Evaluating the Effect of the Various Directions of Seismic Input on an Irregular Building: The Former Uto City Hall

Abstract

A nonlinear time-history analysis of the main building of the former Uto City Hall is carried out using the recorded acceleration of the foreshock, considering various directions of seismic input. Then the nonlinear first modal response is calculated from the results of pushover analysis to investigate the effect of the various directions of seismic input on the first modal response of this building. The main findings of the study are (a) the angle of incidence where the peak drift at the “flexible” side column is the largest is close to that where the peak equivalent displacement of the first modal response is largest, (b) the equivalent velocity of the maximum momentary input energy of the first modal response is clearly related to the peak equivalent displacement of the first modal response, and (c) the evaluated maximum momentary input energy of the first modal mass per unit mass agrees well with the linear elastic momentary input energy spectrum of the seismic input.

1 Introduction

The main building of the former Uto City Hall, a five-story reinforced-concrete building damaged severely in the 2016 Kumamoto earthquakes, was an irregular building [1]. The earthquake series included a foreshock that occurred on 14 April and a main shock that occurred on 16 April 2016. In a previous study, the present author evaluated the seismic capacity of the building adopting a pushover-based procedure [2]. It was found that the seismic capacity of the building may have been insufficient to withstand the foreshock. However, this conclusion relied on the assumption that the GMRotD50 [3] (the medium of the set of geometrical means obtained using all possible rotations) spectrum represents bidirectional ground motion in a real earthquake. The effect of rupture directivity may not be negligible as the epicenter of the foreshock was not far from the target building.

The effect of the various directions of seismic input on the response of irregular (asymmetric) buildings has attracted interest from researchers (e.g., [4,5,6]). However, the nonlinear response makes it difficult to gain a deep understanding of the behavior. The present author considers that the mode shape changes from the initial (elastic) stage appreciably, making analysis based on the modal response complicated and difficult. Another important point for the understanding of this behavior is the energy input of each modal response: when the energy input of the first modal response were larger, some of local responses would be larger. Therefore, it is expected that the effect of the various directions of seismic input to the local response can be explained in terms of the variation of energy input of the first modal response.

This paper reports on the nonlinear time-history analysis of the main building of the former Uto City Hall using the acceleration recorded during the foreshock, considering various directions of seismic input. Then the nonlinear first modal response is calculated from the results of nonlinear time-history analysis and pushover analysis. The following discussion focuses on (a) the relation of the variation of the peak equivalent displacement of the first modal response and the peak local responses (story drift), (b) the variation of the maximum momentary input energy [7, 8] per unit mass of the first modal response, (c) the relation of the maximum momentary input energy per unit mass of the nonlinear first modal response and the maximum momentary input energy spectrum of linear system [7,8,9].

2 Building and Ground Motion Data

2.1 Building Data

Figure 1 shows the simplified structural plan and elevation of the main building of the former Uto City Hall. This building was a five-story reinforced-concrete building constructed in 1965.

Fig. 1

Simplified structural plan and elevation of the main building of the former Uto City Hall. a Structural plan (level Z₀), b Simplified structural elevation (frame B₁)

Damage to the building was first reported on the morning of 16 April 2016, after the main shock, even though the building experienced the foreshock on 14 April 2016. Details of the building were reported in the previous study [2].

The present nonlinear analysis uses one of the three-dimensional nonlinear frame structural models constructed in the previous study [2] (Model-RuW4-100) because this model may explain the observed structural damage of the main building of the former Uto City Hall. Figure 2 shows the structural model. As shown in Fig. 2a, this structural model is irregular in plan and elevation.

Fig. 2

Structural model [2]. a Overview, b Frame B₁, c Frame Y₅

In the numerical model, a one-component model with nonlinear flexural spring at each end and the shear spring in the middle of the line element is used for all beams and columns. The concrete walls in stair block are modeled as two diagonal braces as shown in Fig. 2c, assuming that the shear behavior is predominant. Nonstructural concrete block walls are neglected in the numerical modeling, due to the lack of drawings. Other details can be found in the previous paper [2].

2.2 Ground Motion Data

Figure 3 shows the acceleration of east–west (EW) and north–south (NS) components of the record of the foreshock (14 April 2016) obtained at the K-Net Uto Station, which is the station closest to the former Uto City Hall.

Fig. 3

EW and NS components of the foreshock acceleration recorded on 14 April at the K-NET Uto Station

3 Nonlinear Time-History Analysis

3.1 Analysis Cases

The two components of the recorded acceleration of the foreshock are scaled by factoring constant λ (λ = 0.8, 0.9, and 1.0) in the present study. The EW and NS components are respectively input parallel to ξ- and ζ-axes, as shown in Fig. 1a. The angle of incidence of the ξ- axis with respect to the X-axis, ψ, varies at intervals 15° from 0° to 345°. Therefore, a total of 3 × 24 = 72 nonlinear time-history analyses are carried out. The actual EW axis is approximately 45° counterclockwise from the X-axis, and the case that ψ = 315° is considered as the actual case in this study.

3.2 Analysis Results

Figure 4 shows the peak interstory drift at columns A₁B₁, A₃B₁ and A₃B₃ in the case that ψ = 315°. The figure also shows the drift limit assumed in the previous study [2] (R = 1/75). The peak responses of the “flexible” side columns (A₁B₁ and A₃B₁) are larger than the peak response of column A₃B₃, which is consistent with the fact that the observed damage to frame B₁ is more severe than that to frame A₃. In addition, the responses of columns A₁B₁ and A₃B₁ exceed R = 1/75 when λ = 0.9 and 1.0.

Fig. 4

Peak interstory drift at columns A₁B₁, A₃B₁ and A₃B₃ in the case that ψ = 315°

Figure 5 shows the distribution of the yield hinges at frame B₁. It is seen that for λ = 0.9 and 1.0, flexural yielding occurs at the top of column A₂B₁ on the third story and the right-side end of beam A₁-A₂ on level Z₃ and the left-side end of beam A₂-A₃ on level Z₃, as shown in the dotted rectangle in the figure. Note that a red triangle at a beam end indicates that the yielding of the beam–column joint occurs. Therefore, in the cases that λ = 0.9 and 1.0, the yielding of the beam–column joint and the column end occur simultaneously at the top of column A₂B₁ on the third story.

Fig. 5

Distribution of the yielding hinges at frame B₁ in the case that ψ = 315°

The analysis results presented here suggest that the main building of the former Uto City Hall suffered some level of structural damage during the foreshock. This finding is consistent with the results obtained in the previously conducted pushover-based evaluation [2].

Figure 6 shows the peak responses of columns A₁B₁, A₃B₁ and A₃B₃ on the second story for various directions of seismic input. As shown in plots (a) and (b), the peak responses of columns A₁B₁ and A₃B₁ are larger in the ranges of 120° ≤ ψ ≤ 180° and 300° ≤ ψ ≤ 360°. In contrast, the peak response of column A₃B₃ is larger in the ranges of 0° ≤ ψ ≤ 75° and 180° ≤ ψ ≤ 255° as shown in plot (c). This implies that the damage to frame A₃ is more severe than that to frame B₁ if the direction of the seismic input is different from that of the actual earthquake.

Fig. 6

Peak response at columns on the second story for various directions of seismic input. a Column A₁B₁, b Column A₃B₁, c Column A₃B₃

4 Evaluation of the Effect of Various Directions of Seismic Input on the Peak Drift

4.1 Calculation of the Nonlinear First Modal Response

The nonlinear first modal response is calculated from the pushover analysis results following a procedure proposed by Kuramoto [10] for planar frame analysis.

Figure 7 compares parameters between an N-story irregular building and the equivalent single-degree-of-freedom (SDOF) model representing the first modal response [2]. Here, vectors _nd and _nf_R are the displacement and restoring force vectors obtained from the pushover analysis results while M is the mass matrix determined from the mass and mass moment of inertia, m_j and I_j respectively, for the jth floor. Meanwhile, _nD_1U^* and _nA_1U^* are respectively the equivalent displacement and acceleration of the first modal response, and M_1U^* is the effective first modal mass with respect to the principal axis of the first modal response (U-axis).

Fig. 7

Comparison of parameters between an N-story irregular building and the equivalent SDOF model. a An N-story irregular building with deformation similar to the first mode shape, b Equivalent SDOF model representing the first modal response

Figure 8 shows the flow of the calculation of the nonlinear first modal response. In this calculation, displacement-based mode-adaptive pushover (DB-MAP) analysis [2, 11] is applied to consider the change in the first mode shape at each nonlinear stage. Here, vectors d(t) and f_R(t) are the displacement and restoring force vectors obtained from the results of nonlinear time-history analysis.

Fig. 8

Flow of the calculation of the nonlinear first modal response

Knowing the first mode vector corresponding to D_1U^*_peak, $\Gamma_{1Uie} {\mathbf{\varphi }}_{{{\mathbf{1ie}}}}$, the tangent of the angle of incidence of the U-axis from the X-axis, tan ψ_1ie, is determined as.

$$\tan \psi_{1ie} = - {{\sum\limits_{j} {m_{j} \phi_{Yj1ie} } } \mathord{\left/ {\vphantom {{\sum\limits_{j} {m_{j} \phi_{Yj1ie} } } {\sum\limits_{j} {m_{j} \phi_{Xj1ie} } }}} \right. \kern-\nulldelimiterspace} {\sum\limits_{j} {m_{j} \phi_{Xj1ie} } }}$$

(1)

$${\mathbf{\varphi }}_{{{\mathbf{1ie}}}} = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\phi_{X11ie} } & \cdots & {\phi_{XN1ie} } \\ \end{array} } & {\begin{array}{*{20}c} {\phi_{Y11ie} } & \cdots & {\phi_{YN1ie} } \\ \end{array} } & {\begin{array}{*{20}c} {\phi_{\Theta 11ie} } & \cdots & {\phi_{\Theta N1ie} } \\ \end{array} } \\ \end{array} } \right\}^{{\mathbf{T}}}.$$

(2)

4.2 Effect of Various Directions of Seismic Input on the Nonlinear First Modal Response

Figure 9 shows the results of calculating the peak of the first modal response (D_1U^*_peak and A_1U^*_peak) from all results of time-history analysis. Plot (a) compares the peak of the first modal response and A_1U^* − D_1U^* relationship obtained from the pushover (DB-MAP) analysis. Those peaks of the nonlinear first modal response fairly agree to A_1U^* − D_1U^* relationship obtained from the pushover analysis. This confirms that the nonlinear first modal response is properly calculated by the procedure presented herein. Plot (b) shows the peak equivalent displacement D_1U^*_peak for various directions of seismic input. As shown in plot (b), D_1U^*_peak is larger in the ranges of 120° ≤ ψ ≤ 180° and 300° ≤ ψ ≤ 360°.

Fig. 9

Calculation results of the peak of the first modal response. a Equivalent acceleration-equivalent displacement (A_1U^* − D_1U^*) relationship, b D_1U^*_peak for various directions of seismic input

A comparison of Figs. 9b and 6 shows that the range in which D_1U^*_peak is largest corresponds to the range in which the peak responses of columns A₁B₁ and A₃B₁ are largest, while this is not so for column A₃B₃.

Figure 10 shows the relation of the peak equivalent displacement of the first modal response and the peak drift of the columns on the second story. The figure compares the results of nonlinear time-history analysis and pushover analysis. For columns A₁B₁ and A₃B₁, the results of nonlinear time-history analysis agree well with the results of pushover analysis, as shown in Fig. 10a, b. In contrast, there is an appreciable difference between the results of nonlinear time-history analysis and pushover analysis for column A₃B₃ as shown in Fig. 10c.

Fig. 10

Relation of D_1U^*_peak and the peak drift of columns on the second story. a Column A₁B₁, b Column A₃B₁, c Column A₃B₃

From this observation, the drift responses of columns A₁B₁ and A₃B₁ on the second story can be approximated using the first modal response, while the contributions of the second and higher modal responses are appreciable for column A₃B₃. The variation in the peak drift on the second story at the “flexible” side columns (A₁B₁ and A₃B₁) due to the direction of the seismic input can therefore be explained in terms of the variation of the first modal response.

4.3 Momentary Energy Input of the Nonlinear First Modal Response

The momentary input energy of the first modal response per unit mass is next calculated. The present study follows studies by Inoue and coauthors [7, 8]. The energy input during a half cycle of the structural response is considered. The momentary input energy of the first modal response per unit mass, ΔE_1U^*/M_1U^*, is defined as.

$$\frac{{\Delta E_{1U}^{*} }}{{M_{1U}^{*} }} = - \int\limits_{t}^{t + \Delta t} {a_{gU} \left( t \right)\dot{D}_{1U}^{*} \left( t \right)dt}$$

(3)

where a_gU(t) is the ground acceleration component along the U-axis while t and t + Δt are the beginning and end times of a half cycle of the structural response. The maximum momentary input energy of the first modal response per unit mass, ΔE_1U^*_max/M_1U^*, is defined as the maximum value of ΔE_1U^*/M_1U^* over the course of the seismic event.

Figure 11 illustrates the definition of ΔE_1U^*_max/M_1U^*. In the figure, the case in which the largest D_1U^*_peak occurs (λ = 1.0, ψ = 0°) is shown as an example. As shown in the figure, ΔE_1U^*_max/M_1U^* is the input energy from t = 24.34 s (beginning of a half cycle as shown in (a)) to t + Δt = 25.06 s (end of the half cycle).

Fig. 11

Definition of the maximum momentary input energy of the first modal response per unit mass. a Hysteresis loop of the first modal response, b Time-history of the momentary energy input

For the following discussions, the equivalent velocity of the maximum momentary input energy of the first modal response, V_ΔE1U^*, is defined as.

$$V{_{\Delta E1U^{^{*}}}} =\sqrt{2{\Delta}E{_{1U^{*}}}\,\text{max}/M{_{1U^{*}}}}$$

(4)

Figure 12 shows the results of calculating D_1U^*_peak and V_ΔE1U^* from all results of time-history analysis. Plot (a) shows a clear relation between D_1U^*_peak and V_ΔE1U^*, as was pointed out by Inoue and coauthors for the nonlinear SDOF model [7, 8]. Plot (b) shows the peak equivalent velocity V_ΔE1U^* for various directions of seismic input. As shown in plot (b), V_ΔE1U^* is larger in the ranges of 135° ≤ ψ ≤ 180° and 315° ≤ ψ ≤ 360°. The trend of V_ΔE1U^* is consistent to that of D_1U^*_peak shown in Fig. 9b. Therefore, the variation in the peak drift on the second story at the “flexible” side columns (A₁B₁ and A₃B₁) due to the direction of the seismic input can be explained in terms of the variation of V_ΔE1U^*.

Fig. 12

Calculation results of D_1U^*_peak and V_ΔE1U^*. a Relation of D_1U^*_peak and V_ΔE1U^*, b V_ΔE1U^* for various directions of seismic input

4.4 Comparison of the Maximum Momentary Input Energy with the Linear Spectrum

Next, the maximum momentary input energy of the first modal response shown above is compared with the linear spectrum. In this study, the following linear spectra are calculated: (i) the maximum and minimum linear unidirectional V_ΔE spectra with viscous damping ratio h = 0.10 [7, 8] considering various angle of incidence of seismic input, and (ii) linear bidirectional V_ΔE spectrum [9] with viscous damping ratio h = 0.10. Since the angle of incidence of the U-axis from the X-axis varies gradually as the equivalent displacement D_1U^*_peak increases due to the nonlinearity, the intervals for calculating the maximum and minimum V_ΔE spectrum should be smaller. Therefore, the maximum and minimum V_ΔE spectrum is calculated for a for a natural period of the linear SDOF model T when the angle of incidence of the seismic input ψ varies in 1° intervals from 0° to 359°. Figure 13 compares the V_ΔE1U^* normalized by the scaling factor λ and linear V_ΔE spectrum with viscous damping ratio h = 0.10. In the figure, the notations Max and Min respectively indicate the maximum and minimum V_ΔE, while the notation Bidirectional is the bidirectional momentary input energy spectrum calculated using time-varying function [9]. The response period T´ is defined from the time for a half cycle of the structural response at ΔE_1U^*_max/M_1U^* as.

Fig. 13

Comparisons of the normalized V_ΔE1U^* and linear V_ΔE spectrum

$$T^{\prime} = 2\Delta t$$

(5)

In the figure, most plots of the normalized V_ΔE1U^* are within the band between Max and Min V_ΔE spectra. In addition, the bidirectional V_ΔE spectrum approximates the upper bound of the plots of the normalized V_ΔE1U^*. Therefore, for the conservative prediction of the peak response, the bidirectional V_ΔE spectrum formulated in [9] may be used as the seismic intensity parameter.

5 Conclusions

A nonlinear time-history analysis of the main building of the former Uto City Hall was carried out, considering various directions of seismic input. The effect of various directions of seismic input on the peak response was then discussed based on the nonlinear first modal response. The main conclusions of the study are as follows.

• (a) The angle of incidence where the peak drift at the “flexible” side column (A₁B₁ and A₃B₁) is the largest is close to that where the peak equivalent displacement of the first mode, D_1U^*_peak, is largest.

• (b) The equivalent velocity of the maximum momentary input energy of the first modal response, V_ΔE1U^*, is clearly related to D_1U^*_peak. This is consistent to the results by Inoue and coauthors for the nonlinear SDOF model. In addition, the trend of the variation of V_ΔE1U^* is consistent to that of D_1U^*_peak.

• (c) Most plots of the evaluated V_ΔE1U^* are within the band between the maximum and minimum spectra obtained from the linear elastic analysis considering all possible angles of incidence. In addition, the bidirectional V_ΔE spectrum formulated in [9] approximates the upper bound of V_ΔE1U^*.

Based on these findings, the presenting author considers the effect of the various directions of seismic input to the local response at “flexible” side can be explained in terms of the variation of energy input of the first modal response. In addition, the equivalent velocity of the bidirectional maximum momentary input energy may be one of the possible seismic intensity parameters available for discussion of the nonlinear peak response under bidirectional excitation. Further investigation is needed for confirmation.