Performance of inherent damping models in inelastic seismic analysis for tall building subject to simultaneous horizontal and vertical seismic motion

In recent seismic analyses, it is considered that the structural damping ratio should be treated as independent of frequency, for safety side estimation. Therefore, frequency‐insensitive damping is required for realistic seismic simulations. This paper investigates the performance of various sparse matrix damping models (extended Raleigh, capped viscous and uniform damping) in the inelastic seismic analysis of a 35‐story moment‐frame steel building. These sparse matrix damping models were compared with Rayleigh, tangent Rayleigh, and Wilson‐Penzien (modal) damping models to provide insight into damping models suitable for large‐scale inelastic response history analysis (RHA). First, the necessity of frequency‐insensitive damping in large‐scale analysis is illustrated via numerical simulations. Then, the vibration characteristics with simultaneous inputs of horizontal and vertical ground motion are analyzed using the abovementioned damping schemes, and their results are compared. The comparisons are analyzed by focusing on horizontal displacement/acceleration, story drift angle, beam‐end ductility factor, the amplitude due to beam vibration, and the associated vertical acceleration. Finally, the computation speeds are compared. As a result, it is shown that although these sparse matrix damping models are practically useful, they are not yet sufficient and present challenges.

should be conservative. Rayleigh damping makes it difficult to keep the damping ratio constant in the 0.2−10.0 Hz range. ℎ = 10.0(Hz)/0.2(Hz) = 50.0, and the only damping models that can be used for this are constant modal damping 7 and uniform damping. 19 This study will also verify about the effects of vertical seismic motion for beam vibration.
This study first provides an overview of recently published frequency-insensitive damping models. Then, the results of the various responses of these damping models when applied to a 35-story steel-framed building subjected to horizontal and vertical movement are compared, and the properties of a desirable damping model are discussed.
The damping models used in the study are Rayleigh, tangent Rayleigh, extended Rayleigh, constant modal, 7 capped viscous, 6,13,30 and uniform damping. 19 Finally, the time spent in the analysis by using the various damping models is measured.
The author codes the nonlinear direct-integration time-history response program used in this study. Program code is in C++ and compiled by applying the -O3 optimization level in Intel C++ Compiler. Previous papers on the damping model [16.30] [16.11] have been written using this program.
The modeler for the studied structural model is SS7 23 ; this software is a commercial program commonly used as a structural design tool in Japan.

OVERVIEW OF FREQUENCY-INSENSITIVE DAMPING MODELS
This section provides insights into the vibration characteristics of modal (Wilson-Penzien), 7 capped, 30 and uniform, 19 Rayleigh, tangent Rayleigh, and extended Rayleigh 18 damping.
The dynamic equation formulation is as follows: where is the mass matrix, is the restoring force, ( ) is the damping force and ( ) is the external force. The following outlines how the damping forces ( ) are modeled for each damping model.

2.1
Individual model descriptions

Wilson-Penzien damping
Wilson and Penzien 7 expressed the modal damping matrix as follows: is the damping ratio; and are the undamped natural vibration mode vector and natural frequency of the th mode, respectively; and is the generalized mass, . From Equation (2), if the mass is zero, then no damping force is generated in the massless degree of freedom (DOF). In general, the nodes of the frame model are assigned concentrated masses in the translational direction, but inertial masses in the rotational direction are most often ignored. Therefore, theoretically, the plasticization of the beam end moment does not generate spurious damping force due to rotational damping force. Chopra et al. 12 suggested the use of constant modal damping as a countermeasure to spurious damping forces, but it increases the computational load because it uses a dense matrix. To avoid this problem, Chopra et al. 12 proposed the use of temporary damping matrix * which is partial damping matrix for overlap with stiffness matrix in effective stiffness matrix and solve the displacement sparsely; The balance with the exact modal damping force evaluated by is resolved by the Newton-Raphson method in the dynamic equation of motion. Then, this idea was incorporated into OpenSees. 24 However, since the equations of motion require a dense matrix , its practicality in large-scale analysis is questionable. Therefore, it is not considered in this study.
Wilson-Penzien damping is similar to the problem of Rayleigh damping (e.g., 6,[33][34][35], if the foundation is not fixed (e.g., in uplifting and base isolation structures) or there is large peeling between elements (e.g., cracks in concrete), deformation will be underestimated, and the model may become nonconservative.
According to Luco and Lanzi,25 in modal damping, after the nonlinear element transition from the elastic state to the plastic state, the DOF without mass becomes an unintended velocity response because of numerical artifacts and the absence of damping terms. Model damping can be improved by adding infinitesimal stiffness-proportional damping to avoid this problem.
This damping model is abbreviated as WP in this study.

Capped viscous damping
The capped viscous damping model was proposed by Hall 6 for use in complex seismic simulations involving highly nonlinear behaviors. This model can be applied to the sliding or uplifting of a foundation, among others. Although tangent Rayleigh damping is effective in such an analysis, it has little physical basis for reducing the damping force and overestimates damping due to the influence of the mass term.
In the case of initial stiffness-proportional damping, the restoring force and damping force due to the arbitrary story stiffness have the following relationship: If the spring is elastic and oscillated in the first mode, the peak values of the restoring and damping forces will maintain a ratio of 2 , with a phase difference of π/2. However, if the spring will exceed the yield displacement, and the will peak at yield strength , the ratio of the to the will exceed 2 . Hall 6 reported that the maximum damping force is 60% of the yield strength of a building, but this response result is nonconservative. This problem can be avoided by setting the capped viscous damping force as follows: Here, needs to be set before analysis, but due to the lack of any established idea of determining its value, it should be set individually by designers according to their judgment. In Hall 6,13 and Qian et al. 26 as examples of the setting of , the yield strength of stories was assumed and determined by calibration. The authors proposed the improved capped viscous damping model, 30 whose element damping force is capped by the peak restoring force of the inelastic element. This damping model is frequency insensitive when a single major vibration mode dominates, but its accuracy is poor when multiple modes are excited.
This damping model is abbreviated as CP in this study.

Uniform damping
Huang 19 proposed a new damping model that is based on rate-independent damping theory and achieves frequencyinsensitive damping over a wide frequency range. ( ) is replaced by a set of N filtered force histories ( ), n = 1 to N.
where is the damping ratio; is the cutoff frequency at which the damping ratio is fitted to the target value within the assumed frequency range; is the adjustment factor vector, where = [ 1 , … , ], for adjusting the damping ratio to be uniform within the assumed frequency range. Huang et al. 19 calculated by optimization using the least-squares method.
The damping ratio is generally constant over the wide frequency range. However, it is important to note that this damping model increases dynamic stiffness. This is also described in the LS-DYNA User Manual. 21 Since the apparent natural period of the building changes, its effect on the response analysis cannot be ignored. There is no effective countermeasure for this, and the user manual recommends that the user directly correct the building stiffness with Young's modulus or other factors. In this study, the period of vibration was measured from the time between amplitudes of the free vibration results. The ultimate strength is also a significant issue; however, it is not mentioned anywhere. Therefore, this study experimentally compares three different cases: • The stiffness and ultimate strength are not corrected (UN0).
• Both the stiffness and ultimate strength are corrected (UN2).
The implementation method of the program used in this paper is shown in Appendix 1.

Rayleigh damping
The Rayleigh damping matrix is often used for its ease of use. It is a classical damping model expressed by damping terms proportional to the mass matrix and the initial stiffness matrix [Equation (7)].
Hence, if coefficients 0 and 1 are known, then the th-mode damping ratio can be determined as follows: The stiffness-proportional term 1 of Equation (7) is a model created based on an initial stiffness assumed regardless of the nonlinearity of RHA. However, Ref. 8 reported that an unintended spurious damping force is generated in plastic hinges, and the generation mechanism was explained later by Ref. 9. This phenomenon occurs when a stiff nonlinear element is explicitly incorporated into the end of a beam. Furthermore, Léger and Dussault 10 and Charney 11 stated that nonlinearity may change natural frequencies.
The tangential stiffness matrix is used instead of the initial stiffness matrix to avoid these problems. * = 0 + 1 Many studies have pointed out this model's lack of physical basis. Although the new Rayleigh damping model, which acts only on elastic velocity, 27 has been proposed to improve it, a conclusion has yet to be reached as to what modeling is appropriate.
Rayleigh damping model is referred to as RI, and tangent Rayleigh damping model is referred to as RT in this study.

Extended Rayleigh damping
Although Rayleigh damping is simple and useful, its major challenge is that the range where the damping can be considered constant is extremely narrow. In contrast, the extended Rayleigh damping model replaces the stiffness proportional damping with causal damping, 17 which allows for a wider adaptive frequency range. According to Nakamura, 18 damping force vector ( ) due to the extended Rayleigh damping can be expressed as Equation (10).
TA B L E 1 Characteristics of frequency-insensitive damping models.

Extended Rayleigh Wilson-Penzien
Capped viscous Uniform G: good, P: poor, D: depends on condition.
The coefficient matrix of the first terṁ( ) corresponds to the Rayleigh damping force, and the second term is the time-delay component due to the causal damping force. is the element's restoring force vector, which means that past responses ( − ) and ( − 2 ⋅ ) will affect the current damping force ( ) with a time delay. 0 , 1 , and 2 are the values that are optimized to become stable and constant in the adaptive-frequency range.
is the upper limit of the frequency range to be kept constant, and a stable constant damping ratio is generated around the middle value /2 in this frequency range.
This damping model is referred to as EXR in this study.

Organizing the characteristics of individual models
A good damping model must satisfy the following criteria: 1. Accurate fitting for any number of modes 2. Continuously varying damping force 3. No excessive damping against rigid body motion (e.g., uplifting, sliding) or collapse 4. No spurious damping force due to nonlinearization of inelastic spring 5. Damping force effect should weaken after element plasticity 6. Ability to compute easily and efficiently 7. Ability to model with a small amount of memory Their main characteristics, advantages, and disadvantages are listed in Table 1. Extended Rayleigh damping is rated "D" for (1) because the adaptive frequency range is wider than that of Rayleigh but not as wide as that of uniform damping, making it difficult to accommodate some of the objectives.
Uniform damping is rated "D" for (1) because the damping ratio drops off outside the adaptive frequency range, so there is a region without damping. However, damping can be considered for higher-mode vibration by adding slight stiffnessproportional damping. Rayleigh and Wilson-Penzien are rated "D" for (5)  that viscous damping must be appropriately reduced after plasticity. However, no scientific evidence shows that the elastic damping disappears after plasticity. Young's modulus of steel is 205,000 (N / mm 2 ). The yield strength of the steel is 358 (N / mm 2 ). Assume that the building is constructed on rigid ground and that there is no effect of soil-structure interaction. The building is shaped and has a longitudinal direction length is 72.0 (m), shorter direction is 36.0 (m). The height of each story is uniformly 4.3 (m) for all stories.

OVERVIEW OF ANALYZED BUILDING
Evaluating axial column forces or assessing beam vibration due to horizontal and vertical movements are important in structural design (e.g., 28,29 ). The main vertical vibration modes of the overall building strongly influence the main vibration modes of the columns. On the other hand, beam vibrations are influenced by some modes, such as the main vertical vibration modes of the overall building and local self-vibration modes [e.g., 22 ]. However, there is little knowledge about the effect of vertical movement on beam vibration, so it is difficult to determine which mode to focus on to set the damping.

Model for analysis
The study model in Figure 2A is the plane model of Y6 frame shown in Figure 1. The red circles indicate nodes. The nodal masses are generally the same at each floor: 12.4 (t) at the side column location m 1 , 10.6 (t) at the center location m 2 of the main beam, and 22.9 (t) at the middle column location m 3 . The horizontal natural first mode period is 4.85 (s), the second mode period is 1.62 (s) and third mode period is 0.93 (s). The beam is divided into small sections for the direct evaluation of beam vibration so that local vibration modes are generated in the beam. By using a split beam in this way, the displacement and acceleration response inside the beam can be obtained by analysis results. Figure 2B shows a cumulative graph of the horizontal and vertical effective mass ratios versus frequency. The vertical effective mass ratio of s-order mode ( = 2 ⋅ ⋅ ⋅ ∕ ) is defined as the ratio of the effective mass 2 ⋅ ⋅ ⋅ to the total mass of building derived from the participation vector ⋅ and mass matrix . The same procedure is used to obtain the horizontal effective mass ratio. In this sample model, the effective mass ratio for horizontal direction accounts for 77% in the first mode and over 90% in the third mode. The effective mass ratio for the vertical direction accounts for 77% of the vertical first mode (overall 7th mode) and slowly increases for higher modes. Unlike the major modes in the horizontal mode, those in the vertical mode are distributed over a wide frequency range. Figure 2C shows damping ratio vs. frequency of several Rayleigh damping applied to this analysis. represents the ratio of the damping rate created by the damping model to the target damping rate. In the elastic analysis, vibration analysis can be performed separately for horizontal and vertical motions; for vertical motions, Rayleigh damping ratios need to be set only at two frequencies about the 7th and 40th modes. In this range, a stable constant damping ratio is obtained. On the other hand, the inelastic analysis must be performed with simultaneous horizontal and vertical inputs. If Rayleigh damping is set at the first and 40th modes, there will be a region of a quite low damping ratio between the two modes. Figure 1D shows the main vertical vibration modes. The display scale of each mode vector is normalized so that the maximum displacement is the same amount for all modes. The overall 7th mode of the vertical first mode is dominated by column deformation, but beams can also be seen to be slightly deformed. This means that the 7th mode is a deformation mode that involves not only vertical displacement but also slight rotation of the beam end nodes. The 7th−12th modes, due to the columns' compression and tension, are important column vibration modes.
The validity or effectiveness of the compared damping models is verified based on the seismic response analysis results. Benchmark tests are then conducted on their required storage spaces and computation times.
Columns are assumed to be elastic, and beams are assumed to have bilinear hysteresis properties at beam-ends. As discussed by Chopra and McKenna 12 and Hall, 13 the inelastic beam element is modeled as a series element of a rigidplastic spring and an elastic beam to avoid generating spurious damping forces due to the plasticization of the rigid-plastic springs.

Compared damping models
The relationship of the damping ratio of each damping model with frequency is discussed in this chapter. The damping models used for numerical simulation are RI, RT, EXR, CP, UN, and WP models. The target damping ratio is set to = 2%. The relationship between the damping ratio and frequency of each damping model is evaluated from the elastic free vibration analysis of a single-mass system; The damping ratios are calculated from the logarithmic decrement of free vibration amplitude of 100 single-DOF (SDOF) vibration models (individual natural frequencies of 0.1, 0.2, 0.3, . . . , 10 Hz), which were subjected to a unit impulse shown in Figure 3A. The relationship of damping ratio vs. frequency  for each damping model is shown in Figure 3B. The accuracy of the damping ratio of each mass point evaluated from the free vibration results with respect to the target damping ratio is checked by calculating their ratio = ∕ . Each damping model is set as follows: • RI and RT models The target damping ratios are set at the frequencies of the first mode (0.2 Hz) and the seventh mode (2.9 Hz). This model is named RI (1-7) and RT (1-7); = 1 = 2 in the first and seventh modes. Similarly, RI (1-40) and RT (1-40) are set at the frequencies of the first mode (0.2 Hz) and the seventh mode (7.2 Hz). As seen in Figure 3, Rayleigh damping (1−40) has a region with a low damping ratio (between the first and 40th modes) and Rayleigh damping (1−7) has a high damping ratio outside the adaptive region.

• EXR model
The upper limit of the adaptive frequency is set to 4.0 (Hz), and a high-accuracy model presented in Nakamura 18 is used. As shown in Figure 3, a region of constant damping ratio is formed between 0.2 (Hz) and 3.0 (Hz). However, in the frequency range above 3.0 (Hz), the damping ratio increases significantly in the frequency range above 3.0 (Hz). Therefore, excessive damping is given to 12th modes and higher.

• CP model
The initial damping coefficient is stiffness-proportional damping proportional to the first mode. That is, Equation (4) becomes = 2 ∕ 1 . Depending on the frequency, the damping ratio is 1.5 to 2.0 times the target for second modes and higher.

• UN model
The cutoff circular frequency is set to 1.00, 4.64, 21.54, and 100 rad/s, and the optimization results for are 1.417, 0.918, 0.918, and 1.417, respectively. As seen in Figure 3, The damping ratio is generally constant between 0.2 and 10 (Hz), showing high frequency independence. For countermeasures against the increase of dynamic stiffness, the period of vibration was measured from the time between amplitudes of the free vibration results, and the result was 4.73 (s). The period ratio to the undamped period of 4.85 (s) is 4.73/4.85 = 0.974, and the stiffness ratio is (0.974) 2 = 0.9508. Therefore, the apparent period was adjusted by multiplying the material Young's modulus of this building by 0.95 (corresponding to the case UN1 shown in Section 2.3). The case UN2, where the ultimate strength is corrected in addition to the material Young's modulus, is also multiplied by 0.95. It should be noted that the time increments must be set appropriately small for a sufficiently accurate, as discussed in Appendix 1.
• WP model The damping ratio is set to 2% for all modes. The damping matrix is calculated from the elastic eigenvalue results, and the damping matrix is calculated without reflecting member nonlinearities during the analysis. It is confirmed in preliminary comparisons that the difference due to member nonlinearities during is slight.
In recent years, the vertical seismic motion has been recognized as important and argued to be included in seismic analysis of structures (e.g., 28,29 ). High frequencies of vertical seismic motion can stimulate column axial forces, which can strain steel column splices and affect the flexural and shear strength of concrete columns. If vertical seismic motion is included in the seismic analysis, damping against vertical vibration must be properly set. However, the damping of buildings against vertical vibration is even less well established than the damping against lateral vibration. The reason for this is that it is recognized that most major damage to buildings is caused by horizontal motion earthquakes, and vertical motion has not been studied in as much detail as horizontal motion. Even today, little is known about the damping ratio for vertical vibration. Therefore, in comparing various damping models in this study, WP, which is a constant damping for all modes, is used as a criterion for evaluation. This is not to argue that WP is the most realistic model, but rather that it should be considered conservatively given our limited knowledge.

Ground motion and analysis conditions
The input ground motions are considered as once in 50 years (Ground motion scale factor = 0.2) and once in 500 years (Ground motion scale factor = 1.0). The buildings are assumed to be within elasticity for an earthquake once in 50 years and to allow plasticity of the members for an earthquake once in 500 years; the ductility factor at the beam-ends is generally limited to about 4.0. The intention of examining the two cases is to confirm the difference in behavior between the elastic and inelastic cases. Seismic motion is inputted simultaneously in horizontal and vertical directions. Figure 4 shows the simulated earthquake motion and the response spectra of acceleration and displacement. The figure also shows the location of the horizontal first mode and the vertical first mode (overall seventh mode). The effect of seismic motion on the vertical modes is significant for the acceleration response but minor for the displacement response. The Newmark-β method (β = 1/4) is used for time integration. The integration time interval Δ was changed for each damping model. Δ = 0.01 ( ) was used for RI/RT, EXR, and CP because stable results were obtained even with rough time increments. Δ = 0.005 ( ) was used for WP because the solution diverged at Δ = 0.01 ( ) but stabilized at Δ = 0.005 ( ). For UN, fine time increment is required, and the solution was stable at Δ = 0.0005 ( ), while solutions diverged at rougher increments. Accuracy verification of each damping model is presented in Section 6. In general, columns are assumed to be elastic under seismic motion once in 500 years, so columns are assumed to be elastic, and beams are assumed to be inelastic in this study as well.

RESULT
The purpose of this section is to observe the effect of each damping model with different relationships between frequency and damping ratio on the various peak response results. Figures 5 and 6 show the elastic peak response result at the scale factor = 0.2 and Figures 7 and 8 show the inelastic peak response result at the scale factor = 1.0. The beam amplitude is calculated by the difference between the nodal vertical displacement at the center of the long-span beam and the average of the vertical displacements at the nodes at both ends of the long-span beam. The vertical acceleration is the vertical response at the center node of the long-span beam. The shear coefficient is the value obtained by dividing the story shear force at each story by the weight above that story. The side column axial force ratio is a graphical representation of the ratio of the side column response by each damping model to the side column response of the WP. Since the inelastic analysis showed no significant difference between RI and RT, RT is omitted from this report. The results for elastic peak response in Figure 5 show that the horizontal displacements are similar for all models because the low-order mode is dominant. On the other hand, the RI (1-40) is larger than WP, possibly because RI (1-40) underestimates the damping of 40 modes from the second mode. UN1 tends to be closer to WP than UN0. This is because UN0 had an increase in dynamic stiffness due to damping, whereas UN1 had that increase cancelled out by reducing the stiffness of the main structure.
The horizontal acceleration response is varied among the models, and CP is a slightly smaller response. The response of RI (1-7) and RI (1-40) is larger than WP because the horizontal higher mode damping after the second mode is underestimated. The vertical acceleration responses of RI (1-7), EXR, and CP are smaller than WP because higher vertical mode damping is overestimated. There is no marked difference between UN0 and U01.
From the results in Figure 6, the story shear force coefficient and story drift angle are larger for RI (1-7) and RI (1-40). This is due to underestimating the horizontal higher mode damping after the second mode. For the same reason, the results for RI (1-7) and RI (1-40) are also larger for the story drift angle. The side column axial force ratio is influenced by not only vertical vibration modes, but also over-turning-moment due to horizontal motion. Therefore, response of RI (1-40) is larger than other damping model reason by underestimating the horizontal higher mode damping. RI (1-7) is also slightly larger than WP, this indicates that the effect of vertical higher mode vibration is not so significant. The response of CP is slightly smaller in the upper stories. This may be due to an overestimation of horizontal higher mode damping. TA B L E 2 Data size of test model.

Model information Size
Nodes 756 Independent DOF 1785 TA B L E 3̃matrix sizes. In Figure 7, the vertical beam amplitudes of the CP and UN0 have a smaller response than the WP. UN1 is closer to the WP response than UN01, and UN2 is even closer. Horizontal displacement and acceleration show the same trend as during elasticity.

Matrix type Size
The inelastic responses shown in Figure 8 also behave in the same trend as the elastic response. However, the ductility factor of UN0 is greatly smaller than other damping models. Although UN1 is slightly closer to WP and UN2 is much closer, the difference between WP and UN2 is significant.

COMPUTER PERFORMANCE
The execution speeds of the models are compared to evaluate the computational load they require for RHA. The damping models to be compared are RI, EXR, CP, UN0, and WP. The number of nodes and DOFs of the analytical model are listed in Table 2. Table 3 lists the 1D array sizes required to represent the effective stiffness matrix̃. The array sizes are for the dense matrix for WP, which depends on the array size of the damping matrix , and the sparse (skyline) matrix for RI, EXR, CP, and UN0, which depends on the stiffness matrix . The difference in array size between the two matrices is about a factor of 38, which is the reason for the large difference in calculations. The calculations are performed on the workstation server equipped with large computation nodes. A Load-Sharing-Facility system installed in the operating system (Red Hat Enterprise Linux version 7.6) provides a stable computing environment, even during multiple computation processes. The computer used is a Lenovo ThinkSystem SD530 with an Intel Xeon Gold 6246 3.3 GHz CPU. Figure 9 shows the relationship between integral time interval Δ and the peak response value of the story drift angle, comparing the results of WP (Δ = 0.0005 ( )) except for WP. The results of WP for Δ = 0.01 ( ) are not shown because of divergence. The results for UN0 are also not shown because of divergence at time increments rougher than Δ = 0.0005 ( ).  Table 4 lists the computation times for the use of each damping model. RI, EXR and CP showed stable and accurate results even at rough time increments. UN0 spent a lot of CPU time because it requires sufficiently fine integration time intervals to obtain stable results. Since WP is a dense matrix, it requires a lot of CPU time for the Cholesky decomposition of effective stiffness matrix̃.

DISCUSSION
Focusing on the horizontal response, RI (1-7) and RI (1-40) result in underestimating horizontal higher mode damping and thus larger story drift angle, ductility factor, and acceleration. Focusing on the vertical response, RI (1-7) calculated in the horizontal first mode and vertical first mode results in a slightly larger response than the WP, while RI (1-40) results in a larger displacement response. This indicates that the vertical first mode is the primary source of the displacement amplitude of the beam caused by the seismic motion. This is very different from walking-induced vibration, in which the vibration of the beam itself is the main factor. However, vertical acceleration overestimates damping due to the greater influence of vertical higher modes.
EXR overestimated the damping for vertical acceleration but indicated good frequency independence for other responses. To achieve a constant damping ratio over a wider range of frequencies, the velocity term and the two time-delay terms proposed in Equation (10) could be improved, and the time-delay term could be made multinomial.
Although CP is a simple modeling approach and the theory is clear, it is difficult to predict in advance how it will affect the response because it is not possible to set an exact damping ratio for the target frequency.
UN is expected to be frequency-insensitive over a wide frequency range, but the issue is that the dynamic stiffness is increased by damping force. Although Huang 19 insists this characteristic is realistic based on material experiments, to the author's knowledge, any evidence that the actual building has this characteristic is unclear. Since the natural period is one of the important factors in RHA, most structural engineers may tune analytical origin stiffness to avoid increased dynamic stiffness. Furthermore, it is not clear how to treat structural yield strength. If the stiffness is simply reduced, the yield deformation increases. Therefore, the ductility factor of UN is smaller than other damping models. The behavior was analyzed experimentally by reducing the yield strength by the same value as the stiffness reduction ratio. A slight improvement was observed compared to the case without the strength reduction. However, this adjustment also has no basis, and is merely a reconciliation of the analytical results.

CONCLUSION
Although the mechanism of structural damping remains unclear, its impact on earthquake simulation is significant. Therefore, how to consider such physical quantities is an important engineering issue. In practice, this would incorporate assumptions on the safe side, but may overestimate or underestimate certain vibration modes when the frequency range under consideration is wide. This study provided insight into the frequency independence of damping models through a numerical example of a high-rise building subjected to simultaneous horizontal and vertical earthquake motion. At present, there is no damping model that can be commonly applied to all problems. Therefore, structural engineers need to select and use an appropriate damping model according to the purpose of the study. The findings are summarized as follows: • The situations in which frequency-insensitive damping is required in practical structural design are limited to special cases, for example, where the horizontal and vertical response accelerations need to be evaluated simultaneously. In a specific case, the vibration mode of the displacement amplitude generated in the beams of a tall building subjected to real-phase earthquake motion can be dominated by the building's first vertical vibration mode. Therefore, when evaluating the horizontal and vertical displacement response under simultaneous horizontal and vertical motion input, Rayleigh damping (RI (1-7)), in which damping ratios are set for horizontal and vertical first modes, can be used to obtain good results that are generally independent of the vibration frequency, although the horizontal higher mode damping is underestimated. However, when the horizontal and vertical acceleration responses are evaluated simultaneously, the effects of higher-order vertical vibration modes cannot be ignored, and it becomes difficult to set appropriate damping ratios with Rayleigh damping. This is evident when one considers the frequency components in realistic input seismic motions and the vibration modes excited by them, based on the response spectrum. • Although extended Rayleigh damping (EXR) can increase the frequency independence of horizontal higher mode damping compared to Rayleigh damping, it overestimates the vertical higher mode damping and is difficult to apply in the same way as Rayleigh damping when horizontal and even vertical accelerations are evaluated simultaneously. Attempts to expand the range of adaptive frequencies for extended Rayleigh damping should be a subject for future work. • Limited to horizontal response, capped viscous damping (CP) tends to be less dependent on frequency. However, it should be noted that when higher mode damping needs to be carefully considered (e.g., evaluate higher mode acceleration conservatively), there is a tendency to evaluate damping ratios with a large value. • Since there are few practical examples of Uniform damping (UN) and many unknowns, one of the objectives was to verify its potential. UN has a good performance for frequency-insensitiveness. The issues are that the apparent building period changes significantly because of the increase in dynamic stiffness. Unless there is evidence of this as a real phenomenon, it is doubtful that a damping model with a significant change in period can be used in practical structural design. And it is not suited for implicit solution method because sufficient time increment (Δ = about 0.0005 s) is required to demonstrate sufficient accuracy. • Of the sparse matrix damping models compared, extended Rayleigh damping (EXR) appears to offer the best balance between frequency independence and computational cost, but neither model is sufficient to account for the range from the horizontal first mode to the vertical higher mode.

A C K N O W L E D G M E N T S
The authors have nothing to report.

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 on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

A.1 Time increments and accuracy of uniform damping
The implementation of the program, which is shown in Ref. 13, is followed in this paper as follows: Equation (5) assumes the following approximate relationship between micro times: