# Significance of non-stationary characteristics of ground-motion on structural damage: shaking table study

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## Abstract

This paper reports the results of a set of benchmark medium-scale shaking table tests to investigate the significance of the non-stationary characteristics of ground-motion on nonlinear dynamic responses and the structural damage of reinforced concrete (RC) columns. To examine the influence of ground-motion characteristics, four RC columns are tested under (1) near-field without pulse, (2) near-field pulse-like, and (3) far-field ground-motions. These ground-motion records were spectrally matched by the reweighted Volterra series algorithm without changing non-ergodic characteristics. To explore the confinement effects, two sets of column specimens are designed to represent the modern well-confined and older lightly-confined RC columns. Each column is tested in slight, extensive and complete damage limit states. Then aftershock excitations are conducted to investigate the performance of severely damaged RC columns. Low amplitude white-noise tests are conducted on pristine columns and after each damage limit state experiment to detect natural frequency variant of damaged columns using transfer function estimate. Furthermore, using time–frequency analysis, the real-time variant frequency of test specimens is estimated. The significant duration of ground-motions accounting for the effect of non-stationary characteristics of ground-motion is also estimated by time-cumulative damage analysis of the test results. Finally, the time-variant stiffness degradation of RC columns is estimated.

## Keywords

Shaking table Ground-motion duration Low-cycle fatigue Time–frequency analysis Transfer function estimate Inelastic buckling## 1 Introduction

In performance-based design of reinforced concrete (RC) structures, ground columns in building structures, and piers in bridge structures are usually the most vulnerable components. In bridges, the seismic performance of the whole structure is governed by the performance of the piers when they are subjected to large earthquakes. This is because the current modern seismic design codes (CEN 2010; Caltrans 2013) rely on a proper detailing of the plastic hinge regions where most of the inelastic deformations are expected to occur. Hence, the structural performance is greatly influenced by the structural details (e.g. sufficient confinement for concrete) and material performance (e.g. ductility of reinforcing steel) (Mander et al. 1988; Chang and Mander 1994; Kunnath et al. 1997; Lehman 2000). Furthermore, ground-motion characteristics (frequency content, amplitude, duration etc.) have a significant effect on the nonlinear seismic response and cumulative damage of RC columns (Hancock and Bommer 2005, 2007; Chandramohan et al. 2016; Kashani et al. 2017a). Therefore, several researchers have developed numerical analysis methods to study the behaviour of RC components with different detailing when subjected to different types of ground-motions (Domizio et al. 2017; Kashani et al. 2017a; Nojavan et al. 2017; Su et al. 2017). Other researchers (Kunnath et al. 1997; Laplace et al. 1999; Lehman 2000; Elwood 2004; Johnson et al. 2006; Chen et al. 2008; Phan et al. 2007; Carrea 2010; Brown and Saiidi 2011) conducted several experimental studies on the impact of load history on structural damage of RC columns using quasi-static cyclic and dynamic shaking table tests.

Furthermore, quantifying the influence of a particular ground-motion time-series, on the nonlinear structural response and cumulative damage, by a few salient parameters is a challenging problem. Several researchers (Kramer 1996; Cornell 1997; Iervolino et al. 2006; Hancock and Bommer 2007; Raghunandan and Liel 2013; Sarieddine and Lin 2013) have employed a range of parameters (amplitude, energy, averaged frequency content, duration and envelope shape measures etc.) to investigate the most significant factors that govern the nonlinear dynamic response. Notably, it is apparent that there are differences in near/far field, with/without pulses, time-series. The differences between these ground-motions are typically expressed in terms of their power spectral content, which is statistically stationary (time-invariant) (Chatfield 2003). However, there are also differences in ground-motion time-series caused by the non-stationary (time-variant) statistical characteristics, which are not captured by power spectral alone (Han et al. 2017). These, non-stationary statistical characteristics (Chatfield 2003), include the envelope shape and the presence of large, time localised, pulses in the ground-motion time-series.

Moreover, the effect of ground-motion duration has been widely studied, but resulted in mixed findings, suggesting that the effect of ground-motion duration depends on system response measure (i.e. damage measures). Hancock and Bommer (2006) reported that comparing the responses to spectrally compatible long and short duration ground-motions may not yield significant differences in the maximum peak displacement response. However, long-duration ground-motions will affect the accumulated damage on the structure due to low-cycle high-amplitude fatigue degradation of materials (Hancock and Bommer 2007). Furthermore, several duration metrics have been suggested by various researchers for assessing nonlinear structural response (Bommer and Martinez-Pereira 2000; Kempton and Stewart 2006; Foschaar et al. 2012). A recent study conducted by Chandramohan et al. (2016), proposes a methodology to quantify the effect of ground-motion duration on the probability of structural collapse. They concluded that the probability of structural collapse is larger under a long-duration ground-motion than a spectrally equivalent short-duration ground-motion. Their finding is in contrast to most other previous studies (Hancock and Bommer 2006) which concluded that ground-motion duration does not influence peak structural deformations. These mixed conclusions are mainly due to limitations in numerical models (low-order nonlinear spring models, fibre models, 3D continuum models), and not separating out the effects of stationary and non-stationary characteristics of the ground-motions.

More recently, Kashani et al. (2017a) developed a novel numerical approach to quantify the impact of ground-motion types (near/far field, with/without pulses time-series), caused by the non-stationary content (time-varying parameters that are not captured by power spectral content alone), on the nonlinear dynamic response of RC bridge piers, including the effect of material cyclic degradation. They used the new algorithm (known as RVSA) developed by Alexander et al. (2014) to generate a set of artificial spectrally equivalent ground-motions. They used the suggested far-field (FF), near-field without pulse (NFWP) and near-field pulse-like (NFPL) ground-motions in FEMA P695 (2009). The selected ground-motions were different in their stationary and non-stationary components. Using the RVSA, they matched these ground-motions to a target response spectrum (without qualitatively changing the non-stationary ground-motion characteristics, i.e. envelope and pulses) to be used in nonlinear dynamic analyses. With this approach, they isolated the influence of ground-motion envelope and pulses (non-stationary effects) from ground-motion response spectral characteristics (stationary effects). The structural model that they used was developed in earlier research by Kashani (2014), Kashani et al. (2015a, b, 2016) that account for the inelastic buckling and low-cycle fatigue degradation of longitudinal reinforcing bars, concrete cover spalling and core concrete crushing. They concluded that the non-stationary content of ground-motion affects the cumulative damage of structural response and has less significant impact on peak displacement response. However, their conclusions are purely based on a numerical exploration study and have not been verified experimentally. Accordingly, the aim of this paper is to extend the earlier work by Kashani et al. (2017a) and justify the numerical results through a comprehensive experimental shaking table study.

To this end, three medium scale well-confined columns and a lightly-confined RC column are cast. The well-confined columns represent the modern seismically designed RC columns and the lightly-confined column represents older RC columns. Three spectrally matched ground-motions including a NFWP, a NFPL, and a FF ground-motions are selected from the same set of ground-motions that used in numerical analysis conducted by Kashani et al. (2017a) for the shake table experiments. The three well-confined columns are tested under the selected ground-motions at different intensity, and the lightly-confined column is tested under FF ground-motion to investigate the influence of the reinforcement detail. Each column is tested in slight (scale factor (SF) = 0.25, i.e. 25%), extensive (SF = 300%) and complete damage (SF = 500%) limit states. After the complete damage limit state experiments, aftershock excitations (the same ground-motion as the mainshock, SF = 300%) are conducted to investigate the performance of severely damaged RC columns. Low amplitude white-noise tests are conducted on pristine columns and after each damage limit state experiments to detect the change in natural frequency content of the damaged columns using transfer function estimate. Furthermore, using a novel time–frequency analysis technique, the real-time variation in specimen’s response frequency, during tests, is estimated. The significant duration of the ground-motions accounting for the effect of non-stationary characteristics of the ground-motion is estimated by an innovative time-cumulative damage analysis of experimental results and compared with previous numerical studies. Finally, the time-varying stiffness degradation of RC columns is estimated, and the influence of non-stationary characteristics of the ground-motions on peak displacement response of RC columns is studied.

## 2 Experimental programme

### 2.1 Specimen details and preparation

*L/D*= 5 and 12.5, where

*L*is the length of the bar between ties and

*D*is the diameter of vertical bars. Figure 2b, d shows the influence of tie spacing on inelastic buckling behaviour of reinforcement bars. It should be noted that the strain in Fig. 2b, c is the average strain over the buckling length, and hence, it is shown as ‘Mean Strain’ in the figure. The detailed discussion of the experimental testing and strain measurement is available in Kashani et al. (2013). The inelastic buckling of the vertical bars is an important parameter that affects the nonlinear behaviour of RC columns in flexure (Kashani 2014; Kashani et al. 2015a, b, 2016; Salami et al. 2019).

Mechanical properties of tests specimens

Bar diameter ( | 8 mm | 16 mm | |
---|---|---|---|

Yield strain | | 0.00261 | 0.002733 |

Yield stress (MPa) | | 520 | 530 |

Elastic modulus (MPa) | | 200426 | 193913 |

Hardening strain | | N/A | 0.02547 |

Strain at maximum stress | | 0.05660 | 0.164800 |

Maximum stress (MPa) | | 645 | 640 |

Fracture strain | | 0.151800 | 0.227350 |

### 2.2 Experimental test setup

*x*is the horizontal plane of shaking direction,

*y*is the out of horizontal plane direction and

*z*is vertical coordinate.

*x*direction. Two accelerometers are placed on the mass block to record the responses in

*x*and

*y*directions. An accelerometer is also placed on top of the foundation to record the exact input ground-motion to the RC column, and compare it to the recorded ground-motion on the shake table. The detailed layout of the instrumentations is shown in Fig. 4. Eight Linear Voltage Displacement Transducers (LVDT) are placed on front and back of the columns’ base to measure the columns’ curvature and slippage of the columns’ base.

Four cable extension position transducers (Celesco) are used along the height of the columns to measure the lateral displacement of the columns in the *x* direction. An additional Celesco is used to measure the lateral displacement of the columns in the *y* direction.

### 2.3 Ground-motion selection and matching

As explained in the introduction, the purpose of this experiment is to investigate the impact of non-stationary characteristics of the ground-motion and reinforcement detail on the structural damage of RC columns. To this end, three ground-motions are selected that include a Far-Field (FF), a Near-Field without Pulse (NFWP), and a Near-Field Pulse-Like (NFPL) ground-motions.

A number of ground-motion records with different properties are listed in FEMA P695 (2009). For this study, Northridge and Imperial Valley ground-motions are selected for NFWP and NFPL experiments respectively, and Manjil ground-motion is selected for FF experiments.

Experimental test matrix

Specimen types | Ground-motion | Ground-motion intensity | |||
---|---|---|---|---|---|

Slight-25% | Extensive-300% | Complete-500% | Aftershock-300% | ||

1. Well confined | Northridge | NFWP-R1 | NFWP-R2 | NFWP-R3 | NFWP-R4 |

2. Well confined | Imperial Valley | NFPL-R1 | NFPL-R2 | NFPL-R3 | NFPL-R4 |

3. Well confined | Manjil | FF-WC-R1 | FF-WC-R2 | FF-WC-R3 | FF-WC-R4 |

4. Lightly confined | Manjil | FF-LC-R1 | FF-LC-R2 | FF-LC-R3 | FF-LC-R4 |

## 3 Experimental results and discussion

### 3.1 Force–displacement hysteretic response and structural damage

### 3.2 Transfer function estimate of the response

Power Spectral Density (PSD) is normally used to characterise the frequency content of a time-series (Cryer and Chan 2008). Power spectral estimates are useful in a variety of applications, including system identification white noise tests. The PSD can quantify the periodic pattern, if there is any, in a time-series by determining the peaks, in frequency, which corresponds to these periodicities. Given a linear system, with a known excitation and response, we can perform a system identification by estimating the transfer function. This transfer function estimate makes use of auto and cross power spectral densities, using Welch’s (1967) algorithm. This algorithm involves (1) dividing both signals into a number of overlapping sections, (2) weighting each section with a window function to attenuate the influence of leakage, (3) Fast Fourier transforming the windowed section time-series and (4) average all windowed sections which increases the signal to noise ratio. Longer, stationary, time-series permit greater noise reduction and higher frequency resolution of the transfer function estimate.

### 3.3 Time–frequency analysis using Wigner–Ville distribution algorithm

In Sect. 3.2, we used the transfer function estimate to quantify the impact of structural damage due to material nonlinearity and cyclic degradation on the natural frequency of the structure (frequency of first and second mode). However, the transfer function estimate quantifies the change in frequency before and after seismic tests but not during a seismic test. Investigating the effects of the non-stationary content of ground-motion on (1) structural damage and (2) system frequencies during earthquakes, require time–frequency analyses of the response. Exploration of time–frequency content of a time series can be estimated using, for example, the short-time Fourier transform, the Hilbert transform, and the continuous wavelet transform. However, in this paper we found the Wigner–Ville distribution (WVD) (Cohen 1994; Semmlow and Griffel 2009) has the greatest utility as it allowed a higher resolution in both time and frequency.

Additionally, the values of dissipated energy (work) are also shown in Fig. 11. The method used in the calculation of this energy is available in Sect. 3.4 of this paper. This dissipated energy is essentially the integral of the force/deflection loop. The large increases in normalised work (which are computed with both response displacement and acceleration time-series) are well correlated with the large power in the WVD (which are computed using response accelerations alone). In Fig. 11c, d, for columns 3 and 4, the instantaneous frequency estimates suggest that large cracks form in the cover far earlier, in time, than the dissipated energy damage measure implies. This is because the instantaneous frequency estimate reports information about crack formation while the dissipated energy damage measure calculates the work done in opening/closing these cracks and yielding of steel.

Figure 11 also shows that the response frequency can increases again after these first large drops in frequency. It is not clear whether this is an artefact of the WVD algorithm or due to pieces of broken concrete between the cracks. In Fig. 11a the increase in instantaneous frequency from 7 to 10 s is at reasonably high-power levels so is most probably not an artefact of the WVD algorithm. This, therefore, points to a physical explanation. During the column damage process, the concrete cover and part of the core concrete begin crushing. In this cyclic degradation process, parts of concrete pieces remain within the cracks. Therefore, in the subsequent cycles, the concrete pieces between cracks prevent complete crack closure and stiffen the whole structure. Evidence for this phenomenon is seen during the experiment. This has been reported by other researchers (Stanton and McNiven 1979; Kwan and Billington 2003; Lee and Billington 2009). The incomplete crack closure has a major impact on the cross over displacement and residual drift of the RC columns. It is very difficult to be quantified numerically (Lee and Billington 2009). Furthermore, we concluded that the drop in the response frequency of the RC columns is mainly governed by the damage in concrete. Therefore, it is important to investigate the influence of concrete cover thickness. This is an area for future work, and the results of this study provide a set of experimental data set for other researchers in the future research. Additionally, using high-speed/resolution video footage of the plastic zone would be very useful in future work.

### 3.4 Time-cumulative damage and effective duration analysis

*a*

_{g}is the ground-motion acceleration time-series,

*g*is the acceleration gravitational constant and \(I_{A} \left( t \right)\) is the Arias integral at time

*t*of the time-series

*a*

_{g}. The original Arias Intensity (

*I*

_{A}) (Arias 1970) definition was the integral value, Eq. (1), at end of the earthquake time-series. A graph of this integral vs time it is known as a Husid plot. The initial idea (Arias 1970) was to determine the time points

*t*

_{0}and

*t*

_{1}, which corresponds to 5% and 95% of the Arias Intensity. In this case, the effective duration

*D*would be simply defined as

*D*=

*t*

_{1}−

*t*

_{0}. The confidence limits of 5–95% are typically used statistically for Gaussian random processes. However, it would be more reasonable to use these confidence limits on the structural response (a damage measure) rather than the excitation of the structure (an intensity measure). This is because structural engineers are ultimately interested in damage levels. Let us assume that we use confidence limits of 5–95% on some structural response (damage measure) to define an effective duration \(D^{{\prime }}\). What confidence limits should we employ on my intensity measure to get the same effective duration \(D^{{\prime }}\) ? Foschaar et al. (2012) suggested employing 5–75% of the Arias Intensity. However, the stationary/non-stationary content of ground-motion and the structural system may have a large influence the choice of these suggested confidence limits (Chandramohan et al. 2016; Raghunandan and Liel 2013; Kempton and Stewart 2006). In this experiment, the stationary content of the ground-motion for all tests (as they are spectral matched) is very similar. Therefore, we remove the large variance due to the stationary content to quantify the influence of the non-stationary content alone on effective duration estimates.

In this study, the hysteretic energy is used as a damage measure to quantify the cumulative damage in the RC columns. Confidence limits of 5–95% on this damage measure are then employed to determine the time points *t*_{0} and *t*_{1}, hence effective duration can be determined. We will then explore what confidence limits should be used on the Arias Intensity to produce the same effective duration. Finally, we will explore whether there is any evidence to suggest these confidence limits on the Arias Intensity should depend on the non-stationary class of ground-motion (i.e. NFPL, NPWP, FF).

*W*, can be calculated by evaluating the integration in Eq. (3).

*m*is modal mass,

*F*is the base shear,

*t*is time, and

*x*is the response displacement. Chopra (2016) additionally removes linear elastic and damping energy terms. However, it should be noted that in this experimental case, we do not know precisely the damping model, nor the initial stiffness of the column (under larger amplitudes). Additionally, the experimental model is not a single degree of freedom system. Hence, we use an approximation, Eq. (3), and recognise some small oscillations are due to the limitation described in the previous sentences.

*W*and

*I*

_{A}over the whole duration of ground-motion at 300% excitation level. In Fig. 12,

*W*and

*I*

_{A}are normalised to their corresponding maximum values. In order to quantify the effective duration of ground-motions, we choose the confidence limits of 5% to 95% of

*W*(i.e. 5–95% of actual cumulative structural damage). Then, identify time points

*t*

_{0}and

*t*

_{1}, hence effective duration. We then determine the confidence limits on

*I*

_{A}graphs that would result in the same time points and effective duration estimate. The results are shown in Fig. 12b (NFPL) demonstrate that 5–75% of the Arias Intensity provides a good estimate of the effective duration of ground-motion. This is in a good agreement with the initial numerical study by Foschaar et al. (2012). However, Fig. 12a (NFWP) 5–85% might be a better set of confidence limits.

Comparison of Fig. 12a with Fig. 12c, d shows that even though the FF ground-motion is longer (in seconds) its effective duration is smaller than the NFWP. This suggests that the work done in opening/closing cracks occurs over a longer duration for the NFWP record. Furthermore, comparing Fig. 12c, d shows that structural detailing does not have a significant influence on effective duration of ground-motion at this level of excitation.

### 3.5 Time-effective stiffness degradation analysis

*N*point window for the entire ground-motion (Eq. (4)).

*F*

_{i}are the base shear (that is determined by inertial forces \(F_{i} = - m\left( {\ddot{x}_{i} + \ddot{x}_{g} } \right)\)) and

*x*

_{i}mass displacement response at the time

*t*

_{i}.

*K*

_{i}is the time-varying stiffness estimate and

*c*

_{i}is the time varying residual displacement at the time

*t*

_{i}. Equation (4) is solved using single value decomposition. The window length over which this averaging is performed is taken as 1 s. Thus,

*N*=

*f*

_{s}/2 where

*f*

_{s}is sampling frequency.

Comparing Fig. 13c with Fig. 11c and Fig. 13d with Fig. 11d underlines that (1) the instantaneous frequency estimates (Sect. 3.3, based acceleration responses alone) is a good measure of loss of stiffness, (2) this loss of stiffness is due to crack formation (3) this crack formation occurs before the large energy dissipation due to open/closing cracks and steel yielding.

Figures 13 shows that the minimum stiffness in all the columns (regardless of ground-motion type and reinforcement detail) after the earthquake is about 20% of the uncracked elastic stiffness of the pristine column. This is very important finding and can be used in numerical/analytical studies using simplified low-order models (single/multi degree of freedom models with nonlinear springs) to account for stiffness degradation of the structure.

### 3.6 Influence of ground-motion type and duration on peak response

Several researchers investigated the influence of ground-motion duration on peak displacement response of RC structures (Hancock and Bommer 2006). Hancock and Bommer (2006) concluded that the ground-motion duration does not have a significant influence on the peak displacement response. In contrast, Chandramohan et al. (2016) concluded that peak displacement response of RC structures is higher in long-duration ground-motion than spectrally equivalent short-duration ground-motions. In this section, we discuss if the ground-motion type and duration have any influence on peak displacement response of RC columns using experimental results.

These results confirm that although all of these ground-motions are spectrally matched, the ground-motion type, duration, and reinforcement detail have a significant influence on nonlinear behaviour, structural damage, and subsequently peak displacement response of RC columns at large amplitude excitations. These results are in contrast with the numerical results obtained by Kashani et al. (2017a), where the same NFWP and NFPL ground-motions showed smaller peak displacement responses. The main difference between the numerical study (Kashani et al. 2017a) and the present experimental testing is the cross-sectional shape of the columns. The columns studied in numerical analysis (Kashani et al. 2017a) are a circular cross section, but the columns in the current experimental programme are square cross-sections. In another study by authors (Kashani et al. 2017b), it is shown that cross section shape of RC columns has a significant influence on the nonlinear flexural response and failure mode. Considering the complexities and uncertainties in ground-motion characteristics and limitations in numerical models, it is very difficult to find a solid correlation between the ground-motion duration and peak displacement response. Therefore, there is a need for further detailed numerical and experimental studies in future research to close this issue. However, the results of this experiments provide a good insight into this problem and allow other researchers to use these experimental data as a platform for verification of numerical models.

## 4 Conclusions

- 1.
The transfer function estimate shows that concrete cover spalling is the main factor governing the natural frequency drop in RC columns. This shows that yielding and plasticity in reinforcing steel does not have any significant influence on the inelastic frequency of the RC columns. However, if the RC column is poorly detailed and suffers from the inelastic buckling of vertical bars, then the concrete cover spalls more quickly and core concrete crushes soon after bar buckling.

- 2.
The time–frequency analysis WVD and time-varying stiffness analyses show a small increase in the response frequency after the main drop due to crack formation. This is probably due to the influence of concrete pieces trapped within cracks, which prevent the complete crack closure. This phenomenon also has a significant impact on the cross over displacement and residual drift of RC columns. This conclusion is in good agreement with the results reported by other researchers who conducted static cyclic experiments and numerical modelling of RC columns (Stanton and McNiven 1979; Kwan and Billington 2003; Lee and Billington 2009).

- 3.
The time-cumulative damage (dissipated energy based on force–displacement integral) and the Arias integral (squared acceleration response integral, Eq. (1)) are well correlated with this simple structural system. Obtaining suitable confidence limits for effective duration estimate is difficult as it appears to be dependent on both structure and ground-motion time-series. The results show that confidence limits from 5 to 75–85% are reasonable, which is in partial agreement with initial numerical studies conducted by Foschaar et al. (2012).

- 4.
The experimental results show that regardless of ground-motion type all RC columns experience a stiffness drop to about 20% of their elastic uncracked original stiffness.

- 5.
Results indicate that (1) the instantaneous frequency estimates (Sect. 3.3, based acceleration responses alone) is a good measure of loss of stiffness, (2) this loss of stiffness is due to crack formation (3) this crack formation occurs before the large energy dissipation due to open/closing cracks and steel yielding.

- 6.
It was found that the ground-motion type and duration affect the peak displacement response owing to the excitation amplitude.

## Notes

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