Wall‐diaphragm interactions in seismic response of building systems I: Parametric models and elastic response

The objective of this paper is to study the elastic seismic response of simplified building models that include considerations of the vertical lateral force resisting system (walls) and the horizontal lateral force resisting system (floor diaphragms) through parametric study. The assumption of rigid diaphragms is often used in the seismic design of buildings; however, research suggests that the diaphragm may influence the fundamental building period and the seismic building response. In this paper, a simplified model is presented for single and multi‐story buildings utilizing two degrees of freedom per story for the vertical system, for example, the walls, and one degree of freedom per story for the floor diaphragm. The number of stories, the stiffness of the walls and diaphragm(s), and the mass distribution between the walls and diaphragm are varied to represent a large range of potential building compositions. Eigen analyses of the models indicate that the wall and diaphragm interact when at similar periods resulting in elongation of the overall building period. Response of elastic time‐history analyses of the building models are presented in this paper where it is shown that the force demands in both the wall and diaphragm(s) are dependent on the relative stiffness of the walls and diaphragm(s), and the mass‐distribution between these two systems. In the companion paper “Wall‐Diaphragm Interactions in Seismic Response of Building System II: Inelastic Response and Design” the parametric building study is extending to nonlinear time‐history analyses including different levels of assumed inelasticity in both the walls and diaphragm.


Wall-Diaphragm Interactions in Seismic Response of Building Systems I: Parametric Models and Elastic
Response -This paper uses reduced order multi-story mass-spring models to study the interactions between the vertical lateral force resisting system (LFRS) and the horizontal LFRS (i.e., the diaphragm). -A large parametric study of eigen-and time-history analyses of the models was performed.
-The stiffness of both the vertical and horizontal LFRS was varied, resulting in a large variety of building types being studied. -Interactions between the vertical and horizontal LFRS are found when the period of the two systems coincide.
-Force and drift demands are elevated when the two systems interact.
resisting systems (LFRS) such as shear walls, moment-resisting frames, or braced frames. These vertically distributed systems are often the main component in the lateral force design. However, seismic forces are generated from the inertial forces caused by the mass on or in the floors. In a multi-story building, a typical floor may be made of concrete slabs, or concrete-filled steel decks, either of which is heavy and may account for the majority of the weight of each story. Other non-structural self-weight is often located on the floor system as well, such as non-structural internal walls, thereby adding extra mass and increasing the seismic inertial forces. Therefore, the floor system, besides being designed for gravity loads, must also transfer inertia forces to the LFRS and from there to the foundation. As such, the floor diaphragm is an integral part of the LFRS and one may distinguish between the walls as the vertical portion of the LFRS using the abbreviation vLFRS and the diaphragm as the horizontal portion of the LFRS, or the hLFRS.
Besides distributing gravity and lateral loads, the hLFRS (the diaphragm) also provides lateral support for the vertical elements and is an important element to resist torsional deformations of the building. Traditional seismic building design aims for the hLFRS to remain essentially undamaged and stay in the (near) elastic range for the ultimate limit states. If inelasticity is to occur in the structure, the vLFRS is regarded as the main inelastic element. 1,2 Design codes, such as ASCE 7, 3 incorporate the inelastic response of the vLFRS (the walls) into the seismic design demands of both the vertical and horizontal LFRS by reducing the demands with a response modification coefficient . The value of is based solely on the characteristics of the vLFRS. This reduction is based on two mechanisms occurring when the system responds inelastically: (1) period elongations because of the reduced stiffness of the yielding system, and (2) energy dissipation from the yielding elements, which result in hysteric damping. 3 Longer periods often result in smaller seismic accelerations and therefore demands, and structural damping is also associated with reduced seismic demands. 4 The combined effect is known as the ductility reduction of the system. Notations mass distribution ratio total system collapse uncertainty diaphragm displacement of story i wall displacement of story i ductility Ω 0 overstrength factor unitless eigen frequency wall frequency diaphragm design acceleration coefficient at story x seismic response coefficient vertical distribution factor diaphragm force , diaphragm force at story x, i predicted diaphragm force demand according to ASCE 7-10 ELF method at story x predicted diaphragm force demand according to  Alternative method at story x , wall force at story x, i , lateral seismic force at story x, i yield diaphragm force at story i yield wall force at story î unitless stiffness matrix , diaphragm stiffness of story i, x , wall stiffness of story i, x ′ elements in the diagonal of the stiffness matrix wall stiffness at story n, top floor ′ (n,n) element in the stiffness matrix diaphragm length total seismic masŝ unitless mass matrix total seismic mass above story x normalization factor for ground motion record i peak ground velocity of record from the PEER NGA database seismic response modification coefficient ductility coefficient, ductility portion of the response modification coefficient diaphragm seismic response modification coefficient ( ) spectral acceleration of unscaled ground motion record i DE design spectral response acceleration parameter at short periods 1 DE design spectral response acceleration parameter at 1s period MCE design spectral response acceleration parameter at short periods spectral responses accelerations at MCE level 1 MCE design spectral response acceleration parameter at 1s period scaling factor for ground motion record geometric mean spectral response acceleration of normalized ground motion records period building period diaphragm period mean period across typical floors diaphragm period at story i long-period transition period short-period transition period wall period wall period of mode i 0 transition period base shear diaphragm width stiffness factor of floor gravitational acceleration ℎ the height above the base to story x top story stiffness exponent to approximate higher mode effects diaphragm stiffness normalization factor mass of a typical floor , total mass of story i, x top story mass , floor mass at story i, x , wall mass at story i, x number of stories in the building eigen mode vector Floor diaphragms (hLFRS) have traditionally been designed with simplified demands based on the vLFRS and assumed to remain near elastic and thus to maintain the internal transfer of lateral loads in the structure. Nevertheless, some diaphragms have performed insufficiently in terms of maintaining the load paths for in-plane force transfer when significant amounts of ductility or inelastic demands are required. 1 During the 1994 Northridge earthquake, several concrete parking structures collapsed, which may be attributed to inadequate strength and stiffness capacities of the diaphragms. 5,6 Also, failures of rigid wall flexible diaphragm buildings were observed in the wall-to-diaphragm connections. 7,8 Even diaphragm connections in highway bridges were observed to cause failure in the 1994 Northridge earthquake. 9 And finally, diaphragms with longer spans, that is, more flexible diaphragms, were more likely to be damaged. 10 The design of buildings often relies on the assumption that the diaphragm behaves rigidly. This is reflected in the formula for estimating the fundamental period of a building, which for example in ASCE 7 only depends on the type of vLFRS and the height of the building. However, research suggests that the diaphragm does influence the fundamental period of the building, [11][12][13] and also the response of the building is influenced, or in some cases dominated, by the response of the diaphragm. 7,14 Additionally, work on flexible roof diaphragms [15][16][17] indicate that the diaphragm may respond inelastically and they provide design recommendations for such flexible roof diaphragms.
Traditionally, seismic design demands are based on multiple simplifications, one such method is the equivalent lateral force (ELF) method. ELF utilizes the elastic response of a single degree of freedom (DOF) system that matches the building characteristics to establish the seismic base shear demand. However, the single DOF conceptual model used in ASCE 7 ignores the hLFRS, and that may be an inaccurate representation of buildings with semi-to long span diaphragms, where the response may be influenced or even potentially dominated by the hLFRS.
Past researchers have included flexible diaphragm considerations in their seismic building simulations. The sophistication of the hLFRS model ranges from linear elastic single DOF to complex 3D models with nonlinear hysteric characterizations. In the work of [18][19][20][21] , the walls were modeled with a single DOF for each story and an extra single DOF at each floor level was added to represent the independent behavior of the floor and roof diaphragms. In 20 and 21 , the floor and roof diaphragms were modeled with inelastic springs to study the response of the vertical system as the diaphragm responded nonlinearly. In other cases, the floor and roof diaphragm was modeled with a beam element and the walls were essentially modeled as a single DOF for each story. [22][23][24][25][26][27][28] The seismic response of buildings was studied for both elastic and inelastic diaphragm models in 23,25 and 12 . In 23 it was found that with both inelastic walls and roof diaphragm, the roof diaphragm experienced distributed yielding spread through the beam element, while for elastic walls with inelastic roof diaphragm, the yielding was concentrated at the ends of the beam element, for example, the connection between walls and roof diaphragm. Cohen et al. 13 conclude that a model with at least 2 DOF, one for the diaphragm and one for the walls, is needed to accurately model the response of single-story buildings with flexible roof diaphragms.
For single-story warehouse buildings, more complex models were introduced by Koliou, et al., 7,11 and Medhekar et al., 17,29 which included inelastic behavior of the vertical and horizontal LFRS. They found that the roof diaphragm may experience yielding, and both made design recommendations for single-story buildings with flexible diaphragms. Numerous rigid wall flexible diaphragm buildings were analyzed in Koliou et al.,7,8,11,30 and specific design recommendations, adopted by FEMA P-1026, 31 were made in Koliou et al. 16 for rigid wall flexible diaphragm buildings that are based on a method of distributed yielding in the diaphragm with higher strength end zones.
Finally, research on the seismic response of 3D building models has also been performed, where the rigid diaphragm assumption is found to be an unconservative assumption. 13 [34][35][36] and Schafer 15 studied the whole building and/or diaphragm behavior, where the diaphragm was designed for seismic demands according to the alternative diaphragm design procedure in ASCE 7-16, 3 where diaphragm forces are reduced by a response modification coefficient specific for the diaphragm, for example, in addition to the reduction by for the vertical LFRS. The alternative design procedure for diaphragms in ASCE 7-16 is based on the work in. [38][39][40][41][42][43] The alternative method is based on an assumption of a rigid diaphragm in the building models, however, the work in 7,[11][12][13][14]32,33 indicate that flexible diaphragms influence the behavior of the building and should be included in the analysis of the structure.
With the advent of design methods that now explicitly consider inelasticity in the walls (e.g., through R) and the diaphragm(s) (e.g., through R s ), it is not known how inelasticity in these two competing systems will interact and influence the force and drift demands. In this first of two companion papers, simple reduced-order mass-spring models of singleand multi-story buildings capable of examining these questions are introduced. The force response of the reduced-order models from nonlinear time-history analyses are compared to complete 3D building simulations by Foroughi et al. and Wei et al., 34,35 to provide confidence in the overall approach. This is followed by a parametric study of the vibration response and elastic time-history response of the mass-spring building models. The parametric study is extended to include various levels of inelasticity in both the walls and diaphragm(s) in the companion paper: "Wall-Diaphragm Interactions in Seismic Response of Building System II: Inelastic Response and Design."

SEISMIC DESIGN DEMANDS ACCORDING TO ASCE 7
To establish the variables utilized in the parametric study and to provide a basis for comparison against the developed reduced-order building models, ASCE 7′s seismic prediction methods are summarized and cast into a common notation for this and the comparison paper. ASCE 7 Chapter 12 prescribes seismic force demands for both the vertical and the horizontal LFRS. ASCE 7-16 3 provides one method for determining the seismic demands in the vLFRS and two methods for the demands in the hLFRS (i.e., the diaphragm). Similar to all three methods is that they are tied to a base excitation that is specific to the building and location characteristics, that is, damping, source effects (soil), seismic risk, return period, etc. Moreover, to account for different sizes of ground motion magnitudes and shaking intensity levels, ASCE 7 defines a maximum considered earthquake (MCE) and a design earthquake (DE). The frequency and intensity of an earthquake vary with location and soil conditions; therefore, the MCE ground motion intensity is soil and site-specific, in addition to the response of a structure with a natural period. For this study the MCE curve, ( ), is defined for a building site in Irvine, CA with soil class D using the parameters in ASCE 7-2010 44 Section 11.4.
Given the desire to explore inelastic response the work presented in this and the companion paper 45 compares estimated demands between ASCE 7 and the multi-story mass-spring building models excited at ground motions equivalent to the MCE ( ) level, as opposed to the design demands at DE ( ) level in typical design. Furthermore, the MCE curve ( ( )) employs no lower limits, as is otherwise prescribed in ASCE 7. Subsequently, to make a direct comparison of demands from the mass-spring models and the design demands according to ASCE 7, the same MCE response spectrum is used for the time-history analyses and establishing the design demands.

Wall demands
In ASCE 7, the ELF method was initially used to determine the seismic demands in the vLFRS, but was later expanded to estimate the demands in the hLFRS (i.e., the diaphragm). The force levels in both walls and diaphragm(s) are tied to the base shear ( ) that is proportional to building weight and the pseudo acceleration at the building period, that is, the wall period: Where is the total weight of the structure and is defined here (different than ASCE 7) as the MCE curve divided by the ductility portion of the response modification coefficient ( = ∕Ω 0 ): The distribution of forces along the building height is a ratio of the mass and height of the story to the total mass and height of the building: Where is the vertical distribution factor, and ℎ is the story mass and height from ground to story , and is an exponent in the range of 1-2 dependent on the period (i.e., the wall period ) to account for higher-mode effects.
The story shear and the design demand of the vLFRS of story is the sum of the lateral force, , above the ℎ -story: Where is the mass of the diaphragm at story . Note, the traditional diaphragm method bounds the hLFRS demand with upper and lower limits: The alternative method (ASCE 7-16 Section 12.10.3) accounts for the inelasticity of the vertical and horizontal LFRS with response modification coefficients and , in addition to an alternative distribution of seismic accelerations across the height. The hLFRS force is defined as: With a lower bound of 0.2 applied to the hLFRS demand. The design acceleration coefficient is dependent on the type of vertical LFRS and the height of the building, for example, for buildings with more than three stories, second mode effects are accounted for in the hLFRS demand. and subsquent coefficients and factors can be found in ASCE 7-16 (and ASCE 7-22) Section 12.10.3.2.

MODEL SETUP
In the following, 1, 2, 4, 8, and 12-story building models are assessed through a parametric study, where the models are defined by , , and to specify different building typologies. The single-story mass-spring model introduced in the work of Fischer and Schafer 47 is here expanded to multi-story building models undergoing lateral deflections. Each floor of the model is represented by three masses, two for the walls (subscript ) and one for the floor/roof diaphragm (subscript ) and connecting shear springs, as depicted in Figure 1. The model only undergoes shear deformations, that is, deformations perpendicular to the plane of the frame, with the degrees of freedom (DOF) aligned with the lumped masses, resulting in 3 DOF for a -story building model.
The floor mass is assumed constant across the height: = for ∈ [1, − 1], where denotes the floor weight of story , is the weight of a typical floor and denotes the number of floors in the building. The mass of the roof is assumed as half the weight of a typical floor = ∕2. The lumped wall and diaphragm masses in the model are defined from a mass ratio, , and from the total mass of floor , :

F I G U R E 1 Mass-spring model of a n-story building used for the parametric study of wall-diaphragm interactions in multi-story buildings. (A) Model with mass and stiffness notations and (B) degrees of freedom assumed in the model
The spring stiffnesses of both the vertical and horizontal LFRS (walls and diaphragms) are defined from the period of the isolated vertical and horizontal structures. The isolated wall period, , is the first period of vibration for the structure with rigid diaphragms ( = ∞), and is the isolated diaphragm period, defined as the period of vibration of the structure when the walls are rigid ( = ∞). The hLFRS stiffness is defined from , assuming the period of the diaphragms is constant across the height of the building: Note, the stiffness of the roof diaphragm is therefore half as stiff as a typical floor diaphragm. The wall stiffness is defined via: where is the stiffness factor of floor based on the actual stiffness variation of the archetype building models developed by Foroughi et al. 48 and Wei et al. 35 (see Section 4 where Equation 13, , is validated). The top story wall stiffness is denoted as and is a function of the isolated wall period, and the eigen problem of the isolated wall structure: Where ⋅̂is the stiffness matrix of the isolated vertical system and ⋅̂is the mass matrix of the system.̂andâ re unitless matrices representing the stiffness and mass variations in the model. The unitless eigen frequency is defined as:̂= √ ∕ (15) F I G U R E 2 Design response spectra from ASCE 7-10 and FEMA P695 and mean and mean plus one standard deviation of response accelerations for a SDOF model of the 22 unscaled ground motion record sets Combining Equation 15 with the relation between the isolated wall eigen frequency and isolated wall period: = 2 ∕ , one can write as: Wherêis found from the eigen problem in Equation 14. Finally, Equations 12 and 16 are combined to define the wall stiffness:

Earthquake excitation levels and ground motions set
In addition to ASCE 7′s DE and MCE level, FEMA P695 37 provides an additional scaling level for earthquakes based on the composite uncertainty of the building model and an acceptable conditional probability of collapse under MCE level ground motions. This level of ground motion intensity is used for collapse evaluation of new or existing buildings and is here referred to as the collapse evaluation earthquake (CEE) level. The CEE level is defined based on the MCE spectral response acceleration ( ) and adjustment factors for the acceptable collapse probability of 10%. Spectral acceleration responses at MCE, DE, and CEE levels following ASCE 7-10 44 are illustrated in Figure 2, for building sites in Irvine, CA with soil class D, ductility = 3.0, and with a fair rated total system collapse uncertainty ( ). The work presented in this paper uses the definition for MCE level given in ASCE 7-10, see 49 for further discussion on ASCE 7-10 and ASCE 7-16 MCE levels.
For the time-history analysis of the mass-spring models in this paper, the FEMA P695 37 far-field earthquake record set is selected for the excitation. The earthquake record set is composed of 22 ground motions of both horizontal recordings from 14 earthquake events with a moment magnitude larger than 6.5. The mean and one standard deviation above the mean spectral response accelerations of the 22 unscaled far-field ground motion record sets are illustrated in Figure 2 alongside the code-specified earthquake levels.

Ground motion scaling
There exist a large variety of methods to scale ground motions, in this paper the scaling method used in FEMA P695 is employed. It comprises three steps: (1) normalization of each ground motion set, (2) determining the pseudo accelerations spectrum for each of the normalized ground motions of a single degree of freedom model with 5% viscous damping, and (3) scaling of the entire suite of ground motions to the target response spectrum at a period, .
The ground motion records are normalized using their peak ground velocity (PGV) and the median PGV of the suite of records: Where PGV PEER is the peak ground velocity of record from the PEER NGA database. 50 Resulting in a normalization factor for each of the ground motion sets. This normalization removes some of the discrepancies between records such as event magnitude, distance to the fault, fault type, and site conditions but still keeps the aleatory of each ground motion. 37 This is followed by determining the response spectra for each of the normalized 22 far-field ground motion record sets on a single DOF model with 5% viscous damping. The geometric mean (or logarithm mean) is determined as 51 : Where is the pseudo acceleration of each unscaled ground motion record and is the normalization factor for record .
The geometric mean spectral response acceleration ( ( )) of the 44 normalized ground motion records with 5% viscous damping is determined. Followed by the determination of a uniform scaling factor for the entire set of ground motions for matching either the MCE, DE, or CEE level accelerations curves at building period : Where, ( ) is the acceleration at DE, MCE, or CEE level. Note that the building period is that of the model subjected to the time-history analysis. The geometric mean of the normalized record set is set equal to the target response spectrum at period .

SDII ARCHETYPE MODELS AND MODEL VALIDATION
In this section, characteristic values of the simplified building models are established for a set of archetype buildings designed for the SDII project 52 by Torabian et al. 53 These characteristic values serve two purposes: (1) define the variation of stiffness across the height of the building models and (2) provide a set of simplified models that represent specific buildings in the larger parametric study. This is followed by a comparison of the non-linear time-history response between the simplified models and detailed 3D building models of the same archetype buildings to understand the strengths and limitations of the reduced order mass-spring model. Foroughi et al. 48 and Wei et al. 35 created 3D models of the archetype buildings in OpenSEES and subjected the models to eigen, pushover, and time-history analysis (including collapse assessment according to FEMA P695). The in-plane shear response of the diaphragm is modeled as truss elements with a Pinching4 non-linear material 54 that is fit to cyclic cantilever diaphragm tests of both bare and concrete-filled steel decks and scaled to match the archetype expected strength. The vertical LFRS is either designed with buckling restrained braces (BRB) or concentrically braced frames (CBF), both calibrated in the models to experimental cyclic tests of BRB or CBF braces as appropriate. The layout of the building is constant for the two types of vLFRS and the different number of stories in the building models, see Figure 3 for a typical eight-story archetype building model and typical framing plan in the models. The archetype building models are investigated to find the building and isolated wall and diaphragm periods ( , , and ) and the stiffness in both vertical and horizontal LFRS across the height of the buildings for use in comparing to the parametric studies conducted herein and in the companion paper. To compare the hLRFS forces in the time-history analysis between the 3D OpenSEES building models by Wei et al., 35 Foroughi et al., 48 and the reduced-order mass-spring models utilized here, yielding forces for the vertical and horizontal LFRS are also estimated for the archetype building models. Together, this provides the necessary characteristics to depict the 3D building models as reduced order mass-spring models as utilized here.

4.1
Characteristic parameters for equivalent mass-spring model

Eigen-periods of building, walls, and diaphragm
The fundamental building period in the short direction is given in Table 1, found through elastic eigen analysis of the BRB and CBF models with traditional diaphragm design, as described in. 35,48 The first isolated vertical period ( ), defined as the vertical system with rigid diaphragm behavior, is found by increasing the stiffness of the components in the floor system, such as the beams, girders, and trusses, representing the diaphragm deck, this provides close to rigid diaphragm action. Eigen analysis of the rigid diaphragm models results in the first eigen period for the isolated walls ( ) listed in Table 1. The isolated diaphragm period of each floor, , is determined with eigen analysis, by restraining the walls against displacements, such that only the floors are free to deform/vibrate. The isolated diaphragm periods are also listed in Table 1.

Stiffness estimates of wall and diaphragm
Estimating the stiffness of both vertical and horizontal LFRS was completed by a static step analysis, inside the elastic portion of the building model's response. Additionally, restraining parts of the model against translational movements ensures isolated deformations in the story of interest, such that specific stiffness can be extracted from the models.
Determining the vLFRS stiffness, the floors above and below the floor of interest are restrained, concentrated loads are applied at the locations of the braced walls. The diaphragms, beams, and girders are modeled as rigid (described above). The elastic deformation is recorded and the stiffness of floor is the load applied divided by the displacement ′ = ∕2 . This stiffness estimate is an entry in the diagonal of the stiffness matrix (in Equation 14). The wall stiffness is found from the entries in the stiffness matrix: where is the number of stories in the building model. TA B L E 1 Stiffness for the vertical and horizontal LFRS (walls and diaphragms) for each story in the CBF and BRB SDII archetype building models, and first mode periods of the building (T b ) and the isolated wall (T w ) and diaphragm (T di ) periods The horizontal LFRS stiffness is determined with restraints applied to the floors above and below the floor of interest, in addition, the diaphragm at = 0 and = (see Figure 3B) are restrained to isolate all displacements to that floor. Loads are applied proportionally to the shape of the first eigenmode on the floor of interest. The hLFRS stiffness of floor is the sum of the applied load divided by the recorded displacement at mid-span times a factor for conversion from a beam with uniform mass and stiffness to a mass-spring model: To convert the stiffness from a uniform beam model to a mass-spring model, the natural period of a simply supported uniform beam model and a mass-spring model are set to be equal, making a direct relation between mass and stiffness of the mass-spring model , , and the beam model , . The stiffness of the beam model is determined by a simply supported beam with a sinus load applied, the stiffness is in terms of the total load applied and the displacement at midspan. Final assumption, is that a portion of the diaphragm weight is added to the weight of the vLFRS and the rest to the hLFRS. The stiffness relation between the mass-spring and beam model can then be written to the format in Equation 22.
The estimated stiffnesses for the vertical ( ) and horizontal ( ) LFRS are listed in Table 1 for each story for both the BRB and CBF building models, along with the first building period ( ), the first isolated wall period ( ), and the isolated diaphragm periods for each floor ( ). Figure 4 illustrates the variation of the stiffness of the vertical and horizontal LFRS across the height for the different archetype building models. The vLFRS stiffness is normalized by the top story stiffness , where is the mean diaphragm period across the typical floors ( ∈ [1, − 1]). Based on the observed stiffness, the hLFRS stiffness is assumed constant for typical floors and half as stiff for the roof diaphragm, as depicted in Figure 4. The vLFRS stiffness variation is approximated by a linear variation across the height, based on a best linear fit across all archetype models. The linear variation is given in Equation 13. The approximated stiffness for both the vertical and horizontal LFRS is depicted in Figure 4 with darker solid lines.

Capacity estimates and model validation
The mass-spring models use an elastic perfectly plastic material model for the springs of the vertical and horizontal LFRS. This is a simplification of a realistic behavior of diaphragms and BRB frames and the behavior modeled in the 3D SDII archetype building models. The 3D archetype building models use a Pinching4 material model for the truss elements in the diaphragm. The capacity of the two hLFRS springs in the mass-spring models is approximated by assuming an Equivalent Energy Elastic-Plastic (EEEP) model 55 with equal initial stiffness and matching energy at 80% post-peak stress value. Similar approaches are used to determine the capacity of the braced frames: Assuming the BRB will reach yield stress in both tension and compression, while only the tension brace will reach the yield stress in the strength capacity of the CBF. Based on the above assumptions, yield capacities of the spring of the vertical and horizontal LFRS' springs are estimated for the models and listed in Appendix A, Table A1 and A2 for the BRB and CBF building, respectively. The mass-spring models with 1, 4, 8, and 12-stories were modeled with stiffness and yield force estimates of the BRB and CBF building models and the models were subjected to nonlinear time-history analysis using the 22 ground motion sets scaled to DE, MCE, and CEE level using the P695 scaling method described in Section 3.2. The models use a Rayleigh damp-TA B L E 2 Comparison of the median diaphragm forces at three different earthquake levels from the mass-spring model (MSM) and the 3D BRB and CBF archetype building models (3D) from, 35 ing of 5% at the first two modes. Median peak hLFRS forces recorded in the time-history analysis are listed in Appendix under the title MSM and are compared to the median peak diaphragm forces recorded in the time-history analysis of the SDII BRB buildings in 35 and the SDII CBF buildings in. 36 Table 2 summarizes the comparison between the reduced order models and the archetype building models from Table A1 and A2. The mean ratio of the normalized forces between the 3D BRB building models and the reduced order mass-spring models is 0.94; however, the standard deviation is relatively high at 0.16 reflected in the range of the median ratios across all considered models of 0.94 as shown in Table 2. Larger discrepancies are observed between the mass-spring models and the 3D CBF building models, with a mean ratio of normalized forces at 1.32 and a standard deviation of 0.29. One reason for the discrepancies between the mass-spring model and the 3D building models is the estimation of the yield forces of the vertical and horizontal LFRS. The BRB frames have a material response similar to the elastic-perfectly plastic springs in the mass-spring models, resulting in a similar response to the BRB frames; while the CBF's material model shares less resemblance with the elastic-perfectly plastic material model, resulting in larger differences between the massspring models and the 3D CBF building models. In addition, the 3D building models simulate the diaphragm with many individual truss-elements that can reach peak capacity at different times, which is not the case in the mass-spring model, where the hLFRS springs are symmetric in response and will not allow for a distributed yield pattern as in the 3D models. With limitations noted, the level of difference between the two sets of models is deemed acceptable for the broader goal of studying wall vs. diaphragm inelasticity in seismic building response and the reduced order mass-spring models are advanced for linear and non-linear time history analysis and interpretations of building responses.

ELASTIC RESPONSE
In this section, elastic results from eigen analysis and the time-history analysis of the mass-spring models are presented for variations of wall and diaphragm periods (variation in the stiffness of the vertical and horizontal LFRS), different mass distribution factors, and buildings with different numbers of stories. General inelastic response is examined in the companion paper. 45 The wall stiffness is varied through the wall period, , in the range of a stiff response at =0.1s to a flexible building with = 2s. In a similar manner, the diaphragm stiffness is varied through the diaphragm to wall period: ∕ , from stiff diaphragm response at 0.

Building period elongation
Eigen analysis of the multi-story model has been performed for the 1, 2, 4, 8, and 12-story building models, following the definitions in Section 3, where the diaphragm period, , the wall period, , and mass ratio, are varied. First F I G U R E 5 Building period for a four-story building with varying wall and diaphragm stiffness and mass distribution mode eigen analysis results of the four-story model are illustrated in Figure 5, where the isolated wall mode with a rigid diaphragm is illustrated at ∕ = 0.1 and the isolated diaphragm mode with rigid walls at ∕ = 10. An identical figure was produced for the single-story building model with similar normalization of the axes in, 47 and similar figures can be generated for the 2, 8, and 12-story models. Observe in Figure 5 that the first building period , convergences to either the isolated wall or diaphragm period ( or ), when the isolated wall and diaphragm periods are more than a magnitude apart ( ∕ < 0.1 or ∕ > 10). By contrast, the building period increases for similar wall and diaphragm periods. In the case where = the first building period can increase by up to 40% of the isolated periods for the heavy diaphragm models ( = 0.9) and up to 15% for = 0.1, i.e., the light diaphragm models.

Higher vibration modes
The period elongation and vibration mode interaction between the diaphragm and the first mode of the vLFRS also occur between the diaphragm and higher modes of the vLFRS, as illustrated in Figure 6A.  Figure 1B). In addition, the models have different modes of the vLFRS, that is, the traditional modes of a building. The first building modes discussed above are modes with inphase sway and the first mode of the vertical system. The curve in Figure 5 can describe all the higher vertical modes with in-phase sway, see Figure 6A, where peak elongation occurs when the isolated diaphragm period equals the isolated wall period of mode , = . Note that the building period is normalized using the isolated diaphragm and isolated wall period of mode : max( , ). The four-story model has four vertical modes, resulting in four peak period elongations as illustrated in Figure 6A. The building period for the out-of-phase sway and torsional sway modes can also be normalized with max( , ) to generate the same curves shifted such that peak period elongation occurs at = , see Figures 6B and 6C.

Elastic force response of SDII archetype models
A series of 96,800 elastic time-history analyses have been executed in OpenSEES, to examine the response of 1, 2, 4, 8, and 12-story mass-spring models with varying characteristics across 22 far-field ground motion record sets scaled according to FEMA P695. The models utilize 5% Rayleigh damping on the first two modes. Forces in both the walls and diaphragm (vand hLFRS) are recorded during the time-history analysis, with the peak forces being of interest, that be the maximum absolute force an element will see over a ground motion. The geometric mean of the elastic peak forces in the v-and hLFRS for the subset of building models, with rigid diaphragm(s) ( ∕ = 0.1) and an isolated wall period of = 0.4 , are illustrated in Figure 7. Solid lines indicate the geometric mean of the peak forces and one standard deviation above and below the geometric mean is indicated with F I G U R E 6 Higher building periods for a four-story building with varying wall and diaphragm stiffness and mass distribution, (A) in-phase sway mode, (B) out-of-phase sway mode, and (C) torsion sway mode shaded area. The standard deviation is only illustrated for the heavy diaphragm models, = 0.9. Predicted elastic design forces according to ASCE 7 are illustrated with black lines and hatched area. Figure 7A illustrates the vLFRS (wall) force distribution across the height, along with the predicted ELF design demands according to ASCE 7 in Equation 5, illustrated with black lines. The P695 scaling method ensures that the geometric mean excitation is equal to the magnitude corresponding to MCE level, but preserving the variation in the individual ground motion records. This can be observed, as the geometric mean base shear is not equal to the design base shear (the vLFRS force demand at level 1, 1 , predicted by ASCE 7 expressions at MCE level), and each model will have variations in the v-and hLFRS demands caused by the variability of the ground motions and the wall-diaphragm interactions. Overall, the vLFRS forces are generally in agreement with the design demands for the various building heights.
The hLFRS (diaphragm) demands are illustrated in Figure 7B as the geometric mean of peak forces across the height with an indication of the standard deviation above and below the geometric mean. The diaphragm design demands using the alternative method from  Figure 7B. It is worth noting that the bounds are highly unconservative and non-effective in bounding the diaphragm forces for the models illustrated in the figure-indicating at a minimum that ASCE 7 implicitly expects inelasticity to reduce the demands. From the figure, it is observed that the alternative method predicts the hLFRS forces reasonably accurately for the subset of models with rigid diaphragms. The histograms in Figure 8 illustrate the distribution of the peak forces from each of the 44 ground motions in the fourstory building model with = 0.4 , ∕ = 0.1, and heavy floors = 0.9. The vLFRS force demands in Figure 8A are normalized with the total mass of stories above level x times the MCE acceleration at the wall period : ( ). Alongside the histograms are the log normal distributions illustrated with blue lines, they have mean and standard deviation matching that of the histograms. The distribution of vLFRS forces for each floor in the four-story model follows a lognormal distribution, which supports the notion to report the geometric mean of peak forces or ductility demands. The hLFRS forces also follow a log-normal distribution for each floor in the model.

5.3
Elastic force response in a larger set of models Figure 9 illustrates the vLFRS (wall) force demands in a larger set of building models from the parametric study. The larger set of models includes rigid, semi-rigid, and flexible walls and diaphragms. From the rigid diaphragm models in Figure 9A one can observe small differences between the heavy floor models ( = 0.9) and the light floor models ( = 0.2). In addition, the base shear demand increases slightly with more flexible walls, going left to right in Figure 9A. For increasing diaphragm flexibility, Figures 9B and 9C, the vLFRS demands decrease for the heavy floor models ( = 0.9) and increases for the light floor models ( = 0.2). The exception is the model with = 0.1 and ∕ = 1, the right subfigure in Figure 9B, where smaller vLFRS demands are observed for the light floor models ( = 0.2). The geometric mean of the shear demand varies between 0.5 to 2 times the design base shear for the models shown in Figure 9C for flexible diaphragms. Observe that higher-order effects occur for taller buildings with ≥ 4 with longer wall periods, ≥ 0.5 . Higher mode effects are accounted for in the ELF method through the exponent in Equation 4. However, this method for including higher-order effects is not conservative, and as a result, the vLFRS forces in the upper stories are underestimated by the ELF method. Generally, the distribution assumed by the ELF is in good agreement with the 1-to 12-story elastic vLFRS force results, although the magnitude of the base shear is often not. Figure 10 shows the geometric mean peak elastic hLFRS (diaphragm) forces, the alternative force predictions (Equation 8), and the lower and upper bounds of the traditional diaphragm design method (Equation 6). Note that both the stiffness of the horizontal and vertical systems influence the magnitude and distribution of diaphragm forces across the height of the models. Recall that the alternative method ignores the diaphragm in its estimate of the building periods, essentially only considering the wall period, for predicting the elastic diaphragm forces. The traditional diaphragm method is also based solely on the period of the vertical LFRS, but is restricted by upper and lower bounds. The RWFD method of ASCE 7-22 Section 12.10.4 takes the opposite approach, using the diaphragm period independently to determine the diaphragm force demands.
The alternative method underestimates the hLFRS demands in many instances. Figure 10A with = 0.1 shows that the hLFRS demands are in close proximity to the alternative design demands. In Figure 10B for both stiff walls and flexible walls ( = 0.1 and = 2 ), the results are in fairly good agreement with the alternative method predictions. In the case with ∕ = 1 in Figure 10C, the hLFRS demands are accurately predicted for = 2 , in the other cases, the hLFRS demands in the lower floors are experiencing demands of a similar magnitude as the roof diaphragm, and are therefore underestimated with the alternative design method. For ∕ = 1 in Figure 10C, the demands in the upper floors are reasonably predicted for the heavy floor models ( = 0.9), while the lower floors are being underestimated. Generally, the light floor models are experiencing larger demands and cannot be predicted accurately with the alternative method. In the case with very flexible diaphragms, ∕ = 10 in Figure 10D, and wall periods > 0.1 , the hLFRS demands are small, and both the traditional bounds and the alternative method makes conservative estimates.
The ELF-based diaphragm force demand has upper and lower bounds, as illustrated by the grey areas in Figure 10. It is worth noting, the bounds of [0.2, 0.4] dx MS are unconservative and non-effective in bounding the diaphragm forces found in the model. Use of the bounds thus implicitly require some inelasticity/ductility in the system to be relevant (see the companion paper 45 for more on this topic). However, the traditional bounds are effective for = 2 in Figure 10C and for ≥ 0.5 in Figure 10D. The light floor models ( = 0.2) usually have larger diaphragm demands than the same heavy floor models.

Elastic drift levels in a larger set of models
The elastic displacements in the models are linear related to the forces developed in the model and may be read from Figures 9 and 10 using these relations: F I G U R E 1 0 Elastic diaphragm force demands across the height of the building models (colored lines), one standard deviation above and below the demand (colored shaded area) for a subset of building models. The SDII CBF building models with characteristic values listed in Table 1  Where and are the forces in the vertical and horizontal LFRS (diaphragm and walls), and are the stiffness in the vertical and horizontal LFRS defined in Equations 11 and 17. Similar to the force demands, the drift demands are increased in certain cases and extra details may be needed to meet the elevated demands.

Wall force demands
The ELF force predictions for the vLFRS are compared to the geometric mean demands found in the mass-spring building models. The mean difference between the base shear developed in all the elastic models and the code specified base shear is illustrated in Figure 11A as a function of the number of stories in the model and the diaphragm to wall period, ∕ . The models with rigid diaphragms, ∕ ≤ 0.5, develop base shear demands similar to the code specified demands, regardless of the mass distribution. However, for models with flexible diaphragms, ∕ > 1, the base shear demand depends on the mass distribution between the diaphragm and walls. Models with heavy floors develop base shear demands much smaller than the code specified demands, while light floor models develop increased base shear demand, with demands 55% above the code predicted base shear.
In Figure 11B, the mean difference between the base shear in the models and predictions are illustrated as a function of the number of stories and . From the figure, it can be observed that with increasing wall periods the base shear demand increases compared to the predicted base shear. The increase is higher in the taller building models. Finally, one can observe from Figure 11 that there is a stronger correlation between base shear demands and the diaphragm to wall period ∕ than the wall period .

Diaphragm force demands
In the following, the hLFRS (diaphragm) force demands found in the mass-spring models are compared to the traditional (ELF) diaphragm method and the alternative diaphragm method in ASCE 7-16. In Figure 12, the error is shown between the elastic hLFRS demands in the model and the predicted demands using the alternative method and the traditional method. The error is normalized by the diaphragm mass and the peak of the design spectra and is illustrated as a function of ∕ . Variations between the different numbers of stories and mass distribution in the error can be observed in the figure.
In Figure 12A, the elastic forces from the model are compared to the alternative design method. The comparison is the mean difference across values and across all the stories in the models. Observe that for all models with flexible diaphragms, i.e., ∕ ≥ 2, have conservative force predictions. Single-story models with heavy floors are in agreement with the force predictions for rigid diaphragms ( ∕ < 1). For all other models with rigid diaphragm(s) ( ∕ < 1), the predicted demands become unconservative, especially for the light floor models. For single and two-story models with light floors ( = 0.2), interactions between the vertical and horizontal system occur at ∼ . For models with more than four-stories, vertical and horizontal system interactions occur at shorter diaphragm periods that coincide with higher modes of the vertical system: ∼ 2 . The number of stories in the model influence the hLFRS demands. The hLFRS demand in the one-story heavy floor model is accurately predicted with a difference of 0.4%, while the 12-story model is unconservative with an underestimation of 47% at ∕ = 0.2. The change of behavior with the increase in the number of stories can be associated with higher mode effects, as is discussed in Calvi et al., 56 where floor spectra are developed for multi-story buildings, and increased accelerations are observed when the component period equals the building period at first and higher modes, similar to the observation here.
Elastic hLFRS force comparison to the traditional ELF predictions is illustrated in Figure 12B. In general, the traditional method underpredicts the hLFRS demands across the buildings, only for very flexible diaphragms ( ∕ ≥ 5) is the demands in the model lower than the estimate. The traditional method consistently underpredicts the diaphragm demands for semi-and rigid diaphragms, ∕ ≤ 2.

DISCUSSION
The study in this paper indicates that interactions between the vertical (walls) and horizontal (diaphragm) lateral force resisting systems occur and that these interactions have an impact on building periods and the elastic demands in both the vertical and horizontal LFRS. The seismic base shear demand in the mass-spring models is highly dependent on the stiffness of the vertical and horizontal LFRS and the mass distribution between the diaphragm and walls. For rigid diaphragm models ( ∕ < 1), the base shear demand can be determined with the characteristics of the vLFRS, as design codes prescribe. However, when the diaphragm becomes more flexible ( ∕ > 1), the base shear demand is impacted by the diaphragm stiffness and the mass distribution, with increased base shear demands for light floor models ( = 0.2), and decreased base shear when the models have heavy floors ( = 0.9). This impact on the base shear demand from the horizontal LFRS could be considered in the seismic design demands of buildings.
The ELF method for predicting the force demands in the vLFRS depends on the base shear, and as discussed above varies significantly between the models. However, the general vertical distribution of lateral forces according to the ELF method, disregarding the base shear variation, is in reasonable agreement with the forces developed in the mass-spring models. Higher-mode effects appear in taller models, ≥ 4, causing higher force levels in the vLFRS in the upper floors. This is especially pronounced for long wall periods, ≥ 0.5 and light floor models, = 0.2. Higher-mode effects are considered in the ELF method, through an exponent on the height of the floor. This does increase the design demands in the upper stories, but in many instances in the parametric study, this procedure is still unconservative. Alternatively, the predecessor of ASCE 7, UBC 1997, 57 instead uses a concentrated force at the top story to mimic higher-mode effects in the vertical distribution of lateral seismic demands. This method of handling higher-mode effects is more in line with the results found in this parametric study.
Predicting the elastic hLFRS force demands is challenging. The hLFRS demands are dependent on the stiffness of the vertical and horizontal LFRS, the mass distribution, and the interaction between the higher modes of the vertical system and the diaphragms. All this plays into the response and the elastic force demands developed in the horizontal LFRS.
From the building period analysis, interactions between the horizontal LFRS and the first and higher modes of the vertical LFRS are evident. This is further confirmed in the diaphragm force comparison in Figure 12, where increased hLFRS demands occur for short ∕ values that coincide with the location of higher-mode interactions in the eigen period analysis in Figure 6. The single-story model has interaction between the horizontal and vertical system when ∼ . Whereas multi-story models have interactions when the diaphragm period coincides with the wall period any mode, . The hLFRS demands have increased demands when the diaphragm period coincides with the higher modes of multi-story models, as was found in the study of Calvi et al. 56 on non-structural components. Based on observations from Figure 9 of the vLFRS force demands, the vertical system develops insignificant higher-mode response at short wall periods, < 0.5 . However, structures with a stiff diaphragm may cause the diaphragm to interact with the higher modes of the vertical system, especially for vLFRS with a wall period longer than 0.5 s, causing increased hLFRS demands in these structures. This structural situation should be accounted for in design, and avoided if possible.

CONCLUSION
In this paper, reduced-order mass-spring models of single-and multi-story buildings are introduced and a detailed investigation of the interactions and forces in both the vertical (walls) and horizontal (floors and roof diaphragm) lateral force resisting systems (LFRS) during earthquakes is completed through an extended parametric study employing time-history analysis. The model is composed of three lumped masses per floor, one for the diaphragm and two for the walls, and connecting springs. The reduced-order mass-spring models in this paper are characterized with a set of parameters: stiffness of vertical and horizontal LFRS, and mass distribution between walls and diaphragms. Sets of characteristic parameters are determined such that the mass-spring model matches a set of archetype building models. From nonlinear time-history analysis, the force responses are compared between the reduced-order mass-spring models and the three-dimensional archetype building models. Some variations between the two models' responses are found that are caused by the estimation of yield capacities of both the vertical and horizontal LFRS, and the lack of distributed yielding in the reduced order mass-spring models. However, the level of differences between the two models are considered acceptable and it is deemed that the reduced-order mass-spring models can represent the more detailed models, particularly for studying situations where inelasticity may occur in either the wall or diaphragm.
The stiffness of the vertical and horizontal LFRS and the mass distribution are varied in the parametric study. Elastic eigen analysis of the models shows that the building period elongates when the period of the diaphragm and that of the vertical system, including higher modes, are similar. The elastic base shear demand recorded in mass-spring models varies across the models and is dependent on the diaphragm to wall period and the mass distribution in the model. In general, the base shear demands are similar to the ASCE 7-16 predictions for rigid diaphragms, while for increasing diaphragm flexibility, the base shear demand is either increasing for light floor models, while heavy floor models reduce the base shear demand. The vertical distribution of force demands in the vertical LFRS is estimated reasonably well with the ELF method. Slightly unconservative consideration of higher mode effects occurs in the upper stories using this method; nevertheless, the ELF method is recommended for predicting the distribution of forces across the height.
Elastic horizontal LFRS force demands depend on the stiffness of the vertical and horizontal LFRS and the mass distribution in the building models. Flexible diaphragms have small horizontal LFRS demands that are in close proximity to the predicted demands using either the bounds of the traditional diaphragm method in ASCE 7 or the alternative diaphragm method in ASCE 7-16. Rigid diaphragms experience large force demands caused by interactions between the horizontal system and higher modes of the vertical LFRS, which exceed the predicted demands from both the traditional and alternative methods.
The parametric study is extended to include inelastic responses in both the vertical and horizontal LFRS in the companion to this paper. 45 Inelastic vertical and horizontal LFRS demands and ductility demands are presented from the parametric nonlinear time-history analysis. Vertical and horizontal LFRS forces are compared to current seismic demands and recommendations for use of design methods are made. Future work includes prediction methods for the diaphragm demands using various seismic design methods for non-structural components.

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, 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.