Effect of damping model and inelastic deformation on the prediction of vertical seismic acceleration demand on steel frames

This study addresses the modeling of different energy dissipation mechanisms for numerical prediction of the vertical acceleration demand in regular moment-resisting steel frame structures. One of the issues discussed is the consideration of viscous damping in the structural model. It is shown that well-established Rayleigh-damping may highly overestimate the damping of the vertical modes, resulting in much too low vertical acceleration response predictions. A study with different damping models provides an appropriate damping modeling strategy that leads to reasonable predictions of both horizontal and vertical frame acceleration demands. Another open question is the effect of inelastic material behavior on the vertical acceleration demand on the considered regular structures. The results of a shell model of a frame structure exposed to high intensity ground motion excitation demonstrate that inelastic material behavior has virtually no impact on the vertical acceleration demand, while structural inelasticity leaves the horizontal acceleration response significantly smaller compared to the elastic demand. This leads to the conclusion that common frame models that capture the inelastic horizontal response but behave elastic in the vertical direction are suitable for the computation of both the horizontal and vertical acceleration demand.


Introduction
The importance of assessing the absolute acceleration response of earthquake-excited structures follows from the direct correlation of this response quantity with the damage of very stiff non-structural building components (Taghavi and Miranda 2003).
A large part of the studies reported in the literature deals with the prediction of the horizontal acceleration response (Flores et al. 2015;Perrone et al. 2018), while the vertical component is often neglected because it is assumed that buildings behave practically 1 3 rigidly in vertical direction. Contrary to this widespread assumption, recent studies have shown that even for regular structures, the vertical acceleration demand should not be neglected (Moschen et al. 2016;Francis et al. 2017;Gremer et al. 2019;Manoukas 2021;Valdés-Vázquez et al. 2021;Xiang et al. 2022;Lachanas et al. 2022). However, the influence of energy dissipation due to damping modeling and inelastic material behavior on the numerical prediction of vertical acceleration demand is still an open question.
Many papers are dedicated to a realistic estimation of the damping coefficients and investigate the effect of damping modeling, but only considering the horizontal component of the seismic response (Satake et al. 2003;Fritz et al. 2009;Bernal et al. 2015;Cruz and Miranda 2017;Luco and Lanzi 2019). For instance, Cruz and Miranda (2021) addresses soil-structure interaction and its impact on effective damping coefficients or Kazantzi et al. (2020) investigates the impact of damping on the acceleration demand deduced from instrumented buildings. Since these studies focus only on the horizontal seismic response, one objective of the present study is to propose a damping model that is reasonable to use when the vertical acceleration component of regular steel frame structures is also considered. Preliminary studies on this issue were conducted on a twelve-story frame by the authors Gremer et al. (2020). As will be shown below, well-established damping models can significantly overestimate the damping of the vertical modes, resulting in a substantial underestimation of the vertical seismic acceleration demand.
At this point, it should be noted that it would be equally important to know the actual values of the damping ratios to be applied to the vertical modes. Recent studies by Cruz and Miranda (2021) have shown that for very stiff structures, the effective damping ratio for the first horizontal mode of vibration can be as high as 8-15% when soil-structure interaction is taken into account. It is therefore reasonable to assume that the effective damping ratio of the vertical mode can also be large if the soil-structure interaction is significant. However, this issue is beyond the scope of the present study but should be investigated in the future.
The prediction of the response of non-structural components attached to inelastic structures is addressed in various studies, such as (Adam and Fotiu 2000;Sankaranarayanan and Medina 2007;Taghavi and Miranda 2012;Anajafi et al. 2020). In these studies, only the horizontal acceleration demand is investigated. The influence of inelastic deformation on the vertical acceleration response of regular structures, however, is not yet fully understood.
The objective of the present contribution is to investigate the influence of energy dissipation due to damping modeling and inelastic material behavior on the prediction of vertical acceleration response of steel frame structures subjected to earthquake excitation. For this reason, response history analyses are performed on regular steel frame structures with two different modeling strategies of damping-Rayleigh damping and modal damping. In addition, the impact of inelastic material behavior on the seismic acceleration response is studied. The numerically predicted horizontal and vertical acceleration demand from different ground motions is statistically evaluated, and the results based on these different models are compared and evaluated. This paper is structured as follows. After a brief description of the ground motion set on which the response history analyses conducted in this study are based, the modeling strategies for the considered steel frame structures, i.e., elastic and inelastic models discretized with beam and shell elements, respectively, and the five assessed damping models are explained in detail. The horizontal and the vertical maximum acceleration response amplification of representative nodes is presented in diagrams as a function of the normalized structural height. The response of different models subjected to a single ground motion is compared in both the frequency and time domains. The paper concludes with a discussion on the effects of damping modeling and inelastic material behavior on the numerical prediction of the seismic response, with emphasis on the vertical acceleration demand. As a result of this study, an appropriate modeling strategy is proposed that can be used to predict both the elastic and inelastic horizontal and vertical acceleration demand of regular steel frame structures.

Ground motion record sets
The study is based on the results of response history analyses, where both the vertical and horizontal components of the earthquake records simultaneously excite the considered frame models. The earthquake records used were assembled for the site of Century City, Los Angeles, in Moschen et al. (2016) to the so-called vertical ground motion (VGM) set with focus on the characteristics of the vertical component of ground motion. The application of the multi-objective optimization procedure described in Moschen et al. (2019) considering the constraints • moment magnitude M W ≥ 5.5 • site classification D according to NEHRP (stiff soil, shear wave velocity 180 m/s ≤ v s30 ≤ 360 m/s) • Joyner and Boore distance (Joyner and Boore 1981) less than 30 km • fault mechanism: strike-slip, reverse, reverse-oblique • same scale factor for all ground motions yielded 90 vertical ground motion records from the PEER-NGA database (PEER 2010) and a scale factor α = 2.10 For each of these vertical records, one of the two simultaneously recorded horizontal components was selected to form a vertical/horizontal ground motion pair.
In the period range from 0.025 s to 0.30 s the median of the vertical components corresponds to the normalized vertical response spectrum (NEHRP-spectrum (FEMA P-1050-1, 2015) and the dispersion matches the predefined target dispersion σ t = 0.80 (Moschen et al. 2016). The NEHRP target median response spectrum was used to ensure that the record set is suitable for estimating the vertical seismic response for structures whose vertical fundamental period (T 1,v ) is in the range 0.025 s ≤ T 1,v ≤ 0.3 s. The vertical fundamental period of all frame models considered falls within this period range. Figure 1 shows the response spectra of both the horizontal and vertical ground motion components, with thin gray lines denoting the response spectrum of a single record and bold blue lines referring to the statistical measures (median, 16th percentile, 84th percentile). The solid black line in Fig. 1c shows the target spectrum, i.e., the normalized vertical NEHRP-response spectrum (FEMA P-1050-1 2015).

Structural models
The seismic acceleration demand of one-, two-, four-, eight-, twelve-, and 20-story frames with three bays based on the FEMA P-695 archetypes (FEMA P-695 2009) is studied. The structural properties of the models are adopted from the steel-moment 1 3 resisting frames proposed as part of NIST GCR 10-917-8 (2010) Gremer et al. (2020). (ii) In Abaqus, a much more sophisticated and computationally demanding inelastic shell element model ("shell model") of the eight-story FEMA P-695 frame was built to gain insight into the influence of inelastic material behavior on the vertical acceleration response and to validate the results of the much simpler beam models. (iii) Building on the conclusions drawn from the outcomes of the employed shell model, the inelastic acceleration response is studied using modified OpenSees lumped plasticity beam models that account for inelastic structural behavior ("inelastic beam models"). The inelastic material behavior is concentrated in plastic hinge domains at the ends of the beams and columns, while the beam and column elements were modeled as elastic elements.

Elastic beam models
The design of the shorter frame models (one-, two-and four-story frames) is based on the equivalent lateral force (ELF) approach, while the response spectrum analysis (RSA) approach is used for the taller frames (eight-, twelve-, and 20-story frames). All frame models are designed for the maximum spectral acceleration D max (NIST GCR 10-917-8, 2010). 28.6% of the total story mass (blue tributary area in Fig. 2) is distributed to the corresponding beam of the actual frame structure, and the remaining 71.4% of the mass (green tributary area in Fig. 2) is lumped to leaning columns on both sides of the frames at a given level to achieve symmetric frame structures. Leaning columns, which have zero flexural stiffness, are used to account for P-Delta effects from the gravity loads in the tributary area shown in green in Fig. 2, as illustrated in Fig. 3. The seismic active mass of moment-resisting steel frame tributary area for gravity loads applied to the leaning column tributary area for gravity loads applied to the frame the actual frame is distributed along the beams and the beam-column connections as illustrated in Fig. 3a as both the vertical and horizontal acceleration demands of columns and beams are studied. Unlike the horizontal seismic response, where the distribution of the masses per story has virtually no influence, the vertical acceleration response is significantly affected by the number and location of the lumped masses (see Gremer et al. 2019). The cross-sections and moment of inertias of the beam components were taken from NIST GCR 10-917-8 (2010), and the Young's modulus of steel (E = 2.0 × 10 11 N/m 2 ) is used. In contrast to the models of the preliminary studies (Gremer et al. 2019(Gremer et al. , 2020, the models of this contribution consider reduced beam sections (RBS). The geometry of the considered RBS according to NIST GCR 10-917-8 (2010) can be seen in Fig. 4, where the beamcolumn connection of the studied shell model is shown. To model the RBS in the elastic beam model, each floor consists of beam elements that extend between the nodes of the RBS and beam elements that connect the RBS to the column lines. Between these beam elements are linear elastic rotational springs, whose stiffness depends on the reduced crosssection of the RBS to model the structural behavior of the RBS (Appendix B of Ibarra and Krawinkler 2005). In the elastic beam models, some simplifications have been made, such as using a centerline model, considering only the structural framing, and assuming linearelastic material behavior. Furthermore, no panel zones are taken into account in the elastic beam models. A more detailed description of the elastic frame models with RBS omitted can be found in Gremer et al. (2019). The structural response is shown for the nodes along the vertical axes A-A, B-B, C-C, and D-D in Fig. 3a.

Shell model
For the eight-story FEMA P-695 frame (FEMA P-695 2009), an additional shell model was created in Abaqus (Dassault Systèmes 2015), whose structural properties were also adopted from NIST GCR 10-917-8 (2010). Reduced beam sections (RBS) were realized by changing the geometry of the flanges. The flanges and webs were discretized with S4R shell elements. These are general shell elements with four nodes using reduced integration with hourglass control and bilinear interpolation of displacements and rotations. The crosssections consist of extruded planar shell elements assigned the appropriate shell thickness. Beam-column connections were modeled according to ANSI/AISC 341-16 (2016) with continuity plates and stiffeners, which were also discretized with shell elements. The beamcolumn connections, stiffened with eight bolts, were not modeled directly. Instead, tie constraints were used to connect the components at the locations of the bolts. In Fig. 4a, which shows the left beam-column connection of the first floor in detail, the constrained surfaces are highlighted in red. The stiffeners have the same shell thickness as the web of the associated beam. Doubler plates were modeled by increasing the shell thickness of the column web in the area between the continuity plates. The column splices are located 1.2 m above the beam-column joint, according to ANSI/AISC 341-16 (2016). A structured mesh with quadratic elements is used for most shell elements. Figure 4b shows the RBS, stiffeners, diagonal column connection (discretized with beam elements), and the mesh of the exterior left beam-column connection of the first story.
In contrast to the beam models, a consistent mass formulation is used for the structural elements. The payloads and additional dead loads from the floor construction are distributed uniformly over the surface of the top flange of the beams. The remaining dynamically effective floor mass (indicated by the green area in Fig. 2) is story-wise assigned to two leaning columns in the form of lumped masses, as shown in Fig. 3b. The leaning columns consist of B13 elements representing a linear two-node beam element. The connection of the leaning column beam element to the frame model is implemented by kinematic coupling constraints. The out-of plane-motion of the model is prevented by out-of-plane displacement boundary conditions in all elements. The frame model shown in Fig. 3b has a total of 283,114 degrees of freedom.
Since the principal effect of inelastic material behavior on the acceleration response is being investigated, rather than focusing on the precise modeling of this behavior with respect to softening and damage, the simplest possible model is used to capture the inelastic behavior. That is, the bilinear elastic-plastic model with the yield criterion of von Mises, implemented in Abaqus, with a yield strength f y = 379 × 10 6 N/m 2 and an isotropic strain hardening ratio of 1% is used.

Inelastic beam models
The elastic OpenSees model is modified to account for inelastic material behavior in terms of a lumped plasticity model according to NIST GCR 10-917-8 (2010). Therefore, the components of the inelastic OpenSees beam models are modeled as elastic elements and the inelastic behavior is concentrated in plastic hinge domains at the ends of the beams and columns. Again, reduced beam sections (RBS) are considered, while panel zones are not. Thus, each floor consists of elastic beam elements extending between the nodes of the RBS and elastic beam elements connecting the RBS to the column lines. The lumped masses are equally spaced along the horizontal beam elements. Bilinear rotational springs with suitable strength and stiffness properties, located between the beam elements (more specifically, between the beam element connecting the column to the RBS and the adjacent beam element of the bay), model the inelastic structural behavior based on the Young's modulus of steel E = 2.0 × 10 11 N/m 2 , yield strength f y = 379 × 10 6 N/m 2 , and 1% strain hardening. The properties of the spring elements represent the flexural stiffness of the associated elastic beam element and were determined with the approach described in Appendix B of Ibarra and Krawinkler (2005). This approach considers that the structural properties of the (entire) beam, connecting the two columns are a combination of the properties of the socalled subelements (beam elements and springs). Thus, the properties of the subelements 1 3 are a function of the stiffness of the beam. The other structural parameters are the same as for the elastic beam models listed in Gremer (2020). Figure 3c shows the left half of the symmetric eight-story lumped plasticity frame model.

Modal parameters of the structural models
The regular layout of the considered frames implies that horizontal and vertical degrees of freedom are decoupled, and thus, horizontal and vertical structural periods can be specified independently. In Table 1, the horizontal (T 1,h ) and vertical fundamental period (T 1,v ) as well as the period in which the sum of the effective horizontal (vertical) modal masses exceeds 95% of the total mass (T 95%,h and T 95%,v , respectively) of the original frame setups listed in NIST GCR 10-917-8 (2010) (referred to as "FEMA"), the OpenSees beam models ("OS beam"), and the Abaqus shell model ("ABQ shell") are given. Note that the periods of the elastic and inelastic OpenSees beam models ("OS beam") are identical. Since only the horizontal fundamental periods are provided in NIST GCR 10-917-8 (2010), there are no entries for the other periods of the original FEMA frames. In addition, the corresponding natural frequencies are also given in this table, because later the response is also shown in the frequency domain.
As can be seen, the fundamental horizontal period increases with increasing height from 0.42 s (one-story frame T OSbeam    = 0.71s ). In contrast, the models of the taller structures have a larger fundamental horizontal period compared to the FEMA frames. The differences in the fundamental horizontal periods may be explained by modeling inaccuracies and assumed simplifications used in this study. Of the modeled eight-story frames, the Abaqus shell model has the largest fundamental horizontal period ( T ABQshell The fundamental vertical period of the beam models is in the range of T OSbeam 1,v = 0.06s for the single-story structure and T OSbeam 1,v = 0.25s for the 20-story frame. For the eightstory frame, the fundamental vertical period is T OSbeam 17s , which is almost identical to the period of the Abaqus shell model ( T ABQshell The period T 95%,h ranges from 0.12 s to 0.97 s and the period T 95%,v from 0.02 s to 0.06 s. Figure 5 shows the first horizontal and the first vertical mode shape of the OpenSees single-story and the OpenSees eight-story beam model. The fundamental mode shape illustrated in the first column is dominated by horizontal displacements and vertical deflections are negligible. In the first vertical mode shape depicted in the second column, vertical/ transverse beam displacements control the dynamic response, thus, confirming the assumption of virtually decoupled horizontal and vertical degrees of freedom for the investigated frame models. The first vertical mode shape of the single-story frame shows the largest deflections in the center of the beams, while the displacements of the nodes of the column lines are virtually nonexistent. In contrast, the maximum vertical displacement of the first vertical mode shape of the eight-story frame occurs at the exterior column lines. Similar mode shapes are also found in the other frame models studied.

Damping models
The influence of damping modeling on the horizontal and vertical acceleration response is analyzed by means of two different damping modeling strategies, i.e., Rayleigh damping and modal damping. In earthquake engineering, Rayleigh damping is commonly used to account for viscous damping in the structural model. Rayleigh damping assumes that the damping matrix C is proportional to the mass matrix M and the stiffness matrix K (Chopra 2015), The proportionality coefficients α R and β R are derived from the modal properties of two predefined modes (denoted mth and nth modes) according to (Chopra 2015) where ζ m and ζ n are the damping coefficients and ω m = 2π/T m and ω n = 2π/T n the natural angular frequencies of these modes. In earthquake engineering, commonly the damping coefficient of the fundamental mode (m = 1) and the damping coefficient of the mode (n = n 95% ) where the sum of the effective modal masses exceeds 95% (sometimes 90%) of the total mass, i.e., ζ 1 and ζ 95% , are specified. The remaining damping coefficients ζ k with k = 2, …, n 95% − 1, n 95% + 1, …, N are then given by the relation (Chopra 2015) (1) In contrast, with modal damping it is possible to assign damping coefficients to each mode independently. However, in some commercial software packages this damping model is not implemented for nonlinear models.
Three variants of Rayleigh damping and two variants of modal damping are considered below. For all damping models based on Rayleigh damping, the damping coefficient of the fundamental horizontal mode and of the second mode considered is 5%. In the first implementation of Rayleigh damping (referred to as damping model 1 or R 95%,h ), the second damping coefficient is assigned to the mode where the sum of the horizontal effective modal masses exceeds 95% of the total mass, as is common in earthquake engineering (EN 1998(EN -1 2013. For the second implementation (damping model 2 or R 1,v ), the fundamental Mode shapes of the elastic OpenSees single-story frame model (first row) and the elastic OpenSees eight-story frame model (second row) (Gremer et al. 2019) vertical mode is adopted for the second damping coefficient. The third implementation (damping model 3 or R 95%,v ) uses the mode in which the sum of the vertical effective modal masses exceeds 95% of the total mass for the second damping coefficient. This damping model was employed in Gremer et al. (2019) to investigate the vertical acceleration demands of frame models. For damping model 4 or MD const , a constant modal damping of ζ k = 0.05, k = 1, …, N, is assumed for all modes. In a second implementation of modal damping (damping model 5 or MD lin ), the damping coefficient increases linearly with decreasing period. As proposed by Cruz and Miranda (2017), the damping coefficient increases by 1% per second period in this variant. Furthermore, the damping coefficient of the fundamental mode is not the same for all frames but decreases as the number of stories increases, as discussed in previous studies (Satake et al. 2003;Fritz et al. 2009;Bernal et al. 2015;Cruz and Miranda 2017). Thus, in damping model 5, the first mode of the single-and two-story frame is damped by 5%, the four-story frame by 4.5%, the eight-story frame by 4%, the twelve-story frame by 3.5%, and the 20-story frame is damped by 3%. Figure 6 shows the damping coefficient for the five assessed damping models as a function of period using the elastic eight-story beam model as an example. This figure illustrates that in Rayleigh damping there is a large difference in damping coefficients depending on which second damping coefficient is chosen. If the mode with horizontal period T 95%,h is chosen for assigning the second damping coefficient, the first and third mode of the eight-story frame are damped by 5%. Since the first vertical mode corresponds to the seventh global mode of this frame, the latter is already damped by 12.5%, as shown by the black line in this figure. Assigning the second damping coefficient to the mode with vertical period T 95%,v results in unreasonable low damping of the horizontal and vertical modes in the period range T 1,h < T < T 95%,v , cf. the blue line. The last variant of Rayleigh damping considered, where the first vertical mode is used to assign the second damping coefficient (illustrated by the orange line), leads to damping ratios between the previously discussed Rayleigh damping variants. For damping model 5 with a linear increase in modal damping Fig. 6 Damping coefficients of the eight-story frame as function of the period for different damping models with decreasing period, the damping coefficients vary between 4.0% at T 1,h and 6.4% at T = 0. Table 2 contains the damping coefficients ζ 1 , ζ 2 , ζ 95%,h , ζ 1,v and ζ 95%,v for all elastic beam models. It shows the large difference in damping coefficients between the considered damping variants. The taller the building is, the larger the difference becomes.

Assessment of the damping models
In the first study, the effect of the damping models on the acceleration demand of the considered elastic beam models is examined. For this purpose, the elastic beam models of the    Fig. 3. As a result of the time history analyses, the peak total floor acceleration components in horizontal and vertical direction (subsequently referred to as PFA h and PFA v , respectively) are divided by the peak ground acceleration component (PGA h and PGA v , respectively) of the corresponding ground motion record. These normalized response variables (PFA h /PGA h and PFA v /PGA v , respectively) are determined for each record and then statistically analyzed node by node with the median, 16th and 84th percentiles. First, a detailed illustration of the acceleration ratios PFA h /PGA h and PFA v /PGA v of the eight-story frame with constant modal damping ζ k = 0.05 (i.e., damping model 4 or MD const ) is presented. These results serve as a reference for the subsequent discussion on the effects of the different damping models on the acceleration response. In Fig. 7, thin gray lines illustrate the peak acceleration ratios for selected nodes over the normalized frame height h rel = h/H, where h denotes the height measured from the base (see Fig. 3a) and H the total height of the frame, as a result of the individual earthquake records. The solid thick red line corresponds to the median of these ratios, the dashed graphs represent the 16th and the 84th percentile values. The horizontal acceleration ratios of the nodes on the left column (axis A-A) are shown in Fig. 7a as representative for the entire structure, since the horizontal response is virtually the same for all nodes on the same floor, as discussed, for instance, in Gremer et al. (2019). In contrast, the vertical response acceleration depends largely on the position of the node within a floor, hence Fig. 7b-e show the ratios PFA v /PGA v separately for different nodes within a story. More specifically, Fig. 7b shows PFA v /PGA v of the nodes of the left column (axis A-A), Fig. 7c of the central node of the left bay (axis B-B), Fig. 7d of the nodes of the left interior column (axis C-C), and Fig. 7e of the central node of the second bay (axis D-D). As observed, in the lower stories the median horizontal PFA ratio is smaller than one, i.e., in many cases the peak response PFA h is smaller than the corresponding peak ground acceleration PGA h . On the roof, the median of the ratio PFA h /PGA h is about 1.14, see Fig. 7a. This response behavior reflects the well-known fact that the higher modes have a significant contribution to the horizontal acceleration response, as observed, for example, in Chaudhuri and Hutchinson (2004). In contrast, the median of the vertical acceleration ratio PFA v /PGA v increases with height, depending on the location of the node within a story. The maximum of the median PFA v /PGA v ratio of 3.69 is attained on the roof in the central node of the second bay, see Fig. 7e. The vertical accelerations are lowest in the nodes of the left interior column line, see Fig. 7d.
In the next step, the response of the eight-story elastic beam model using the other four damping models described in Sect. 4 is compared with the response shown in Fig. 7, which is based on damping model 4 (MD const ). Figure 8 illustrates the median of the dynamic amplification of this model for the five damping models analyzed. The presentation of the results follows the same scheme as in Fig. 7. The red line refers to the median response based on constant modal damping of ζ = 0.05 already shown in Fig. 7 (i.e., damping model 4 or MD const ). The results of the frame with linearly increasing modal damping (damping model 5 or MD lin ) are shown by the green line. The black lines illustrate the results based on Rayleigh damping, where ζ 1 = 0.05 and the second damping coefficient of 5% is assigned to the mode where the sum of the horizontal effective modal masses exceeds 95% of the total mass, as is common in earthquake engineering (damping model 1 or R 95%,h ). The orange lines represent the results for the second case of Rayleigh damping (damping model 2 or R 1,v ), where the provided damping coefficients are assigned to the fundamental horizontal and the fundamental vertical mode (i.e., the 7th global mode), therefore ζ 1,h = ζ 1,v = 0.05. The results for the third case of Rayleigh damping (damping model 3 or R 95%,v , where ζ 1 = 0.05 and the second damping coefficient of 5% is given for the period when the sum of the vertical effective modal masses exceeds 95% of the total mass, i.e., the 28th global period) are shown in blue.
Comparison of the horizontal response ratios presented in Fig. 8a shows that the commonly applied Rayleigh damping model R 95%,h (damping model 1) and the modal damping variant with constant damping of all modes (MD const ) yield median PFA h /PGA h predictions of similar magnitude. The largest difference of about 10% occurs at the second to last floor, which is within the modeling accuracy for earthquake engineering. When the second modal damping model (MD lin ) is used, the median horizontal acceleration amplification is already 30% larger compared to the traditional Rayleigh damping variant (damping model 1). This is due to the smaller damping coefficient of 4% for the fundamental horizontal mode. For damping model 2 R 1,v (i.e., Rayleigh damping with ζ 1,h = ζ 1,v = 0.05) or damping 1 3 model 3 R 95%,v (i.e., Rayleigh damping with ζ 1 = ζ 95%,v = 0.05), the structure experiences even larger horizontal acceleration amplifications over the entire height of the frame with a maximum value of 1.83 for R 95%,v at the top floor. This is explained by the fact that in these models the damping coefficients of the modes between the modes of the provided damping coefficients are too small, and therefore the modal response of the higher horizontal modes is much too large (see Fig. 6; Table 2). The maximum difference between damping model 3 and traditional Rayleigh damping (R 95%,h ) is about 57% and occurs at the second to last floor.
Examining the vertical accelerations shown in Fig. 8b-e, it can be seen that the two modal damping models (MD const and MD lin ) lead to similar outcomes. The median vertical accelerations based on a uniform modal damping of 5% (damping model 4) are only slightly smaller than those based on damping model 5 (MD lin ). The maximum difference of 11.3% can be found on the fourth floor at the central node of the left bay. Damping model 2 R 1,v leads to similar vertical amplifications compared to the modal damping variants, except at the central beam node of the second bay (axis D-D shown in Fig. 8e), where damping model 2 exhibits significantly smaller values over the height of the frame. In contrast, the traditional Rayleigh damping model R 95%,h (damping model 1) yields a PFA v /PGA v prediction in the half order of magnitude. For this model, the maximum dynamic amplification varies between 1.4 (left interior column line, Fig. 8d) and 2.2 (left exterior column line, Fig. 8b with modal damping is due to the (too) high damping of 12.5% of the vertical fundamental period T 1,v , as visualized with a black line in Fig. 6. As discussed earlier for the response in horizontal direction, the third Rayleigh damping variant R 95%,v (damping model 3) considerably overestimates the PFA v /PGA v demand with a peak value in the median of 4.7 (axis D-D).
To support these explanations about the effects of the considered damping models on the acceleration response, the frequency content of one randomly selected ground motion record and the corresponding acceleration response of the central node of the left bay of the top floor (denoted as "B8" in Fig. 3) of the eight-story elastic beam model are examined. Figure 9 shows the frequency content of the (a) first horizontal and (c) vertical component of the ground motion "RSN 1000" (PEER 2010) recorded at the station "LA-Pico&Sentous" during the Northridge earthquake in 1994 with a magnitude of 6.69. The Fourier transform of the horizontal (b) and vertical (d) acceleration response of the central node of the left bay on the top floor for the five damping models is also shown in this figure. To better illustrate the response normalized to the maximum value of one, the frequency axis ("f") is displayed logarithmically. The most important natural frequencies of the structure are highlighted by vertical gray dashed lines. The largest peaks of the frequency response are obviously found at the natural frequencies. These peaks show that not only the first mode contributes significantly to the acceleration response of the eightstory frame but also the higher modes. According to Fig. 9b, the horizontal acceleration is governed by the first three modes. While the first horizontal modal response is the same for all Rayleigh damping models and constant modal damping (damping model 4) (ζ 1 = 0.05 in these models), this figure confirms that for the third Rayleigh damping model R 95%,v (damping model 3), the higher modal response is larger compared to the other models because the corresponding damping coefficients of this model are smaller (see Fig. 6). The most important finding from the vertical response in the frequency domain (Fig. 9d) is that with the first Rayleigh damping model R 95%,h (damping model 1), the first vertical mode at f 1,v = 6.01 Hz is barely excited, resulting in a very small vertical acceleration amplification as observed in Fig. 8. The explanation lies in the excessive damping of this mode, which has already been discussed in detail, see also Table 2. For the other damping models, in particular for damping model 3 (R 95%,v ), this modal response is more pronounced.
In the following, the effect of damping modeling is investigated when the number of floors and thus the modal properties of the frames are varied. In addition to the eight-story structure, the acceleration response of the two-story ((a) and (e)), four-story ((b) and (f)), twelve-story ((c) and (g)), and 20-story ((d) and (h)) elastic beam frame model is examined. The response of the single-story frame was also evaluated but is not depicted here. Figure 10a-d shows the median of the horizontal acceleration ratios PFA h /PGA h of these frames and the median vertical acceleration amplifications PFA v /PGA v of the central nodes of the left bay (axis B-B in Fig. 3) are illustrated in Fig. 10e-h. The vertical response of the remaining nodes is not shown, as this does not lead to any significant gain in knowledge compared to the eight-story frame. As observed, the global response pattern of all frames is consistent with that of the eight-story structure discussed above. This means that traditional Rayleigh damping R 95%,h and constant modal damping of all modes MD const lead to approximately equal horizontal acceleration responses. The second modal damping variant MD lin and the second Rayleigh damping variant R 1,v have slightly larger median horizontal acceleration amplifications, while the third Rayleigh damping model R 95%,v significantly overestimates the horizontal acceleration response. This effect is small for the shorter frames, but the larger the number of stories and thus the larger the contributions of the higher modes to the horizontal acceleration, the larger the difference. The vertical node acceleration is underestimated when the traditional Rayleigh damping model R 95%,h is used and overestimated when the third Rayleigh model R 95%,v is used. The modal damping and the second Rayleigh damping options R 1,v provide vertical peak demands that are in between the two other Rayleigh damping variants. However, it is noticeable that the predicted vertical acceleration decreases with increasing number of stories using Rayleigh damping R 95%,h (damping model 1). While a magnification of 2.79 is observed for the twostory frame on the roof, this magnification is only 1.76 for the 20-story supporting structure on the roof. This is due to the fact that in this model the damping of the first vertical mode T 1,v increases with increasing number of stories, cf. Table 2. This could also be a reason for the fact that in most previous investigations the amplification of the acceleration in vertical direction was not detected.

Effect of inelastic structural deformations on the acceleration demands
In the second study, the effect of inelastic material behavior on the seismic acceleration demands is investigated. Since in previous studies (e.g., López-Garcia et al. 2008;Sankaranarayanan and Medina 2007;Taghavi and Miranda 2012) have already examined this effect on the horizontal acceleration demand in detail, the focus here is on the vertical acceleration response. In a beam model of regular frames (such as the ones studied here), the horizontal response is practically decoupled from the vertical response. Therefore, the representation of inelastic zones in such beam models in terms of common lumped plasticity elements therefore essentially affects only the horizontal response, leaving the vertical response virtually unaffected. However, it is not yet known whether this reflects the actual behavior.
In order to investigate the influence of inelastic deformations on the vertical acceleration of the considered structures, a more sophisticated model is required that more realistically captures the actual stress state. In the present study, the Abaqus shell model of the eight-story frame described in Sect. 3.4 is used for these investigations, which can predict the distribution of the plastic zones and, consequently, also possible effects on the vertical acceleration response. For inelastic models, modal damping is not implemented in the software suite Abaqus, therefore, Rayleigh damping R 1,v (damping model 2) is used for response history analysis. The authors are aware that Rayleigh damping in combination with inelastic structural deformation can lead to spurious damping forces and thus to a distortion of the predicted response (see e.g. Zareian and Medina 2010). However, since this study does not focus on the actual magnitude of the response but only on the principal vertical acceleration behavior in inelastic frame structures, this approach seems appropriate. The 90 records of the VGM record set are scaled with the factor α = 5.0 to ensure that substantial inelastic deformations emerge in the model. First, the response induced by one ground motion record is discussed in detail and subsequently the acceleration response of the whole record set is presented.
In the following, the response of the shell model resulting from the horizontal and vertical component of the scaled ground motion record "RSN 1000" of the PEER NGA database (PEER 2010) is examined, which is representative for the response resulting from the other scaled records. Figure 11 shows the time history of the horizontal acceleration (Fig. 11a) and displacement (Fig. 11b) response of the central node of the left beam of the top floor (denoted by "B8" in Fig. 3). In addition to the inelastic response illustrated by the blue solid line, the response of the elastic shell model is also shown by a red dashed line. The difference between the horizontal response of the inelastic and elastic frame models is evident, both in acceleration and displacement. The elastic horizontal peak floor acceleration ( PFA elastic h = 10.09 m/s 2 ) is 1.47 times larger than the corresponding inelastic peak response ( PFA inelastic h = 6.85 m/s 2 ). The drift in the displacement at the end of the observation period is characteristic for large inelastic structural deformations.
In contrast to the horizontal response, the vertical acceleration response is hardly affected by the inelasticity of the frame, as the results in Fig. 12 demonstrate. This means that the inelastic and the elastic response are almost identical over long time intervals. The inelastic PFA inelastic v is 10.91 m/s 2 compared to the elastic counterpart of PFA elastic v = 10.90 m/s 2 . To support this observation, Fig. 12 shows this response in detail for two time intervals (Fig. 12b, c) in addition to the corresponding vertical acceleration of the considered node through the entire observation period (Fig. 12a). The vertical displacement is not shown because it is negligible. Figure 13 illustrates the distribution of the equivalent plastic strain in the shell model at the end of the observation period to show that plastic deformations are confined to the reduced beams sections and to the columns close to the supports. In the inelastic beam models, springs are implemented at these exact locations to consider inelastic material behavior (see Fig. 3). This result supports that for the structures considered, a lumped plasticity model is reasonable for time-efficient response analysis. Both the inelastic and elastic acceleration response of the considered node in the frequency domain, shown in Fig. 14, confirm that the horizontal acceleration is considerably more affected by inelastic deformation than the vertical acceleration.
Next, the horizontal and vertical response of the inelastic shell model subjected to the 90 records of the VGM ground motion set is examined. The presentation of these results shown in Fig. 15 follows the same scheme as in Fig. 7. The horizontal acceleration response shown in Fig. 15a demonstrates that the median peak ground acceleration is larger than the median of the PFA h in all stories. This decrease of the acceleration can be attributed to energy dissipation due to inelastic material behavior. In contrast, the vertical median peak floor acceleration shown in Fig. 15b-e exhibits an increase over the height of the structure, as previously discussed for the elastic beam model. Figure 16 compares the statistical measures of the acceleration response of the elastic and inelastic eight-story shell model. Figure 16a, illustrates the large impact of inelasticity on the horizontal acceleration demand for the investigated steel frame structures using a scale factor of 5, which is consistent with many previous studies on this field. However, it should be noted that inelasticity might also have an adverse impact on the horizontal acceleration demands or, in other cases, only a minor impact. The maximum difference between the median horizontal PFA of about 75% is at the roof floor. In contrast, the vertical accelerations in Fig. 16b-e of the inelastic and elastic shell model are almost identical, confirming the finding from the shell model subjected to a single ground motion. On the second-to-last floor, the central beam node of the left bay (axis B-B in Fig. 16c) exhibits the largest difference in vertical median PFA of 9.1%. It can be concluded that inelastic material behavior has no significant influence on the vertical acceleration response of the investigated regular frame model. It should be noted that further studies, taking into account three-dimensional effects and the influence of plates, should be performed to validate these results.

Assessment of the inelastic OpenSees beam model
After evaluating the damping models and finding that the vertical acceleration response is only insignificantly affected by inelastic deformations of the considered class of regular frame structures, the accuracy of the acceleration demand from the computationally inexpensive OpenSees inelastic beam model is examined by comparing the results with those Fig. 13 Equivalent plastic strain PEEQ of the eight-story shell model excited by the scaled record "RSN 1000" at the end of response history analysis of the more accurate and elaborate shell model. For this purpose, the response of the eightstory frame structure subjected to the ground motion records of the VGM set scaled with a scale factor of α = 5.0 is examined in more detail as an example. To obtain comparable results, Rayleigh damping R 1,v is used in the inelastic beam model as in the Abaqus shell model. That is, the damping coefficient of the fundamental mode (ζ 1 = 0.05) and the first vertical mode (ζ 1,v = 0.05) is specified.  Fig. 18, the (a) horizontal and (b) vertical acceleration response of node "B8" of both models in the frequency domain is compared (solid blue lines refer to the shell model and red dashed lines to the inelastic beam model). As observed, the frequency domain response of both models is also a good match. Only the high frequency content in the range of 8 Hz ≤ f ≤ 13 Hz of the shell model shows larger vertical response amplitudes compared to the inelastic beam model. This can be attributed to local modes confined to the flanges of the shell model that do not exist in the inelastic beam model.  Figure 19 compares the statistical measures of the acceleration response of the inelastic beam model with those of the shell model for the VGM record set. All results show good agreement between the two different modeling strategies, except for the nodes of the interior column line (axis C-C shown in Fig. 19d), where the maximum difference in median acceleration of 30% occurs in the roof floor. Therefore, it is actually appropriate to utilize a lumped plasticity beam model, commonly used in earthquake engineering, to predict the seismic horizontal and vertical acceleration demand of

Inelastic versus elastic acceleration demand
Understanding that the lumped plasticity beam models are suitable for predicting the horizontal and vertical acceleration demand, the inelastic response is determined for all other structural models considered and compared with the response of the elastic beam models. In particular, the influence of inelastic material behavior is examined when the number of floors and hence the modal properties of the frames are varied. To this end, the computationally inexpensive OpenSees elastic and inelastic beam models with damping model 2 (R 1,v ) are subjected to the 90 scaled ground motions of the VGM record set with scale factor α = 5.0. Figure 20 compares the statistical quantities of the acceleration response of the of the left bay (axis B-B) of the two-story (a and e), four-story (b and f), twelve-story (c and g), and 20-story (d and h) inelastic beam models (blue solid lines) with those of the corresponding structures assuming elastic behavior (red dashed lines). The response of the single-story frame was also determined but is not shown here. The response pattern of all frames is consistent with the one of the eight-story structure previously discussed. As can be seen, the inelastic median ratio PFA h /PGA h for all stories is less than one, i.e., the horizontal peak floor acceleration demand is smaller than the corresponding horizontal PGA. In contrast, the median PFA h /PGA h of the elastic model is larger than one over the entire frame height, with a maximum value of 2.3 at the roof of the two-story structure. The largest difference of 63% between the median response of the two models is observed at the nodes on the roof of the 20-story frame structure (depicted in Fig. 20d). On the other hand, the vertical acceleration response is almost unaffected by inelasticity, resulting in virtually the same vertical acceleration response for models considering elastic and inelastic material behavior. The vertical acceleration amplification of the central nodes of the left bay shown in Fig. 20e-f have a median maximum value in the range between 2.64 (twelve-story frame) and 2.85 (20-story frame). At the roof of the two-story frame, the largest median response difference is 6.0% (see Fig. 20e).

Conclusions
Recent studies have shown that in earthquake-excited elastic steel frame structures the vertical acceleration response is not negligible, which contradicts the previous assumption that buildings behave rigidly in vertical direction under seismic impact. However, since in these studies the predicted acceleration response appears unrealistically large, the damping model used was doubted. In an effort to uncover these contradictory results for the vertical accelerations, in this contribution response history analyses were performed on regular Fig. 20 Median of the horizontal (a-d) and vertical components (e-h) of the normalized peak floor acceleration along the axis B-B for the elastic and inelastic a and e two-story, b and f four-story, c and g twelvestory and d and h 20-story beam model steel frame models using different damping strategies. The objective of this study with different damping models is (i) to highlight the related differences in the computed vertical acceleration response and (ii) to propose a modeling strategy that is easily applicable to response history analyses. Another issue not yet answered is the effect of inelastic deformation on the vertical acceleration of regular frame structures. To address this issue, time history analyses were performed on an inelastic shell model of a frame structure, which (unlike the commonly used lumped plasticity beam models) is closer to reality but also much more complex and computationally expensive.
The findings of the studies presented in this contribution can be summarized as follows.
• The results show that the choice of the damping model has a large impact on the numerically predicted vertical acceleration response. • Rayleigh damping as commonly used in earthquake engineering (the damping coefficients of the first mode and of mode where the sum of horizontal effective modal masses exceeds 95% of the total mass are given), significantly underestimates or even suppresses the vertical peak floor acceleration demand. • However, in the scope of Rayleigh damping, if the second damping coefficient is assigned to the mode where the sum of the vertical effective modal masses exceeds 95% of the total mass, the first few vertical modes are damped (too) low. This damping option results in an unrealistic increase in the well-studied horizontal acceleration response, leading to the assumption that the corresponding vertical maximum floor acceleration of up to five times is also too large. • Rayleigh damping, where the second damping coefficient is assigned to the fundamental vertical mode the acceleration response, provides more realistic response predictions that fall between the other two Rayleigh damping options. • Modal damping with equal damping coefficients over the entire period range also results in horizontal and vertical peak floor accelerations of reasonable magnitude, similar to the response of the Rayleigh damping variant where the second damping coefficient is assigned to the fundamental vertical mode. • Allowing for a slight increase in modal damping with decreasing period, as suggested in previous studies, leads to almost the same response predictions as constant modal damping. • Another possibility is to specify the damping coefficients for the horizontal and vertical modes independently and separately. The problem, however, could be the practical implementation in commercial software. • It can be concluded that the inappropriate choice of the damping modeling lead to the incorrect assumption that regular frame structures behave rigidly in the vertical direction. • The effect of inelastic deformations on vertical floor accelerations is insignificant and can be neglected. • Consequently, no particular inelastic elements need to be considered in lumped plasticity frame models.
Summarizing these findings, the main conclusion is that for the prediction of the inelastic horizontal and vertical peak floor acceleration demand, the common inelastic lumped plasticity frame models without special vertical inelastic elements can be used. However, viscous damping should be considered in the form of modal damping or a variant of Rayleigh damping, where the damping coefficient is provided for the fundamental horizontal mode and the fundamental vertical mode. A suitable damping model is in particular crucial for the prediction of the vertical acceleration demand. It should be noted that the studies performed give an insight into the vertical seismic earthquake acceleration response of flexible steel frame structures and that the results should not be generalized, such as for very stiff structures, reinforced concrete structures or shear walls.
In the next steps, three-dimensional effects and the effects of slabs in the numerical models should be studied. A further open question, which requires a profound investigation, are the actual values of the damping coefficients for the vertical modes, especially when soil-structure interaction plays a significant role. To this end, experimental investigations and evaluation of the recorded responses of instrumented buildings as well as parametric numerical studies on a soil-structure model should be performed.