To begin, we first describe the influence of a central-point shift component, which is an offset or residual deformation in the equation of motion of the displacement response of a structure post-earthquake. Additionally, the physical meaning of the different amplitudes and durations within a displacement response of a nonlinear structure is also discussed, as these features are closely associated with the development of SPLiTS (Enokida and Kajiwara 2020).

When a single-degree-of-freedom (SDOF) linear structure excited by an external force, its equation of motion without the central-point shift is described by: m x ¨ t + c x ˙ t + k x t = f t (1) where {m, c, k} is the set of the mass, damping, and stiffness coefficients, respectively; x is the relative displacement, equal to the inter-storey drift in the case of an SDOF structure; f is the external force and t is the time variable.

An SDOF linear structure having a constant central-point shift x _c in its displacement is described by: m x ¨ t + c x ˙ t + k x t − x c = f t (2)

By employing the coordinate of x * t = x t − x c , Eq. 2 can be rewritten as: m x ¨ * t + c x ˙ * t + k x * t = f t (3)

Apart from the notation of *, Eq. 3 is identical to Eq. 1. In addition, Eq. 3 even holds for a structure having a time-varying central-point shift: x ˙ c t , particularly at x ˙ c t ≈ 0 , because it leads to x ˙ * t ≈ x ˙ t and x ¨ * t ≈ x ¨ t . This indicates that the introduction of the coordinate x * t allows us to handle the structure having the shift component as an equivalent structure without a shift component.

When a displacement response consisting of n sets of half-cyclic waves, as shown in Figure 1, is obtained from a nonlinear structure, the amplitude and duration of each half-cyclic wave contain essential information of the structure, particularly for the case of A ₁≠A ₂≠···≠A _n and T ₁≠T ₂≠···≠T _n . Different amplitudes can indicate a structure’s non-linearity as it is closely related to the amplitude of the responses, whereas, different durations would demonstrate variations of the stiffness within the structure.

These considerations motivated the development of SPLiTS, which minimises the effect of the central-point shift component in displacement responses and handles a nonlinear structure as a set of linearised structures, based on the number of half-cyclic waves in the displacement response. SPLiTS has been developed for estimating the time-varying physical parameters of nonlinear structures using TDI, based on the following four principals:

P1. SPLiTS minimises central-point shift components in the displacement response data x t , and generates new data with minimised components x * t . The residual deformation observed in severely damaged structures and distortion generated by the integral of the measured acceleration data are typical examples of central-point shifts.

P1 and P2 are fundamental for estimating the time-varying physical parameters. P3 and P4 are the criteria for extracting major structural response data and minimising noise within the data for reliable estimation using TDI.

SPLiTS is based on the premise that structural responses of the displacement x t , velocity x ˙ t and acceleration x ¨ t are given; however, in practice, the acquisition of these data is not a simple task. In this regard, SPLiTS4ad is based on the estimation of the velocity response from the measured acceleration and displacement, while SPLiTS4a is based on the estimation of the velocity and displacement from the measured acceleration. As SPLiTS4a only relies on the measured acceleration, its measurement cost is lighter than that of SPLiTS4ad. As a result, SPLiTS4a has more signal processing procedures for estimating the required responses than SPLiTS4ad. These features are summarised in Table 1.

The estimation of physical parameters based on SPLiTS is exemplified by an N-storey structure that can be modelled by a lumped-mass system with N masses. The equation of motion for the ith sotrey can be described by: m i x ¨ i t + f c i t + f k i t = − m i x ¨ g t + f i + 1 t (4) where i = 1, ∙∙∙, N; x _g is the ground displacement; {m _i , x _i , f _ci , f _ki } is the set of the mass, relative displacement, damping force, and restoring force of the ith storey, respectively; and f i t = − ∑ j = i N m j x ¨ j t + x ¨ g t and f N + 1 t = 0 . For a linear system with constant damping and stiffness on each storey, the damping and restoring forces in Eq. 4 are described as f c i t = c i x ˙ i t − x ˙ i − 1 t , f k i t = k i x i t − x i − 1 t , and x ₀(t) = 0.

By introducing the inter-storey drift coordinate to the linear system governed by Eq. 4, the equation of motion for the ith storey becomes: c i x ^ ˙ i t + k i x ^ i t = f ^ i t (5) where f ^ i t = f i t and x ^ i t = x i t − x i − 1 t . When this system exhibits time-varying damping and stiffness on the ith storey, Eq. 5 can be rewritten as: c i t x ^ ˙ i t + k i t x ^ i t = f ^ i t (6)

SPLiTS equivalently estimates the physical parameters c i t ,   k i t from its responses x ^ ˙ i t ,   x ^ i t ,   f ^ i t by piecewise linearisation of the structure governed by Eq. 6. The linearisation is performed on half-cyclic waves in the displacement data containing the minimum central-point shift components. When the response data with minimised central-point shift components x ^ ˙ i * t ,   x ^ i * t are given in addition to its external force f ^ i t , the piecewise linearised structure for the lth half-cyclic wave consisting of n _l (>n ₀) steps is described by: f ^ i t ≈ c i * x ^ ˙ i * t + k i * x ^ i * t   t l   <   t   <   t l + Δ t l , (7) where {c _i ^*, k _i ^*} is the set of damping and stiffness in the linearised structure, respectively, and {t _l , Δt _l (=n _l ∙dt)} is the set of the initial timestamp and duration of the lth half-cyclic wave, respectively. The physical parameters in Eq. 7 can be estimated using the following inversion: P l = Q l T Q l − 1 Q l T F l , (8) where P l R 2 × 1 = c i * k i * ; Q l ∈ R n l × 2 = x ^ ˙ i t l ⋮ x ^ ˙ i t l + Δ t l x ^ i t l ⋮ x ^ ˙ i t l + Δ t l ; F l ∈ R n l × 1 = f ^ i t l ⋮ f ^ i t l + Δ t l ; R is the set of real numbers, and its superscript is the dimension of the matrix. The estimated parameters are stored in c i * t ,   k i * t as the representation of its entire duration, Δ t l . In computations, Eq. 8 should be solved by using a pseudo inversion method to enhance its accuracy.

By repeating the procedure in Eq. 8 to the other half-cyclic waves, the time-varying physical parameters in Eq. 6 can be equivalently obtained as the parameters of the linearised structures, c i * t ,   k i * t .

This parameter estimation is based on the condition that the masses of a structure are known. However, the estimation is possible even without this condition so long as the mass ratios of each storey are known. By dividing Eq. 6 by a reference mass m _s , it can be rewritten as: c i ∗ ′ t x ^ ˙ i t + k i ∗ ′ t x ^ i t = f ^ i ′ t = − ∑ j = i N α j x ¨ j t + x ¨ g t (9) where c i ∗ ′ t = α i c i t m i ,   k i ∗ ′ t = α i k i t m i and α i = m i m s . Note that, when Eq. 9 is employed for parameter estimation of structures, the estimates obtained by Eq. 8 become c i ∗ ′ t ,   k i ∗ ′ t instead of c i * t ,   k i * t .

The original form, SPLiTS4ad, was first introduced along with an estimation of the velocity response from the measured displacement and acceleration responses. This was intended to make SPLiTS feasible for laboratory experiments using dynamic testing facilities, such as shake tables, because they are commonly performed with sensors for acceleration and displacement without measuring velocity.

In SPLiTS4ad, the velocity response was estimated by applying a composite filter (Stoten DP 2001) to the measured acceleration and displacement data, as shown in Figure 2. Then, the set of structural responses, with the minimised central-point shift components, to be used for the TDI is obtained by: x * s = F r c p s ⋅ x s x ˙ * s = F r c p s ⋅ F d s ⋅ x s + F a s ⋅ x ¨ s (10) where F _d (s) = sω _c /(s + ω _c ); F _a (s) = 1/(s + ω _c ), ω _c is the switching frequency to determine the contribution of acceleration and velocity to the estimated velocity; F _rcp (s) (= 1 − F _cp (s)) is the filter to remove the central-point shift components; and F _cp (s) is a low-pass filter with a very low cut-off frequency to extract the shift components, which are mainly associated with residual deformation. ω _c can be determined from attainable ranges of the sensors employed. For example, when accelerometers and displacement transducers are reliable in the ranges of 0.5–5.0 Hz and DC–3.0 Hz, ω _c should be set from the smallest reliable range: 0.5–3.0 Hz.

SPLiTS4a relies only on the measured acceleration and other structural responses (velocity and displacement) must be estimated by integrals of the acceleration. However, the integrals generate unwanted distortions in the displacement and velocity data due to noise within the measured data.

In this regard, SPLiTS is effective as it minimises all the components of central-point shifts, which include the distortion generated at the integrals and residual deformation observed in a severely damaged structure. This feature allows us to minimise these components without separating the residual deformation and distortion derived from the noise. Note that the new form relinquishes estimating the residual deformation in structures, indicating the necessity of other approaches (e.g., onsite visual inspections) to inspect the deformation when it is required.

The distortion caused by noise is much more significant than the residual deformation in structural responses and must be mitigated by a more efficient technique than the filter used for the residual deformation in Eq. 10. The severity of the distortion was more clearly observed in the time domain than in the frequency domain. Thus, this study employs a technique using a moving-average filter that smooths a signal in the time domain to effectively extract the distortion in the integrated data. The filtering technique is shown in Figure 3A, and the structural responses required for TDI are obtained using the procedure shown in Figure 3B.

The performance of a moving-average filter depends on its average duration, which governs the target frequency to be minimised. When the filter with the averaging duration T _a is employed for a signal with the entire duration T _L , this filter becomes a rectangular pulse, as shown in Figure 4A. According to a study on moving-average filters by Smith (1997), the frequency response in the range of 0.0–0.5 Hz can be described by: F a v f a = sin π ⋅ f a ⋅ T a T a ⁡ sin π ⋅ f a , (11) where f _a = df, 2df, ∙∙∙, 0.5; and df = 1/T _L . Note that Eq. 11 when f _a = 0 was set to F _av (0) = 1 to prevent division by zero.

Figure 4B clarifies the relationship between the average durations and frequency components to be minimised, showing some unwanted humps in the curve. This indicates that the frequency components corresponding to the humps were not fully removed by the average filter. However, they can be mitigated by a multi-pass moving-average filter technique, which applies a filter multiple time. The four-pass filter (p = 4) in Figure 4C does not have humps, and the frequency components over 0.1 Hz can be fully removed by the filter.

This study introduces a four-pass moving-average filtering technique based on the features of moving-average filters. In this technique, a moving-average filter is applied to the velocity data twice, and a similar filter is applied to the displacement data twice to minimise their distortions. The filtering technique and structural response estimation for SPLiTS4a are described below.

The velocity data with the minimised central-point shift components x ˙ p * t are obtained by: x ˙ p * t = x ˙ p t − x ˙ p c t x ˙ p t = ∫ 0 t x ¨ ( τ ) d τ (12) w h e r e   x ˙ p c 0 t = 1 T v ∫ t t + T v x ˙ p τ 1 d τ 1 x ˙ p c t = 1 T v ∫ t t + T v x ˙ p c 0 τ 2 d τ 2 (13) and T _v is the average duration for the two-pass moving-average filter.

After obtaining the velocity x ˙ p * t , the displacement data with the minimised central-point shift component x p * t is obtained by: x p * t = x p t − x p c t x p t = ∫ 0 t x ˙ p * τ d τ (14) w h e r e   x p c 0 t = 1 T d ∫ t t + T d x p τ 1 d τ 1 x p c t = 1 T d ∫ t t + T d x p c 0 τ 2 d τ 2 (15) and T _d is the average duration for the two-pass moving-average filter.

Based on displacement x p * t in Eq. 14, the set of structural responses x ¨ * t ,   x ˙ * t for SPLiTS4a is created by: x * s = F L P s ⋅ x p * s x ˙ * s = s F L P s ⋅ x p * s (16) where F _LP is the low-pass filter used to realise the differentiation in Eq. 16. F _LP is applied to the responses of the displacement and velocity in Eq. 16. Subsequently, these responses are employed for physical parameter estimation.

As explained above, the structural responses in SPLiTS4a are governed by the four-pass moving-average filtering technique. Its effectiveness depends on the average durations, and we introduce a guide to decide the durations. To the process in Eq. 12, the duration T _v for the first trial should be determined by Eq. 11 with the target frequency to be removed. When the obtained x ˙ p * t still has some distortion, T _v needs to be adjusted to the one that provides velocity with the minimum distortion. Then, based on x ˙ p * t , the displacement x p * t is calculated by Eq. 14. In the first trial to remove its central-point shift components, T _d should start from a duration equal to or smaller than T _v , because this process further mitigates the distortion remained even after the process on x ˙ p * t . Again, at the following trials, T _d should be adjusted to the one that realises the displacement with the minimum distortion x p * t .

A full-scale three-storey structure in Figure 5 was designed to duplicate a typical steel structure built in Japan in the early 1990s. It consisted of 2-span frames for both longitudinal and transverse directions, and its size was 12.0 m × 10.0 m with a height of 10.8 m. The masses of the first, second, and third storeys were 44.9, 43.9, and 40.6 t, respectively.

In the experiments, servo-accelerometers were placed at five points on each storey (four corners and their centre) as well as the base. Two laser displacement transducers (measurable range: ±250 mm) were placed on the protection stands on each storey to measure each inter-storey drift in the longitudinal direction. The sampling interval was set to dt = 0.001 s.

The structure was uniaxially excited in the longitudinal direction using the two ground acceleration records shown in Figure 6. The one in Figure 6A is the North-South component of the ground acceleration data recorded at the Japan Railway Takatori station during the 1995 Hyougo-ken Nanbu (Kobe) earthquake. The other ground acceleration data in Figure 6A were artificially synthesised at the site of Kobe City Hall for an anticipated Nankai Trough earthquake, which has been a major concern in Japan. These were selected to examine the remaining seismic resistance performance of the structure against the anticipated earthquake, even after it was severely shaken by the Kobe earthquake. These ground motions are referred to as the Takatori and Nankai motions in this study.

According to the response spectra in Figure 6B, the Takatori motion surpasses the Nankai motion over the entire range of ∼3 s. Based on these ground motion characteristics, shake table experiments were performed with the following amplitudes: {40%, 60%, 80%, 100%} for the Takatori motion, and {50%, 100%, 150%} for the Nankai motion.

System identification tests using band-limited white noise were performed before and after each seismic excitation to investigate the modal parameters of the structure. According to the system identification before the first experiment with the Takatori motion, the intact structure was found to have natural frequencies of 1.37, 4.46, and 8.21 Hz.

The experiments were conducted using the Takatori motion with four amplitude cases {40%, 60%, 80%, 100%} and the Nankai motion with three amplitude cases {50%, 100%, 150%}. The maximum inter-storey drift of the structure obtained by these experiments is summarised in Figure 7. Structural responses in the cases of 60% and 100% Takatori motion and 150% Nakani motion are respectively illustrated in Figures 8–10. The hysteresis shown were obtained using the measured inter-storey drift responses and shear forces, which were calculated by the floor acceleration responses and masses. Note that the structural responses of 40% and 80% Takatori motions have been illustrated in detail in the first study on SPLiTS (Enokida and Kajiwara 2020).

In the experiments with the Takatori motion, the 40% excitation caused slight yielding in the structural components (Mizushima et al., 2018), and the 60% excitation expanded the yielded areas along with the increase in inter-storey drift on each storey, as seen in Figure 8C. According to Figure 7A, the 80% excitation resulted in a similar inter-storey drift as the 60% excitation. However, 100% excitation resulted in an extremely large inter-storey drift on the first storey that exceeded the measurable range of the displacement transducer: 250 mm. The twisted hysteresis loop of the first storey in Figure 9 is caused by this exceedance.

Experiments with the Nankai motion were performed on the structure, which had already experienced a large deformation by numerous excitations of the Takatori motion. According to Figure 7B, the inter-storey drifts at the excitation of 150% Nankai motion are roughly in between the Takatori motion’s 60% and 80% excitations. Although the inter-storey drifts of 150% Nankai motion in Figure 10B are greater than those of 60% Takatori motion in Figure 8B, the hysteresis loops in Figure 10C are generally not as voluminous as those in Figure 8C.

System identification tests were performed with a band-limited white noise after every seismic excitation and the modal parameters of the structure were identified using the FRF method (Ji et al., 2011). Figure 11A illustrates the natural frequencies identified by all the identification tests, and Figure 11B illustrates the modal shapes obtained after the major experiments (80% and 100% Takatori motion as well as 150% Nanaki motion). The modal shapes were standardised, such that the maxima became 1.0.

According to Figure 11A, frequency changes are not clearly observed in the identification tests after the Takatori motions of 40%, 60%, and 80%. The 100% values were found to significantly decrease the natural frequencies, indicating the occurrence of severe structural damage. In addition, the decreased natural frequencies remained unchanged with respect to the excitations of the Nankai motion. According to Figure 11B, the modal shapes are not significantly influenced by the Takatori motion for amplitudes up to 80%; however, the modal shapes after 100% differ from those for lower amplitudes. In addition, the first-order modal shape obtained after the 150% Nankai motion also differs from the modal shape after the 100% Takatori motion. This difference is presumably caused by further extensions of structural damage, such as fractures or cracks, by the series of Nankai motion excitations; however, these extensions are not clearly projected into the frequency changes, as shown in Figure 11A.

Structural state indices were estimated using both SPLiTS forms and the Takatori motion. The results are summarised in Table 2, and the detailed results of the 60% and 100% excitations are illustrated in Figures 12, 13. The steady stiffness listed in Table 2 was obtained based on the rates of increase of the structural energy absorptions, E ^* _cv , which can distinguish major structural responses contributing to the absorptions from those that do not.

For the experimental data of the 60% Takatori motion, the four-pass moving-average filtering technique in SPLiT4a effectively extracted the distortion in the displacement and velocity derived from the measured acceleration data, as seen in Figures 12A, B. The displacement and velocity with the minimised central-point shift components by SPLiTS4a in Figures 12C, D are sufficiently close to those obtained by SPLiTS4ad.

Based on the displacement and velocity in Figures 12C, D, the physical parameters for the 60% excitation were estimated, with reasonable similarities between both forms, as shown in Figures 12E, F. In Figure 12G, the structural energy absorptions, E ^* _cd , calculated using SPLiTS4a corresponded to that obtained using SPLiTS4ad, although the former is slightly lower. According to Figure 12H, the rates of increase of the absorptions, E ^* _cv , are large only in limited ranges, indicating that structural responses out of the ranges do not contribute to energy absorption. Regarding the steady stiffness, which is obtained from the structural responses out of the ranges, the values of SPLiTS4a do not significantly differ from those of SPLiTS4ad, as shown in Table 2.

In the experiment with 100% excitation, the inter-storey drift on the first storey exceeded the measurable range of the displacement transducers. The lack of data in the first storey makes the application of SPLiTS4ad difficult, although SPLiTS4a is not affected by this measurement issue. Figures 13A, B highlights the effectiveness of SPLiTS4a in estimating the velocity and displacement data, whereas Figures 13C, D clarifies the measurement issue for SPLiTS4ad by showing unwanted pulse-like waves for the velocity and displacement data.

Based on the data in Figures 13C, D, the physical parameters and structural energy absorptions during 100% excitation were reasonably estimated by SPLiTS4a, whereas SPLiTS4ad failed to do so, as shown in Figures 13E, F. The influence of the measurement issue becomes apparent from the structural energy absorptions in Figure 13G, which show a sharp drop for SPLiTS4ad in the first storey. In addition, Figure 13H shows SPLiTS4ad’s negative increase rate within the first storey, which theoretically should not occur. This examination clarifies the superiority of SPLiTS4a over SPLiTS4ad.

Structural state indices were estimated using both SPLiTS forms and the Nankai motion. These results are summarised in Table 3, and the detailed results for the 150% excitation are illustrated in Figure 14. The steady stiffness values listed in Table 3 were obtained in the same manner as those in Table 2.

Figures 14A, B clearly show the effectiveness of the four-pass moving-average filtering technique in the Nankai motion experiments because distortions in the velocity and displacement are effectively extracted by the technique. The response data with the minimised central-point shift components by SPLiTS4a in Figures 14C, D are close to those obtained by SPLiTS4ad.

As shown in Figures 14E, F, the physical parameters estimated by SPLiTS4a are closely matched with those by SPLiTS4ad, although SPLiTS4a′s structural energy absorptions have become slightly lower than that of SPLiTS4ad in Figure 14G. This difference is mainly because of different processes for structural responses of displacement and velocity.

The physical parameter estimation using both SPLiTS forms provided the structural state indices of the structure (i.e., physical parameters, structural absorptions, and its rates of increase). We discuss the overall results of the estimates by linking them to the structural damage caused by each experiment. Figure 15 summarises the maximum structural energy absorption, steady stiffness, and accumulation of the energy absorption for each storey.

As shown in Figure 15A, the structural energy absorption increases as the ground motion amplitude becomes larger. In general, the structural energy absorption using SPLiTS4a was 90% of SPLiTS4ad, as shown in Figure 15B, excluding the first storey’s absorption at 100% Takatori motion, which had a displacement measurement issue.

In the case of the Takatori motion, the steady stiffness in Figure 15C gradually decreased with an increase in the structural energy absorption. This indicates that the constraints of the storeys are gradually loosened by the increase in structural energy absorption. The 100% Takatori motion clarifies this feature, showing significant energy absorptions at the first and second storeys in Figure 15A along with significant reductions in steady stiffness at the two storeys in Figure 15C. In fact, during the onsite inspection after the 100% Takatori experiment (Mizushima et al., 2018), the first and second storeys were found to have fractures at the end of the beams. The occurrence of the fractures is more clearly comprehensible from Figure 15D, which shows significant accumulation in these two storeys until the end of the 100% Takatori excitation with significantly lower accumulation in the third storey.

In the case of the Nankai motion, the structural energy absorption is much lower than that of the 100% Takatori motion, as shown in Figure 15A. This resulted in an insignificant increase in the accumulated energy absorptions, as shown in Figure 15D. Thus, the steady stiffness on each storey did not change significantly, as shown in Figure 15C. This result indicates that these excitations caused minimal structural damage. In fact, no additional fractures were found during the onsite inspection after 150% excitation of Nankai motion, although some cracks did extend (Mizushima et al., 2018).

This indicates that the structural energy absorption can quantify the deterioration of a structure caused by numerous seismic excitations. Excessive energy absorption is transformed into fractures of structural components, which can be deduced from the time-history storey stiffness. This examination clarified the effectiveness of the structural state indices obtained by SPLiTS for the condition assessments of seismically damaged structures.