Preliminary analysis of response spectra, energy spectra, and energy-input directionality for the 2026 Kumamoto earthquake

1 Introduction

On 28 July 2026 at 16:27 Japan Standard Time (JST), an earthquake with a JMA magnitude (M J) of 7.1 and a moment magnitude (M w) of 6.7 occurred in the Kumamoto region of Japan (National Research Institute for Earth Science and Disaster Resilience (NIED, 2026a). During this earthquake, long-period pulse-like accelerations were recorded at K-NET Yatsushiro (KMM012), one of the strong-motion stations operated by the NIED (2026b). Figure 1 presents the location of KMM012 and the observed acceleration time histories.

Pulse-like ground motions observed in near-fault regions have attracted considerable attention in earthquake engineering (e.g., Somerville et al., 1997). Various approaches have been proposed to characterize such motions in terms of their amplitude and period characteristics (Mavroeidis and Papageorgiou, 2003; Bray and Rodriguez-Marek, 2004; Baker, 2007). In the present study, however, the term “pulse-like” is used descriptively to refer to the observed waveform characteristics of KMM012; no quantitative pulse classification is performed.

The response of building structures subjected to pulse-like ground motions has been extensively investigated since the early 2000s. Alavi and Krawinkler (2004) showed that structures with periods longer than the pulse period exhibit markedly different response characteristics from those with shorter periods. Kalkan and Kunnath (2006) demonstrated the significance of large energy input over relatively few cycles in the nonlinear response of structures subjected to pulse-like ground motions. Kalkan and Kwong (2013) demonstrated that the maximum three-dimensional structural response to near-fault ground motions does not necessarily occur when the ground-motion components are oriented in the fault-normal and fault-parallel directions. In particular, the effects of pulse-like ground motions on long-period structures, such as high-rise and base-isolated buildings, have also been investigated. Güneş and Ulucan (2019) demonstrated that the distribution of story drifts in a 40-story core-wall building subjected to pulse-like ground motions varies substantially depending on the ratio of the pulse period to the first-mode natural period of the structure. Güneş (2022) demonstrated that the superstructure ductility demand of a base-isolated building is sensitive to the pulse period and that the effectiveness of the isolation system decreases considerably as the pulse period increases.

The direction of seismic incidence has long been recognized as an important issue in structural response, particularly for near-fault ground motions with pronounced directional characteristics. Buildings generally have different structural characteristics in different horizontal directions, and their response may therefore depend on the direction of seismic input relative to the structural axes. López and Torres (1997) proposed a response-spectrum-based method for linear structures to determine the critical angle of seismic incidence and the corresponding maximum structural response. Ground motions themselves also exhibit directional characteristics. López et al. (2006) showed that the response spectra of the major and minor principal horizontal components differ and that their spectral ratio depends on vibration period and distance to the fault.

For nonlinear structural response, an energy-based approach provides a useful framework for evaluating seismic response. To investigate the influence of the direction of seismic input on nonlinear structural response, it is therefore important to understand the directionality of the energy input from the ground motion to the structure. Akiyama (1985) proposed a seismic design theory of building structures based on cumulative input energy. Hori and Inoue (2002) introduced the concept of maximum momentary input energy based on the energy balance over a half cycle of structural response. Fujii and Murakami (2020) extended the concept of momentary input energy proposed by Hori and Inoue (2002) to structures subjected to bidirectional horizontal ground motions. Fujii (2022) investigated the influence of the angle of seismic incidence of long-period pulse-like seismic input on an irregular base-isolated building. The results showed that the principal axis of ground motion defined in terms of cumulative input energy is useful for discussing the influence of the angle of seismic incidence on the response of the irregular base-isolated building.

In this study, a preliminary analysis of the strong-motion records obtained during the 2026 Kumamoto earthquake is conducted, with particular attention to the record at K-NET Yatsushiro (KMM012). KMM012 is examined in detail because it exhibits particularly pronounced long-period response and energy-input characteristics among the records analyzed. As shown in the Supplementary Appendix, KMMH14 also exhibits strong long-period characteristics, whereas these characteristics are less pronounced in the other four records. The analyses of these records are provided in the Supplementary Appendix.

First, linear response spectrum analysis is carried out considering bidirectional horizontal ground motions. Second, the cumulative and maximum momentary input energies are examined. Particular attention is paid to the directional concentration of cumulative input energy and the direction of the displacement increment associated with the maximum momentary input energy. The physical mechanism responsible for the observed long-period pulse-like accelerations is beyond the scope of this study.

The analytical methods employed in this study have been established in previous studies (Fujii and Murakami, 2020; Fujii, 2022). The contribution of the present study lies in their timely application to the newly recorded KMM012 ground motion from the 2026 Kumamoto Earthquake, providing a preliminary characterization of its long-period response characteristics, seismic energy input, and energy-input directionality.

2 Ground motion data and analysis methods

2.1 Ground motion data at KMM012

Figure 1A illustrates the location of the epicenter and the KMM012 observation station. According to the NIED F-net website (NIED, 2026a), the epicenter was located at 32.6°N, 130.7°E, with a focal depth of 17 km. KMM012 is located approximately 14 km southwest of the epicenter.

Figure 1B presents the time histories of the recorded accelerations in the EW and NS components at the ground surface (NIED, 2026b). As a simple baseline correction, the mean value of the entire acceleration record was removed as an offset before the response analysis. No additional baseline correction or filtering was applied. The peak ground acceleration (PGA) values of the EW and NS components are 3.50 and 3.10 m/s², respectively. As shown in Figure 1B, large-amplitude accelerations are observed between approximately 15 and 35 s. Figure 1C provides an enlarged view of the recorded acceleration time histories during this interval. As shown in Figure 1C, large-amplitude pulse-like accelerations are observed between approximately 18 and 23 s.

2.2 Analysis methods

2.2.1 Response and energy spectrum analysis

This study considers an isotropic linear one-mass two-DOF model, following Fujii and Murakami (2020). The orthogonal X- and Y-axes are defined in the horizontal plane, and the EW and NS components of the ground motion are simultaneously applied in the X and Y directions, respectively. The equation of motion of the isotropic one-mass two-DOF model subjected to bidirectional horizontal ground motion is given by Equation 1:

m x ¨ y ¨ + 2 m h ω 0 x ˙ y ˙ + m ω 0 2 x y = − m a g X a g Y (1)

In Equation 1, m is the mass, h is the damping ratio, ω 0 (=2 π / T) is the undamped natural circular frequency, where T is the undamped natural period, x and y are the relative displacements in the X and Y directions, respectively, and a g X and a g Y are the ground accelerations in the X and Y directions, respectively.

In this study, the spectral horizontal displacement (S D T , h) is defined by Equation 2:

S D T , h = max t x t 2 + y t 2 (2)

The pseudo-velocity spectrum (S V p T , h) and pseudo-acceleration spectrum (S A p T , h) are calculated using Equation 3:

S p V T , h = 2 π T S D T , h , p S A T , h = 2 π T 2 S D T , h (3)

In this study, the damping ratio is set to h = 0.05 for the calculation of the response spectra.

Next, the cumulative input energy spectrum and the maximum momentary input energy spectrum under bidirectional excitation are defined as follows. Following the study by Akiyama (1985), Akiyama(1999), the cumulative input energy spectrum for bidirectional excitation (V I T) is calculated using Equation 4:

V I T = 2 E I / m (4)

In Equation 4, E I / m is the cumulative input energy per unit mass over the entire duration of the seismic input, calculated using Equation 5:

E I / m = − ∫ 0 t d a g X t x ˙ t + a g Y t y ˙ t d t (5)

In Equation 5, t d is the duration of the seismic input.

Meanwhile, the maximum momentary input energy spectrum (V Δ E T) is calculated using Equation 6:

V Δ E T = 2 Δ E B I , ⁡ max / m (6)

In Equation 6, Δ E B I , ⁡ max / m is the maximum momentary input energy per unit mass (Fujii and Murakami, 2020) and is defined as the maximum value of the momentary input energy per unit mass (Δ E B I / m), which is calculated using Equation 7:

Δ E B I / m = − ∫ t t + Δ t a g X t x ˙ t + a g Y t y ˙ t d t (7)

In Equation 7, t and t + Δ t are the start and end times of a half cycle of the structural response, respectively. As described in Fujii and Murakami (2020), the times t and t + Δ t are determined from the conditions given in Equation 8:

x t x ˙ t + y t y ˙ t = 0 x t x ¨ t + y t y ¨ t + x ˙ t 2 + y ˙ t 2 < 0 (8)

In this study, the damping ratio is set to h = 0.10 for the calculation of the energy spectra, following previous energy-based studies (Akiyama, 1985; 1999; Hori and Inoue, 2002; Fujii and Murakami, 2020). The larger damping ratio used for the energy spectra is intended to approximately account for the additional energy dissipation associated with nonlinear structural response.

2.2.2 Directionality of energy input

First, the definition of the principal axis of horizontal ground motion based on the cumulative energy input is described following Fujii (2022). The orthogonal ξ- and ζ-axes are defined in the horizontal X-Y plane. The ξ-axis is the major axis that maximizes the cumulative input energy per unit mass, while the ζ-axis is the minor axis orthogonal to the ξ-axis. The angle from the X-axis to the ξ-axis, measured positive in the clockwise direction, is denoted by ψ E T.

The cumulative input energies per unit mass along the major and minor axes (E I ξ / m and E I ζ / m, respectively) are calculated using Equation 9:

E I ξ / m 0 0 E I ζ / m = cos ⁡ ψ E − sin ⁡ ψ E sin ⁡ ψ E cos ⁡ ψ E E I X X / m E I X Y / m E I X Y / m E I Y Y / m cos ⁡ ψ E sin ⁡ ψ E − sin ⁡ ψ E cos ⁡ ψ E (9)

In Equation 9, E I X X / m, E I X Y / m, and E I Y Y / m are the cumulative input energies per unit mass in the X-Y plane calculated using Equation 10:

E I X X / m = − ∫ 0 t d a g X t x ˙ t d t E I X Y / m = − 1 2 ∫ 0 t d a g X t y ˙ t + a g Y t x ˙ t d t E I Y Y / m = − ∫ 0 t d a g Y t y ˙ t d t (10)

The angle ψ E T is calculated using Equation 11:

ψ E T = − 1 2 Tan − 1 2 E I X Y / m E I X X / m − E I Y Y / m (11)

The angle ψ E T obtained from Equation 11 is taken within the range from −90° to 90°.

The cumulative input energy spectra along the major and minor axes (V I ξ T and V I ζ T, respectively) are calculated using Equation 12:

V I ξ T = 2 E I ξ / m , V I ζ T = 2 E I ζ / m (12)

The ratio of the cumulative input energy spectra along the minor and major axes (R E I T) is calculated using Equation 13:

R E I T = V I ζ T / V I ξ T (13)

The value of R E I T ranges from 0 to 1. If R E I T is less than 1 / 2 (approximately 0.71), the cumulative input energy per unit mass along the minor axis (E I ζ / m) is less than half of that along the major axis (E I ξ / m).

Next, the direction of the displacement increment associated with the maximum momentary input energy is examined, following Fujii and Murakami (2020). The direction angle of the displacement increment from time t to t + Δ t measured from the X-axis and taken as positive in the clockwise direction (ψ T) is defined as Equation 14:

ψ T = − Tan − 1 y t + Δ t − y t x t + Δ t − x t (14)

Note that the times t and t + Δ t are the start and end times of the half cycle of the structural response corresponding to the maximum momentary input energy per unit mass (Δ E B I , ⁡ max / m).

The value of ψ T directly obtained from Equation 14 ranges from −90° to 90°. When x t + Δ t − x t > 0, no correction is applied to ψ T. When x t + Δ t − x t < 0, ψ T is increased by 180° if y t + Δ t − y t < 0, whereas ψ T is decreased by 180° if y t + Δ t − y t > 0. When x t + Δ t − x t = 0, ψ T is taken as 90° if y t + Δ t − y t < 0 and as −90° if y t + Δ t − y t > 0.

3 Analysis results

3.1 Response spectra

Figure 2 compares the response spectra of the KMM012 record with those of two strong-motion records from previous earthquakes. The records obtained at the Kobe Marine Observatory during the 1995 Hyogo-ken Nanbu (Kobe) Earthquake (KMO1995; Japan Meteorological Agency, 2026) and at KiK-net Mashiki during the mainshock of the 2016 Kumamoto Earthquake (MSK2016; NIED, 2026b) are used as illustrative reference records. KMO1995 is a well-known strong-motion record from a past destructive earthquake in Japan, while MSK2016 provides a reference from the 2016 Kumamoto Earthquake in the same region as the present event. For basic reference, the JMA magnitude, focal depth, and epicentral distance are 7.1, 17 km, and approximately 14 km for KMM012; 7.3, 16 km, and 16.5 km for KMO1995 (Japan Meteorological Agency, 2026); and 7.3, 12 km, and 7 km for MSK2016 (NIED, 2026b). Figures 2a–c show the pseudo-acceleration, pseudo-velocity, and displacement spectra, respectively, with the S A p–S D relationship given in Figure 2d.

The main observations are as follows:

• In the short-period range (T < 1.5 s), the response spectral values of KMM012 are smaller than those of KMO1995 and MSK2016. In contrast, in the longer-period range (T > 1.5 s), those of KMM012 exceed those of KMO1995 and MSK2016.
• The pseudo-velocity spectrum (S V p) of KMM012 reaches the maximum value of 2.59 m/s at T = 3.12 s, while the displacement spectrum (S D) reaches the maximum value of 1.35 m at T = 3.42 s.
• In the S A p–S D relationship shown in Figure 2d, KMM012 reaches an S D value of approximately 1.3 m with S A p remaining above 5.0 m/s².

3.2 Energy spectra

Figure 3 compares the cumulative input energy spectrum (V I T) and the maximum momentary input energy spectrum (V Δ E T) of KMM012 with those of KMO1995 and MSK2016. The main observations are as follows:

• The cumulative input energy spectrum (V I T) of KMM012 is smaller than those of KMO1995 and MSK2016 for periods shorter than 2.0 s. In contrast, for periods longer than 2.0 s, V I T of KMM012 exceeds those of KMO1995 and MSK2016.
• The maximum momentary input energy spectrum (V Δ E T) of KMM012 exceeds that of KMO1995 for periods longer than approximately 1.3 s and that of MSK2016 for periods longer than approximately 1.6 s.
• The cumulative input energy spectrum (V I T) of KMM012 reaches a maximum value of 3.59 m/s at T = 3.14 s, while the maximum momentary input energy spectrum (V Δ E T) reaches a maximum value of 1.95 m/s at T = 2.50 s.

3.3 Directionality of energy input

Figure 4 presents the directionality of energy input for KMM012. Figure 4a compares the cumulative and maximum momentary input energy spectra (V I T and V Δ E T, respectively) with the cumulative input energy spectra along the major and minor axes (V I ξ T and V I ζ T, respectively). Figures 4b,c present the principal-axis angle ψ E T, and the ratio R E I T, respectively. Figures 4d,e illustrate the directional characteristics of energy input in polar coordinates. Figure 4d shows the directional distribution of the normalized cumulative input energy spectrum along the major axis (V I ξ T / max V I ξ T). The plots are supplemented with those in the ranges from −180° to −90° and from 90° to 180°, taking symmetry into account. Figure 4e plots the normalized maximum momentary input energy spectrum (V Δ E T / max V Δ E T) in the direction of the displacement increment associated with the maximum momentary input energy.

The main observations are as follows:

• The cumulative input energy spectrum along the major axis (V I ξ T) is close to the cumulative input energy spectrum for bidirectional excitation (V I T) for periods shorter than 3.2 s. The difference between V I ξ T and V I T increases for periods longer than 3.2 s.
• The cumulative input energy spectrum along the minor axis (V I ζ T) is much smaller than the cumulative input energy spectrum for bidirectional excitation (V I T) for periods shorter than 3.2 s. The difference between V I ζ T and V I T decreases for periods longer than 3.2 s.
• The maximum momentary input energy spectrum for bidirectional excitation (V Δ E T) is larger than the cumulative input energy spectrum along the minor axis (V I ζ T) for periods shorter than 4.6 s. For periods between 0.9 and 1.3 s, V Δ E T exceeds V I T.
• For periods shorter than 0.6 s, the angle between the major axis (ξ-axis) and X-axis (ψ E T) fluctuates considerably. For periods of approximately 1.0–2.0 s, ψ E T remains close to 45°; it then decreases smoothly to around 30° for periods of approximately 2.0–3.5 s, and gradually returns to around 45° for periods longer than 3.5 s.
• For periods of approximately 1.0–3.6 s, the ratio R E I T ranges from approximately 0.20–0.44. The ratio R E I T is 0.222 at T = 3.14 s, where V I T reaches its maximum value.
• The plots of V I ξ T / max V I ξ T are predominantly distributed in directions ranging from southeast to east-southeast (and, by symmetry, from northwest to west-northwest).
• Almost all plots of V Δ E T / max V Δ E T are distributed in the southeast direction.
• The directional distribution of V Δ E T / max V Δ E T with respect to ψ T shows a trend similar to that of V I ξ T / max V I ξ T with respect to ψ E T.

The observation that V Δ E T exceeds V I T for periods between 0.9 and 1.3 s can be explained from Equation 7. The energy input becomes negative when the ground acceleration and the structural velocity have the same direction. This can be interpreted as energy being returned from the structure to the ground. Consequently, the maximum momentary input energy over a half cycle can exceed the net cumulative input energy over the entire ground-motion duration.

4 Discussion and conclusion

The analysis results demonstrate several notable characteristics of the KMM012 record in terms of structural response and energy input.

Figure 2 indicates that KMM012 exhibits larger response spectral values than KMO1995 and MSK2016 in the long-period range. In particular, the S A p–S D relationship shows that S A p remains above 5.0 m/s² while S D reaches approximately 1.3 m. This suggests that the KMM012 record contains a strong long-period component capable of producing simultaneously large acceleration and displacement demands. The illustrative comparison also shows that the period range associated with large seismic demands differs among the records. While KMO1995 and MSK2016 exhibit larger response spectral values in the shorter-period range, KMM012 exhibits larger values in the longer-period range.

The energy spectra shown in Figure 3 also exhibit large values in the long-period range. The cumulative input energy spectrum (V I T) reaches a maximum value of 3.59 m/s at T = 3.14 s, while the maximum momentary input energy spectrum (V Δ E T) reaches a maximum value of 1.95 m/s at T = 2.50 s. These results indicate that KMM012 can produce large energy input to structures with relatively long natural periods. Together with the response-spectrum results in Figure 2, these energy-spectrum characteristics indicate that KMM012 has potentially significant damaging properties for structures with relatively long natural periods.

Figure 4 further reveals strong directionality of the energy input in the period range where the energy input is large. At T = 3.14 s, where V I T reaches its maximum value, R E I T is 0.222, indicating that the ratio of the cumulative input energy along the minor axis to that along the major axis is only approximately 0.222² ≈ 0.049. This demonstrates that the large energy input induced by KMM012 is strongly directional.

The directional distributions shown in Figures 4d,e further clarify the directionality of the energy input. In the period range where the energy input is large, the major-axis direction of the cumulative input energy is predominantly distributed from southeast to east-southeast (and, by symmetry, from northwest to west-northwest). Meanwhile, the direction of the displacement increment associated with the maximum momentary input energy is concentrated predominantly in the southeast direction. These results indicate that the major-axis direction of the cumulative input energy and the displacement-increment direction associated with the maximum momentary input energy exhibit similar directional trends in the period range where the energy input is large.

These characteristics are particularly relevant to structures with relatively long natural periods, including tall buildings and seismically isolated buildings. Previous studies have shown that pulse-like ground motions can impose significant seismic demands on such structures, depending strongly on the relationship between the pulse period and the structural period (Güneş and Ulucan, 2019; Güneş, 2022). The present results further suggest that the effects of ground motions such as KMM012 on long-period structures may depend not only on the magnitude of seismic demand but also on the directionality of energy input.

However, the pronounced long-period characteristics identified at KMM012 should not be regarded as representative of all strong-motion records obtained during the 2026 Kumamoto Earthquake. As shown in the Supplementary Appendix, KMMH14 also exhibits strong long-period characteristics, whereas these characteristics are less pronounced in the other four records examined. The present study therefore focuses on KMM012 as a particularly noteworthy record rather than as a record representative of the earthquake as a whole.

The value S D ≈ 1.3 m should therefore be interpreted as a result of the present preliminary analysis based on the K-NET record after the mean-removal procedure described in Section 2.1; its sensitivity to alternative baseline-correction and filtering procedures has not been examined in this study.

The present study focuses on the characteristics of the observed ground-motion records and does not attempt to identify the physical mechanisms responsible for the observed long-period characteristics and energy-input directionality. In particular, the respective contributions of source directivity, propagation path, and basin/site response remain to be investigated. Because the three records differ in source, propagation path, site conditions, and other seismological characteristics, the illustrative comparisons among these records are not intended to establish a general ranking of ground-motion severity or to attribute the observed differences in response and energy spectra to specific physical factors.

Quantitative identification of the pulse characteristics, including the pulse period, amplitude, and predominant direction, was not performed in the present study and remains a subject for future investigation.

5 Transparency statement

This Brief Research Report is based on a preliminary report previously prepared and made publicly available by the author shortly after the 2026 Kumamoto Earthquake (Fujii, 2026). The preliminary report represents an earlier version of the same study and has been revised, reorganized, and refined for submission as the present Brief Research Report. The principal analyses presented in this manuscript, including the analysis of seismic energy-input directionality, were already included in the preliminary report.

Generative AI statement

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fbuil.2026.1967966/full#supplementary-material

Supplementary material Supplementary material at the source

Footnotes

Edited by: Izuru Takewaki, Kyoto Arts and Crafts University, Japan

Reviewed by: Shinta Yoshitomi, Ritsumeikan University, Japan

Jia Guo, Kyoto University, Japan