The shear and bending stiffnesses of the SB model can be identified by the inverse-mode method, which will be explained in the following section. In order to apply this inverse-mode method, the lowest-mode shape of all degrees of freedom including horizontal displacements and rotation angles of the SB model is required. In this section, the estimation method of the lowest-mode shape of the SB model is proposed using the subspace method based on the observation of transfer functions.

The subspace method is known as an effective identification algorithm to provide a sequence of the state vector of the state space model described for k = 1, 2, …, N_d as (3a)x(k+1)=Ax(k)+Bu(k) (3b)y(k)=Cx(k)+Du(k)+w(k) where u(k), y(k), x(k), w(k), and N_d are the input, output, state vectors, measurement noise caused by the limitation of resolution of sensors, and data length, respectively. Furthermore, A, B, C, and D denote the matrices including the unknown system parameters, which can be derived by the least-squares estimation using the N4SID method. The transfer function G(z) between y(k) and u(k) can be given in terms of A, B, C, and D as (4)G(z)=C(zI-A)−1B+D where z=eiωT0. In this paper, the observed ground acceleration, e.g., earthquake ground motions or micro-tremors, is used as the input vector u(t). On the other hand, the observed floor horizontal accelerations and floor rotational accelerations are used as the output vector y(k). (5)u(k)=u¨g(k). (6)y(k)=u1(k),u2(k),⋯,uN(k),θ1(k),θ2(k),⋯,θN(k).

In this paper, the floor rotational acceleration is evaluated in the numerical simulation using building frame models by dividing the difference of the vertical accelerations, v¨Rj(t) and v¨Lj(t), at the corner columns by the total span length L. (7)θ¨j(t)=v¨Rj(t)−v¨Lj(t)L.

From the viewpoint of the resolution of measurement of floor rotational accelerations in high-rise buildings, the measurement of θ_i(t) or θ¨i(t) may be difficult especially in lower stories. For this reason, the lowest-mode shape for rotational angles is estimated empirically in a non-simultaneous measurement system.

The lowest-mode shape is identified by using the horizontal and rotational transfer functions derived from the subspace method. In this paper, the horizontal and rotational lowest-mode shape is given by the ratio of the transfer function amplitudes at the fundamental natural frequency. The fundamental natural circular frequency ω₁ is determined from the frequency that maximizes the amplitude of the transfer function of the horizontal displacement at the top floor. Figure 2 illustrates the summary of the identification of the lowest-mode shape from the transfer functions. The lowest-mode shape of floor horizontal displacements U and floor rotation angles θ can be evaluated as the vector of the obtained transfer function amplitudes {G₁(ω₁), G₂(ω₁), …, G_2N(ω₁)} at ω = ω₁ described by (8a,b)U=1ω12U¨,θ=1ω12θ¨ where U¨={G1(ω1),G2(ω1),⋯,GN(ω1)} and θ¨={GN+1(ω1),GN+2(ω1),⋯,G2N(ω1)}. These are derived from the transfer functions as shown in Figure 2.

Although the simultaneous measurement of the horizontal and floor rotational accelerations is recommended in the proposed SI method, the SI based on the non-simultaneous measurement system, e.g., the relocation of sensors in the measurement system under micro-tremors, is also investigated for the practical application purpose. The non-simultaneous measurement system has a benefit to reduce the initial and maintenance costs due to the small number of sensors. Figure 3 shows the summary of the lowest-mode shape identification in the non-simultaneous measurement system. As shown in Figure 3, in the non-simultaneous measurement system, the top-floor horizontal acceleration and vertical accelerations at the corner columns are needed to stabilize the transfer function evaluation in the subspace method. This is because the amplitude of the transfer function at the i-th floor may be variable due to the difference of utilized micro-tremors in the non-simultaneous measurement system. Although the simultaneous measurement at the top-floor is necessary, it is unnecessary to measure floor accelerations of all stories at the same time. In this paper, we call this system as the non-simultaneous measurement system.

The lowest-mode shape of the horizontal displacement at the i-th floor is determined by the ratio of the amplitude of the i-th floor transfer function at ω = ω₁ to that for the top floor. Finally, the lowest-mode shape of horizontal displacement in the non-simultaneous measurement system can be derived as (9)Unon-simul=1ω12U1(1)(ω1)UN(1)(ω1),U2(2)(ω1)UN(2)(ω1),⋯,1 where the number in the superscript represents the non-simultaneous measurement number index. On the other hand, since the measurement of vertical accelerations in the slab at lower stories is difficult due to the problem of low S/N ratio and resolution, the lowest-mode shape of the floor rotation is necessary to estimate by using an empirical equation. This procedure will be explained in the following section.

The shear and bending stiffnesses k_s, k_b of the SB model can be derived by substituting the lowest mode of both horizontal displacements and floor rotation angles into the governing equations in the inverse-mode method (Takewaki and Nakamura, 1993; Takewaki, 1999). Figure 4 illustrates the scheme of the inverse-mode method. In the inverse-mode method using the lowest mode of both floor horizontal displacements and rotation angles, these stiffnesses can be determined by (10)ks1,ks2,⋯,ksN,kb1,kb2,⋯,kbNT=S−1F where (11a,b)S=s1−s2000s2−s3⋱0sN00−H1s1ϕ1−ϕ20−H2s2ϕ2−ϕ3⋱⋱−HNsNϕN,F=ω12m1u1⋮mNuNI1θ1⋮INθN (12a,b)sj=uj−Hjθj(j=1)uj−uj−1−Hjθj(j=2,⋯,N),ϕj=θ1(j=1)θj−θj−1(j=2,⋯,N).

As seen in Figure 4, s_j and ϕ_j are the inter-story shear deformation and the inter-floor rotation angle. In addition, H_j is the story height of the j-th story.

In this section, 5-, 10-, and 20-story plane steel building frames are used to investigate the influence of various structural properties, e.g., number of stories, number of spans, story-wise stiffness distribution, on the lowest-mode shape of floor rotation angle. The reference lowest-mode shape of the floor rotation angle of those building models can be derived by using the simultaneously measured floor rotation angle obtained in the abovementioned lowest-mode shape identification based on the subspace method.

Table 1 shows the cross-sectional list of columns and beams, and Table 2 presents the properties of building models. The structural damping (damping ratio: 0.02) is given as the stiffness proportional damping. In 5-story building models, models 1, 2, and 3 are used to investigate the influence of the number of spans. In addition, the effect of the difference of the stiffness distribution is compared in models 1, 4, and 5. Since the stiffness distribution affects the lowest-mode shape of the floor rotation angle, models 6, 7, 8 and models 9, 10 are examined for 10-story and 20-story building models where the stiffness distributions are different.

Figure 5 shows the comparison of the identified lowest-mode shape of floor rotation angles for 5-, 10-, and 20-story building models listed in Table 2. In this figure, the empirically estimated lowest-mode shapes are also shown. It can be observed from Figure 5A that the mode shapes of models 1, 2, and 3 may not be affected by the number of spans and span length. Furthermore, from the parametric analysis for various building frames except those in Table 2, it has also been confirmed that the aspect ratio and the distribution of story stiffness may slightly affect the mode shape of floor rotation angle. Based on these results, in order to obtain the empirical equation for the lowest-mode floor rotation angle, the amplitude of floor rotation angle θ_j at the j-th story is assumed to be given by using the just upper story floor rotation angle θ_j + 1 by (13)θj=θj+1−A⋅θj+1(N−j)2 where A is a shape coefficient. In Eq. 13, the floor rotation angle in the top story can be given by the obtained floor vertical accelerations and Eq. 7. The coefficients A in Eq. 13 is needed to be modified by the number of stories. From the results for various building frames in Table 2 and other models, the coefficients A can be derived by the least-squares approach that can be categorized by the number of stories as (14)A=0.0250(for5-storybuilding)0.0040(for10-storybuilding)0.0012(for15-storybuilding)0.0005(for20-storybuilding).

The shape coefficient A can be derived in an exponential form as (15)A(N)=0.2332×e−0.49N+0.01053×e−0.1537N where N is the number of stories of the objective building.

Consider a 10-story plane building frame. The story height H_j, story mass m_j, and rotary inertia J_j are constant in all stories as 4.0 m, 120,000 kg, 1,620,000 kgm², respectively. The number of spans is four, and the common span length is 3.0 m. The stiffness proportional damping ratio is 0.02. Table 3 shows the cross-sectional list. The output time history data, i.e., horizontal and vertical accelerations at the corner column nodes, are calculated by the response analysis software, SNAP (2015).

The micro-tremor recorded in the Disaster Prevention Research Institute of Kyoto University is used as the input ground acceleration. Figure 6 shows an example of the time history and the acceleration response spectrum of the micro-tremor. The measurement duration of the micro-tremor is longer than 5 min. The sampling frequency is 100 Hz. In the following numerical examples, the band-limited white noise (from 0.1 to 20 Hz) is added to the output data, i.e., horizontal and rotational acceleration, independently at each story. The root-mean-square (RMS) of the noise is determined for the horizontal acceleration and rotation angle, respectively. These amplitudes of the noise are constant at all stories and given so as to make the 10 or 20% RMS of noise to that of the responses at the first story.

First of all, in order to investigate the accuracy of the stiffness identification algorithm, shear and bending stiffnesses of the SB model are identified by using all floor horizontal and rotational responses measured simultaneously under the same input. Figure 7 shows the transfer functions for horizontal accelerations without noise derived by the subspace method. From these transfer functions including floor rotation angles, the lowest-mode shape can be derived from the amplitude of them as explained in Eq. 8. After obtaining the lowest-mode shape, the shear and bending stiffnesses of the SB model can be determined directly without any additional operation. Figure 8 shows the comparison of the identified stiffnesses, including the case considering noise-biased data, with the reference stiffnesses of the objective building frame and the shear stiffness identified as the S model. For identifying as the S model, the inverse-mode method in Eq. 10 can be reduced to N × N matrices. The reference stiffnesses in Figure 8 can be derived from the pushover analysis result using the object building frame. Table 4 shows the comparison of the identified natural periods with the reference ones. As shown in Table 4, it can be observed that the SB model is more appropriate to represent high-rise building frames than the S model.

In this section, the identification accuracy in shear and bending stiffness of the SB model is investigated in the case of non-simultaneous measurement. Furthermore, the influence of noise added to the output data on the identification accuracy is also studied. As explained before, in the non-simultaneous measurement, the horizontal and rotational accelerations at the top story are required simultaneously in addition to the horizontal accelerations at each objective story and at the base. This is because the lowest-mode shape can be predicted more accurately by using the transfer function’s amplitude at the i-th floor normalized by that of the top story. Figures 9A,B show the identified participation vector of both horizontal displacement and floor rotation angle derived from the subspace method. Furthermore, the shear and bending stiffnesses in the non-simultaneous measurement with noise are shown in Figures 9C,D. Table 5 shows the error (ratio) of natural periods of the identified SB models. From Figure 9, the identified bending stiffnesses in the non-simultaneous measurement are a little different from the reference values even in the case without noise. This difference may be caused by the influence of the reliability of the empirical equation on the lowest-mode shape of floor rotation angle.

In practical building frames, the lowest-mode shape of the inter-floor rotation angle is non-monotonic due to irregular column section distributions. In order to estimate the bending stiffness of the SB model more accurately, it may be necessary to consider these factors in the empirical equation. However, as summarized in Table 5, the natural periods of the identified SB model in non-simultaneous measurement using the output data with noise can be obtained in a reliable manner. This may result from several facts. First, the bending stiffness of the SB model has slight influence on the natural periods. Second, the shear and bending stiffnesses of the SB model are influenced by each other. Therefore, as seen in Figure 9, the shear stiffness of the SB model varies in the case of 5 and 10% noise. Figure 10 shows the comparison of the top-story acceleration as the seismic response of the identified SB model with that of the building frame model under El Centro NS (1940). As shown in Figure 10, the seismic response of the frame building model can be predicted in a reliable manner by the SB model identified using the noise-biased output data. Although the required accuracy in the prediction of the seismic response by using the identified physical model may vary depending on the situation, it may be concluded that minimizing the identification error on the modal properties, i.e., natural frequencies including higher modes, is important for this purpose.

A scaled structure was designed so that certain amounts of remarkable shear and bending deformations can be measured, and this model has an appropriate ratio of bending deformation to shear deformation considering high-rise buildings. Figure 11 shows a photograph and schematic diagram of the scaled structure; floor plates, column plates, spacer blocks (SUS304), and angle bars supporting column (steel). The objective direction to identify the story stiffness of the scaled structure is x in Figure 11. It is known that the width of the spacer block is correlated with the bending stiffness. In this experimental study, the width of the spacer block at the first story is 20 mm and those at the both second and third stories are 10 mm. In order to prevent the rocking vibration of the scaled structure, the base plate is fixed at the shaking table.

The servo-type velocity-meter (VSE-15D, Tokyo Sokushin) was used to simulate the micro-tremor measurement. The resolution of velocity of this velocity meter is 10⁻⁴ mm/s. The S/N ratio of this experiment was confirmed to be larger than 60 dB at the base input. Since the mass of this velocity-meter is not negligible compared with that of the floor mass, it was modeled by an additional mass, and the location of the velocity meter was changed so as to simulate the situation of non-simultaneous measurement. Another additional mass was placed both at the first and second floors to reduce the influence of mass variation and shifting of the center of mass. Considering the additional mass, the lumped mass and rotational inertia of each story were given as m₁ = 4.02 kg, m₂ = 3.72 kg, m₃ = 3.05 kg and I₁ = 6.30 × 10⁻³ kgm², I₂ = 5.84 × 10⁻³ kgm², I₃ = 4.78 × 10⁻³ kgm².

Figure 12 shows the time histories of input and output measured data in the non-simultaneous measurement. It can be confirmed that the amplitudes of the measured input micro-tremor are almost the same among these several measurements. It can also be observed that the vertical velocities at the left and right ends of the top story exhibit opposite-phase. From the measured input and output data, the transfer function at each story can be derived by the subspace method. The identified transfer functions of the horizontal and rotational accelerations to the base acceleration are shown in Figure 13. In this figure, the transfer functions of the horizontal acceleration derived by the sweep testing are also shown. As seen in Figure 13, the transfer function identified by the subspace method seems to be derived correctly. It was confirmed that, by applying the subspace method, the transfer functions can be derived appropriately. Finally, the shear and bending stiffnesses of the SB model have been obtained from the identified lowest-mode shape. The comparison of the natural frequencies of the identified SB model with those observed by the sweep-vibration test result is shown in Table 6. Although a slight difference can be seen in the third natural frequency, the SB model does not necessarily provide the actual modal properties (especially higher modes).

In order to investigate the accuracy of the identified SB model, the comparison of the time history records for the random input between the simulated and the measured has been conducted in a scaled structure using a shaking table. By using the identified SB model in the numerical simulation, the horizontal displacements and accelerations can be estimated for the specified input motion. The input acceleration is given by the measured data on the shaking table. Figure 14 shows the time history of the absolute acceleration at the top story together with the base input, and Figure 15 presents the story-wise distribution of the maximum acceleration. In this verification, the target time history of ground acceleration on the shaking table is El Centro NS (1940), where the amplitude is controlled to 10% of the original wave (see Figure 14A). In Figure 15, the maximum error ratio is 15.4% in the first story. In order to predict the acceleration responses reliably, the modal properties including higher modes are needed to be identified correctly. As for the merit of the proposed SB model compared to the S model, it has been made clear before (Minami et al., 2013) that the S model cannot simulate the acceleration in an accurate manner because the S model cannot express the higher natural frequencies of the actual structure accurately. However, it is also true that the accurate simulation of acceleration is difficult even by the SB model due to high sensitivity of acceleration to structural and input motion parameters. Since the present paper uses the modal–physical hybrid method for the SB model different from the previous method (Minami et al., 2013) using a direct physical-parameter SI, the stiffness parameters can be identified in a more stable manner. It may be concluded that the SB model has been identified appropriately by the proposed method and can be used to predict the response of the objective structure even under noise-biased environment.