the vLFRS inelasticity: ≥ . In cases where the hLFRS inelasticity is smaller than the vLFRS inelasticity < , Equation 18 may be used as an approximate ductility demand for the hLFRS in the model, however, there is a large variation between the ductility demand in the first floor and the roof diaphragm.

Wall force demands
In this section, a comparison is made between the ELF force predictions for the vLFRS and the geometric mean demands found in mass-spring building models. The comparison is the differences between the geometric mean model forces Negative comparison values indicate "conservative" design, or over-predicted force levels.  Table 4. From the table, one can observe that the base shear of the model is not equal to the design base shear, even though the ground motions are scaled to MCE level in the elastic case. This is an effect of the diaphragm interaction with the walls. The vLFRS demands are overestimated by 7% and 38% in the elastic case, with a higher overestimation for the heavy floor model. With increasing wall inelasticity, the overestimation of the vLFRS forces is reduced. For the light floor models, the estimates become unconservative while for the heavy floor model, the estimates stay conservative. In cases where < , the vLFRS demands are reduced with in both the models and in the predictions and therefore the differences remain constant. In the case where > , the wall demands in the model are reduced, which are not considered in the ELF prediction. The force reduction caused by is somewhat small, and could be neglected. In the following, vLFRS force comparisons are only reported for = 1, as has a little impact on the vLFRS force demands and the impact is conservative.
In Table 5, the comparison between vLFRS force predictions and design demands are presented for the models given in Table 3 with an elastic diaphragm, = 1. In most instances, the base shear is overestimated, except for the cases when the walls are inelastic ( > 1). The overestimation is especially significant for the elastic diaphragm models and taller building models. From the table, one can observe that higher mode effects occurring in the upper stories for models with more than 4 stories. The higher mode effects are making the vLFRS force predictions unconservative for the affected stories across all ductility levels in the vLFRS, even though the base shear is being overestimated.

Diaphragm force demands
In the following the diaphragm demands found in the mass-spring models will be compared to the traditional (ELF) diaphragm method and the alternative diaphragm method in ASCE 7-16. The measure will be the difference between the predicted method and the force at story in the model normalized by the predicted force: where negative values indicate "conservative" design, or over-predicted force levels. In Table 6, the mean differences between the traditional diaphragm (based on ELF) method ASCE 7 Section 12.10.1, and the alternative diaphragm method ASCE 7 Section 12.10.3, with the model demands are reported for the subset of analyses matching the SDII CBF building models with characteristic values listed in Table 3. Demand comparisons for the 4-and 8-stories models lie between the results presented for the 2-and 12-story models, therefore results for the 1, 2, and 12-story models are a good representation of all the models. Second, some variations across the height are observed in the data, however, this variation is small and forces are reported as a mean value across typical floor diaphragms and the force in the roof diaphragm.
It is observed that in general both the alternative design method and the traditional method make unconservative estimates of the predicted diaphragm demand. However, some cases give reasonable predictions: the traditional method predicts reasonable demands for single-story models with high ductility levels in the diaphragm ≥ 2.5 for light floor models and ≥ 2 for heavy floor models. The alternative method makes reasonable predictions for single-and 2-story models with heavy floors in the case of elastic walls, = 1 and for single-story models with light floors and elastic walls. In other cases with multiple stories, the estimate is highly unconservative.

RESPONSE MODIFICATION FACTOR FOR THE DIAPHRAGM DEMAND
In the preceding, the geometric mean hLFRS (diaphragm) demands were found from the mass-spring models for increasing ductility levels in both the vertical and horizontal LFRS. It was found that the hLFRS forces are reduced when the ductility coefficient for the hLFRS, , is increased, but also by an increase to the ductility coefficient of the vLFRS, . With this in mind, a composite reduction in hLFRS forces as a function of and is found by comparing the hLFRS forces to the elastic hLFRS force found in the parametric study: * = Where is the hLFRS force and is the hLFRS force in the elastic case. The average ductility coefficient * is found across all the models for each inelasticity case, average values can be found in Table 7. Based on these values, a function for * to estimate the design hLFRS forces is proposed: * = ( , TA B L E 6 Summarized mean differences between the traditional (ELF) and alternative diaphragm methods for predicting hLFRS demands and the hLFRS forces developed in the mass-spring models for both the light and heavy roof models, errors are normalized with m S MS and listed in [%] and differences larger than 10% is shaded in grey to help the reader Traditional (ELF) Alternative = .
It was found that the heavy floor models are more influenced by than the light floor models, and may be reduced with a larger * value. A function for * that considers the mass distribution between walls and floors is also suggested. The effective ductility coefficient, * , as a function of the mass distribution, , and the ductility coefficients of the v-and The two proposed equations for the effective ductility coefficient are illustrated in Figure 9 alongside

DISCUSSION
ASCE 7 has adopted provisions for seismic design methods that use separate seismic response modification coefficients for vLFRS (e.g., ) and hLRFS (diaphragms, e.g., ). The study in this paper finds that the seismic design demands in the hLFRS are dependent on both the vertical and horizontal systems' characteristics and the ductility levels. The seismic design methods in ASCE 7 provide conservative results for a variety of one-story models, except when the isolated wall period, , and diaphragm period, , are similar in value, as discussed in Fischer and Schafer. 14 For multi-story building models, increased force demands are observed when the isolated diaphragm period coincides with the isolated wall periods, first and higher modes, as discussed in Fischer and Schafer. 9 The interactions between diaphragm and walls at short diaphragm periods, ∕ < 1, cause a significant increase in the diaphragm demands which is not considered in the ASCE 7 diaphragm demands.
However, if the diaphragm has significant ductility, ≥ 2, then the traditional diaphragm design method provides reasonable predictions no matter the height of the model. While the alternative diaphragm method provides reasonable predictions for short building models with one and two-stories and elastic walls, = 1. In the majority of cases studied, the alternative method consistently underpredicts the hLFRS forces, often by a factor of 2. This is in opposition to the observations made in Foroughi et al., 15 where the predicted diaphragm force levels are reasonable when = 1 with inelastic vLFRS of = 3. The interactions between the isolated diaphragm period and the wall periods at first and higher modes, cause increased diaphragm demands which either the traditional or the alternative diaphragm method in ASCE 7 considers. In Fischer, 16 it is speculated that the diaphragm forces may be predicted with seismic design methods for non-structural components. Different design codes, such as the Chilean seismic code (NCh), 17 New Zealand Standard (NZS), 18 and Eurocode 8 (EN1998) 19 seismic codes all have different predictions methods for the seismic design demand of non-structural components. Each method captures one aspect of the complex nature that is observed in these mass-spring models, but none of the methods can fully predict the hLFRS forces developed in the models.
In Fischer and Schafer, 9 the reduced-order models used in this paper are matched and compared to complex 3D models by Foroughi et al. 20 and Wei et al. 21 It was found that the differences between the simple mass-spring models and the models by Foroughi, et al. and Wei, et al. are reasonable considering the limitations. The complex 3D models experience a distributed yielding pattern in the diaphragm, which the mass-spring models cannot recreate, as it is limited to concentrated yielding in the springs. The simplified models may be able to create a distributed yielding pattern in the diaphragm, by modeling the hLFRS as a non-linear beam element with distributed mass, permitting the beam to experience a yielding pattern similar to the complex 3D models. The mass-spring models in this paper show the diaphragm wall interactions with nothing to shield or dampen the effect of the interactions.

CONCLUSION
A reduced-order mass-spring model of single-and multi-story buildings that provides degrees of freedom for the walls and diaphragms of each building story is developed. An extensive parametric study is conducted where both stiffness and strength are varied in the vertical and horizontal lateral force resisting system (v-and hLFRS) and the resulting building model is subjected to a suite of earthquake ground motions. Considering inelasticity/ductility in the vLFRS (walls) lowers the seismic force demands in the vLFRS and also in the hLFRS (diaphragms). Similarly, inelasticity/ductility in the diaphragm reduces hLFRS force demands and may reduce to a lesser extent the vLFRS force demands. The component, wall or diaphragm, with the larger ductility, i.e., larger or , will generally control the force demands and shield the other component from large force and ductility demands, however, the vLFRS are more effective at shielding demands from the hLFRS than vice-a-versa. ASCE 7-16 has introduced two system-level reductions for seismic design forces for the vLFRS and for the hLFRS. Systems utilizing these reductions must have a certain amount of ductility capacity-often approximated as similar in magnitude to the response modification coefficient, or , reduction itself. The building models herein show that the actual required demands are much more complicated in a system. The ductility demands may be much less than expected if one component (e.g., the walls) shields another component (e.g., the diaphragm) and this tends to happen when and are different-with the greater of the two controlling the inelasticity and hence requiring significant ductility. In addition, one also finds that the dynamics of the system indicate that almost regardless of yielding, ductility demands can be significant for short period diaphragms or buildings.
For results in the subset of models presented in this paper, the vLFRS force predictions according to the equivalent lateral force (ELF) method in ASCE 7-16 and the force demands developed in the models are in reasonable agreement across different levels of inelasticity in both the vertical and horizontal LFRS. However, the hLFRS forces predicted with the alternative diaphragm method in ASCE 7-16 estimate hLFRS forces smaller than those developed in the models. The traditional ELF diaphragm method uses strict upper bounds for the hLFRS force demands, and as such fails to even encompass the demands developed in the models. Only when large ductility levels are present in the model does the upper bound of the traditional method become effective in bounding the hLFRS forces in the heavy floor models, only. Thus, one is left to conclude that some level of inelasticity and ductility is implicit in traditional diaphragm design, even if this is not generally recognized.
Finally, a recommendation for an effective diaphragm ductility coefficient is made based on the hLFRS demands from the time-history analysis. It was observed that the ductility levels in both vertical and horizontal LFRS reduce the hLFRS demands in the model. An effective diaphragm ductility coefficient is a function of the ductility coefficient of both the vLFRS and the hLFRS. Future work includes improved estimates for elastic and inelastic seismic hLFRS demands.

A C K N O W L E D G M E N T S
The authors gratefully acknowledge the financial support funded by the American Iron and Steel Institute, the American Institute of Steel Construction, the Steel Deck Institute, the Metal Building Manufacturers Association, the Steel Joist Institute, and the US National Science Foundation through grant CMMI-1562821. Collaborators in the Steel Diaphragm Innovation Initiative project have provided the authors with assistance throughout the work on this paper, and we acknowledge the ideas and contributions they have given. Any opinions, findings, and conclusions, or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation or other sponsors.

D ATA AVA I L A B I L I T Y S TAT E M E N T
The data that support the findings of this study are available from the corresponding author upon reasonable request.