# Study on Quasi-Static Loading Protocols considering the Action Characteristics of Long-Period Ground Motions

Authors: Ke Yang; Bo Wang; Jiawei Zhang; Zhe Li; Boquan Liu
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

Due to abundant low-frequency components of long-period ground motions (LPGMs), long-period structures are susceptible to severe damage. The corresponding time-history displacement responses have significant “large-displacement” and “long-duration” characteristics. These action characteristics essentially reflect the different loading paths imposed on structures of LPGMs from ordinary ground motions (OGMs). Hence, revealing the influence mechanism of the action characteristics on the seismic performance of structural components is the key to investigating the influence of LPGMs on the whole structure. This paper presents a kind of quasi-static loading protocol considering the action characteristics of LPGMs. Firstly, nonlinear time-history analyses on structural systems subjected to 50 selected representative LPGMs were conducted. Inelastic cycles and corresponding amplitudes of time-history displacement responses under LPGMs were statistically analyzed through the rainflow method. Then, considering two of the most significant factors, structural period and target ductility, a prediction model of cycle number and cycle amplitude was obtained by regression. On this basis, the quasi-static loading protocol considering the action characteristics of LPGMs was developed. Proposed protocols can be directly applied to experimental investigations on the seismic performance for structural components under LPGMs.


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## Body

## 1. Introduction

Long-period ground motions (LPGMs) refer to the far-field ground motions with predominant periods of 1∼10 s or longer in general, whose cause is primarily related to the propagation path of seismic waves [1]. Existing researches [2–4] indicated that the time-history displacement responses of long-period structures under LPGMs have distinct characteristics of “large-displacement” and “long-duration.” It essentially reflects different action paths imposed on structures of LPGMs from ordinary ground motions (OGMs). Plenty of researches [5–16] showed that the loading path has a remarkable influence on the strength and stiffness deterioration, deformation performance, energy dissipation capacity, and failure mechanism of structural components. Therefore, it can be considered that revealing the influence mechanism of the LPGMs action characteristics, “large-displacement” and “long-duration,” on the seismic performance of structural components is the key to investigating the influence of LPGMs on the whole structure. Quasi-static loading protocols considering the action characteristics of LPGMs could provide a possibility for experimental investigation of seismic performance and failure mechanism for structural components subjected to LPGMs.The loading history of ground motions imposed on structures can be regarded as a random loading process. The load time-history can be converted into a series of full or half cycles through a counting algorithm. Among the existing counting algorithms, the rainflow counting algorithm [17–20] is the most widely used one. It presents the consistency of the counting procedure with stress-strain characteristics of materials, the better consideration of cycle distribution of random loading, and reliable mechanical basis. In recent years, some quasi-static loading protocols [21–30] considering the action characteristics of practical ground motions have been developed based on the rainflow method. However, LPGMs were not considered in and the corresponding quasi-static loading protocols could not reflect the action characteristics of LPGMs.Therefore, quasi-static loading protocols considering the action characteristics of LPGMs were proposed in this paper. Two types of structural components were taken as the engineering background: RC pier columns in the long-span bridge, which can be simplified as the single-degree-of-freedom (SDOF) system, and columns on the weak story of high-rise RC frame structure representing the multiple-degree-of-freedom (MDOF) system. Firstly, nonlinear time-history analyses on SDOF and MDOF systems subjected to 50 selected representative LPGMs were conducted. Inelastic cycles and corresponding amplitudes of time-history displacement responses were statistically analyzed through the rainflow method. Then, multiple influence factors of cycle number and cycle amplitude including the ground motion type, structural period, target ductility, strength reduction coefficient, and damping ratio were evaluated. Finally, considering two of the most significant factors, structural period and target ductility, a prediction model of cycle amplitude and cycle number was proposed by regression. On this basis, the quasi-static loading protocols for structural components in SDOF and MDOF systems under LPGMs were developed.


## 2. Nonlinear Time-History Analyses

### 2.1. Selection of Ground Motions

In this paper, the weighted average value of amplitude (βl) between 2 s and 10 s on the spectrum curve of dynamic amplification coefficient [31] was used as the quantitative index to distinguish LPGMs from OGMs. 50 representative LPGMs from Chi-Chi earthquake, WenChuan earthquake, East Japan earthquake, and Tokachi earthquake were selected to conduct the nonlinear time-history analyses. 15 representative OGMs were also selected for the comparison.


### 2.2. Structural Model

RC pier column in the long-span bridge was simplified as the SDOF system, and high-rise RC frame structure was simplified as the MDOF system. The Bouc–Wen–Baber–Noori (BWBN) model [32–35] was adopted to simulate the hysteretic behavior of structural components. It comprehensively considers strength deterioration, stiffness deterioration, and the pinching effect. Hence, the cumulative structural damage caused by LPGMs could be well exhibited.The motion equation of SDOF system is shown as follows:(1)mu¨t+cu˙t+Ft=ft,where m is the mass; c is the linear viscous damping coefficient; u¨tand u˙tare the acceleration and velocity, respectively; F(t) is the restoring force; f(t) is the external excitation.As shown in Figure1, the restoring force of the BWBN model is provided by two parallel springs (linear and nonlinear). Correspondingly, F(t) in equation (2) is the summation of linear restoring force Fe(t) and nonlinear restoring force Fh(t).(2)Ft=Fet+Fht=αkut+1−αkzt,where α is the ratio of stiffness after yielding to stiffness before yielding; k is the initial linear elastic stiffness; u(t) is the displacement; z(t) is the hysteretic displacement, defined by the differential equation with initial condition z(0) = 0, as shown in the following equation:(3)z˙t=hzAu˙−νβu˙zn−1z+γu˙znη,where A is the control parameter of hysteretic amplitude, A = 1; β, γ, and n are the shape parameters of hysteretic curves; υ and η are the strength and stiffness deterioration parameters, respectively.(4)ν=1+δνε,η=1+δηε,where δv and δη are the strength and stiffness deterioration rates, respectively; ε is the cumulative hysteretic energy consumption.(5)ε˙=1−αω02zu˙.Figure 1
Bouc–Wen–Baber–Noori model.In equation (3), h(z) is the function describing the pinching effect, as shown in following equation:(6)hz=1−ζ1exp−zsignu˙−qzmaxζ22,ζ1=ζs1−exp−pε,ζ2=ψ+δψελ+ζ1,zmax=Aνβ+γ1/n,where signu˙ is the symbolic function of velocity. When u˙ > 0, u˙=0, and u˙ < 0, signu˙ takes 1, 0, and -1, respectively. ζs, ψ, δψ, λ, q, and p are the parameters of the pinching effect. Among them, ζs controls the total slippage; ψ controls the range of pinching; δψ is the pinching rate; λ is the interaction coefficient between pinching severity and pinching rate; q controls the starting position of the pinch; p is the pinching slope.The values of all parameters in the BWBN model adopted in this paper were obtained from Reference 35.


### 2.3. Nonlinear Time-History Analyses

Nonlinear time-history analyses on simplified structural models subjected to the selected ground motions were conducted through the iterative procedure shown in Figure2. It should be pointed out that the results of the weak story with maximum drift ratio are required for the MDOF system. The ductility was expected as the control index in developed loading protocols. Therefore, the peak ground acceleration (PGA) of ground motion, as the iterative variable, was increased from 0 with a constant increment of 0.1 gal until the structural components reach the predetermined ductility (target ductility μ). The maximum ductility, which is defined as the ratio of the maximum displacement to the yield displacement (μmax = δu/δy), was used as the target ductility [22].Figure 2
The iterative procedure of nonlinear time-history analyses.


## 2.1. Selection of Ground Motions

In this paper, the weighted average value of amplitude (βl) between 2 s and 10 s on the spectrum curve of dynamic amplification coefficient [31] was used as the quantitative index to distinguish LPGMs from OGMs. 50 representative LPGMs from Chi-Chi earthquake, WenChuan earthquake, East Japan earthquake, and Tokachi earthquake were selected to conduct the nonlinear time-history analyses. 15 representative OGMs were also selected for the comparison.


## 2.2. Structural Model

RC pier column in the long-span bridge was simplified as the SDOF system, and high-rise RC frame structure was simplified as the MDOF system. The Bouc–Wen–Baber–Noori (BWBN) model [32–35] was adopted to simulate the hysteretic behavior of structural components. It comprehensively considers strength deterioration, stiffness deterioration, and the pinching effect. Hence, the cumulative structural damage caused by LPGMs could be well exhibited.The motion equation of SDOF system is shown as follows:(1)mu¨t+cu˙t+Ft=ft,where m is the mass; c is the linear viscous damping coefficient; u¨tand u˙tare the acceleration and velocity, respectively; F(t) is the restoring force; f(t) is the external excitation.As shown in Figure1, the restoring force of the BWBN model is provided by two parallel springs (linear and nonlinear). Correspondingly, F(t) in equation (2) is the summation of linear restoring force Fe(t) and nonlinear restoring force Fh(t).(2)Ft=Fet+Fht=αkut+1−αkzt,where α is the ratio of stiffness after yielding to stiffness before yielding; k is the initial linear elastic stiffness; u(t) is the displacement; z(t) is the hysteretic displacement, defined by the differential equation with initial condition z(0) = 0, as shown in the following equation:(3)z˙t=hzAu˙−νβu˙zn−1z+γu˙znη,where A is the control parameter of hysteretic amplitude, A = 1; β, γ, and n are the shape parameters of hysteretic curves; υ and η are the strength and stiffness deterioration parameters, respectively.(4)ν=1+δνε,η=1+δηε,where δv and δη are the strength and stiffness deterioration rates, respectively; ε is the cumulative hysteretic energy consumption.(5)ε˙=1−αω02zu˙.Figure 1
Bouc–Wen–Baber–Noori model.In equation (3), h(z) is the function describing the pinching effect, as shown in following equation:(6)hz=1−ζ1exp−zsignu˙−qzmaxζ22,ζ1=ζs1−exp−pε,ζ2=ψ+δψελ+ζ1,zmax=Aνβ+γ1/n,where signu˙ is the symbolic function of velocity. When u˙ > 0, u˙=0, and u˙ < 0, signu˙ takes 1, 0, and -1, respectively. ζs, ψ, δψ, λ, q, and p are the parameters of the pinching effect. Among them, ζs controls the total slippage; ψ controls the range of pinching; δψ is the pinching rate; λ is the interaction coefficient between pinching severity and pinching rate; q controls the starting position of the pinch; p is the pinching slope.The values of all parameters in the BWBN model adopted in this paper were obtained from Reference 35.


## 2.3. Nonlinear Time-History Analyses

Nonlinear time-history analyses on simplified structural models subjected to the selected ground motions were conducted through the iterative procedure shown in Figure2. It should be pointed out that the results of the weak story with maximum drift ratio are required for the MDOF system. The ductility was expected as the control index in developed loading protocols. Therefore, the peak ground acceleration (PGA) of ground motion, as the iterative variable, was increased from 0 with a constant increment of 0.1 gal until the structural components reach the predetermined ductility (target ductility μ). The maximum ductility, which is defined as the ratio of the maximum displacement to the yield displacement (μmax = δu/δy), was used as the target ductility [22].Figure 2
The iterative procedure of nonlinear time-history analyses.


## 3. Rainflow Cycle Counting

Rainflow method with an improved preliminary treatment was adopted in this paper. All cycles could be extracted by adopting rainflow cycle extraction only once. Through the improved rainflow method, the cycle counting procedure is simplified and a higher counting efficiency could be achieved. Figure3 shows the rainflow cycle counting process taking a representative time-history displacement response as an example. Detailed counting procedures are as follows:Figure 3
Rainflow cycle counting process.


### 3.1. Convergence Processing

Convergence processing aimed to ensure the overall convergence of time-history displacement response. Reversing two segments separated by the point with the maximum absolute value on the curves and the convergence processing was accomplished.


### 3.2. Extrema Extraction

Extrema is the point whose displacement value on the time-history displacement response curves is larger (or less) than that of all two adjacent points. During the extraction, the extrema points were reserved, while the non-extrema points were discarded. The start and end points were directly reserved since they only have one adjacent point.


### 3.3. Cycle Counting

After convergence processing and extrema extraction, cycles could be extracted directly through the four-point rainflow cycle counting algorithm. The criterion of the cycle counting algorithm is shown in equations (7) and (8). A cycle could be counted when any of these criteria were satisfied, as shown in Figure 4. Corresponding cycle amplitude could be calculated by equation (9). At the end of the cycle counting, only three points were left forming the last cycle. An example of cycle counting is presented in Figure 5. The rainflow counting results of the representative time-history displacement response in ascending order of cycle amplitude is depicted in Figure 3.(7)Di≤Di+2&Di+1≤Di+3,i=1,2,…,(8)Di≥Di+2&Di+1≥Di+3,i=1,2,…,(9)Dn=Di+1−Di+22,i=1,2,…,where Di is the displacement of the i-th point; D(n) is the displacement amplitude of the n-th cycle.Figure 4
The criteria of cycle counting algorithm.Figure 5
The cycle counting example.


## 3.1. Convergence Processing

Convergence processing aimed to ensure the overall convergence of time-history displacement response. Reversing two segments separated by the point with the maximum absolute value on the curves and the convergence processing was accomplished.


## 3.2. Extrema Extraction

Extrema is the point whose displacement value on the time-history displacement response curves is larger (or less) than that of all two adjacent points. During the extraction, the extrema points were reserved, while the non-extrema points were discarded. The start and end points were directly reserved since they only have one adjacent point.


## 3.3. Cycle Counting

After convergence processing and extrema extraction, cycles could be extracted directly through the four-point rainflow cycle counting algorithm. The criterion of the cycle counting algorithm is shown in equations (7) and (8). A cycle could be counted when any of these criteria were satisfied, as shown in Figure 4. Corresponding cycle amplitude could be calculated by equation (9). At the end of the cycle counting, only three points were left forming the last cycle. An example of cycle counting is presented in Figure 5. The rainflow counting results of the representative time-history displacement response in ascending order of cycle amplitude is depicted in Figure 3.(7)Di≤Di+2&Di+1≤Di+3,i=1,2,…,(8)Di≥Di+2&Di+1≥Di+3,i=1,2,…,(9)Dn=Di+1−Di+22,i=1,2,…,where Di is the displacement of the i-th point; D(n) is the displacement amplitude of the n-th cycle.Figure 4
The criteria of cycle counting algorithm.Figure 5
The cycle counting example.


## 4. Influence Factors of Cycle Number and Cycle Amplitude Distribution

Every inelastic cycle contributes to structural damage. The damage degree depends on the inelastic cycle number and cycle amplitude distribution, which are two important indicators for loading protocols. However, these two indicators could not be directly used for the comparison of loading protocols. Thus, the total inelastic cycle numberNall and the maximum cumulative displacement ductility CDDmax [21] were introduced to quantitatively describe the inelastic cycle number and cycle amplitude distribution.The total inelastic cycle numberNall is the sum of all numbers of inelastic cycles. The maximum cumulative displacement ductility CDDmax is the sum of the cycle amplitudes corresponding to all inelastic cycles. Two loading protocols with the same Nall may have different cycle amplitude distributions, while those with the same CDDmax may have different inelastic cycle numbers and cycle amplitude distributions.Multiple influence factors of cycle number and cycle amplitude distribution are evaluated as follows:


### 4.1. Ground Motion Type

Figures6 and 7 present the total inelastic cycle number Nall and the maximum cumulative displacement ductility CDDmax of SDOF and MDOF systems with different periods under different types of ground motion, where the target ductility is 4; the strength reduction coefficient is 4; the damping ratio is 0.05. All the data are the average values.Figure 6
The influence of ground motion type on SDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)Figure 7
The influence of ground motion type on MDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)It can be observed that the ground motion type had a significant influence on the total inelastic cycle numberNall and the maximum cumulative displacement ductility CDDmax. The Nall and CDDmax under LPGMs were significantly larger than those under OGMs. This could be attributable to the “long-duration” action characteristics of LPGMs.It also can be found thatNall and CDDmax of MDOF systems exhibited different trends from SDOF systems when the period was in the range of 4∼6 s. The average values of Nall and CDDmax under LPGMs were visibly larger than those under OGMs, indicating that MDOF systems with a long period were prone to the influence of LPGMs.


### 4.2. Target Ductility and Structural Period

Figures8 and 9 present the total inelastic cycle number Nall and the maximum cumulative displacement ductility CDDmax of SDOF and MDOF systems with different periods and target ductilities under LPGMs, where the strength reduction coefficient is 4; the damping ratio is 0.05; T is the structural period.Figure 8
The influence of target ductility and structural period on SDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)Figure 9
The influence of target ductility and structural period on MDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)It can be observed that the target ductility and structural period had a remarkable influence on the total inelastic cycle numberNall and the maximum cumulative displacement ductility CDDmax. Nall and CDDmax increased with the increase in target ductility and the decrease in the structural period. Nall and CDDmax of SDOF and MDOF systems with different periods presented consistent trends as the target ductility increased. The reduction of Nall and CDDmax was significant when the period was less than 2 s, while it slowed down when the period was in the range of 2∼6 s. CDDmax increased linearly as the target ductility increased, implying that with the increase in target ductility, the structures suffered more serious damage at the later loading stage with larger cycle amplitude.


### 4.3. Strength Reduction Coefficient and Damping Ratio

Figures10 and 11 present the total inelastic cycle number Nall and the maximum cumulative displacement ductility CDDmax of SDOF and MDOF systems with different strength reduction coefficients and damping ratios under LPGMs, where the target ductility is 4; the structural period is 2 s; ξ is the damping ratio.Figure 10
The influence of strength reduction coefficient and damping ratio on SDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)Figure 11
The influence of strength reduction coefficient and damping ratio on MDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)It can be observed that SDOF and MDOF systems showed extremely distinctive characteristics ofNall and CDDmax varying the damping ratio. That might be attributable to the different calculation methods of structural damping in SDOF and MDOF systems. Linear viscous damping was adopted for the SDOF system, while the Rayleigh damping was adopted for the MDOF system.For SDOF systems,Nall and CDDmax had the similar ascending trends increasing the strength reduction coefficient except when the strength reduction coefficient was 6. While compared with Figure 8, the growth of Nall and CDDmax caused by the increase in the strength reduction coefficient was less than that caused by the increase in target ductility and the decrease in the structural period. For MDOF systems, Nall and CDDmax had stable changing trends varying the strength reduction coefficient with the same damping ratio.


## 4.1. Ground Motion Type

Figures6 and 7 present the total inelastic cycle number Nall and the maximum cumulative displacement ductility CDDmax of SDOF and MDOF systems with different periods under different types of ground motion, where the target ductility is 4; the strength reduction coefficient is 4; the damping ratio is 0.05. All the data are the average values.Figure 6
The influence of ground motion type on SDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)Figure 7
The influence of ground motion type on MDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)It can be observed that the ground motion type had a significant influence on the total inelastic cycle numberNall and the maximum cumulative displacement ductility CDDmax. The Nall and CDDmax under LPGMs were significantly larger than those under OGMs. This could be attributable to the “long-duration” action characteristics of LPGMs.It also can be found thatNall and CDDmax of MDOF systems exhibited different trends from SDOF systems when the period was in the range of 4∼6 s. The average values of Nall and CDDmax under LPGMs were visibly larger than those under OGMs, indicating that MDOF systems with a long period were prone to the influence of LPGMs.


## 4.2. Target Ductility and Structural Period

Figures8 and 9 present the total inelastic cycle number Nall and the maximum cumulative displacement ductility CDDmax of SDOF and MDOF systems with different periods and target ductilities under LPGMs, where the strength reduction coefficient is 4; the damping ratio is 0.05; T is the structural period.Figure 8
The influence of target ductility and structural period on SDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)Figure 9
The influence of target ductility and structural period on MDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)It can be observed that the target ductility and structural period had a remarkable influence on the total inelastic cycle numberNall and the maximum cumulative displacement ductility CDDmax. Nall and CDDmax increased with the increase in target ductility and the decrease in the structural period. Nall and CDDmax of SDOF and MDOF systems with different periods presented consistent trends as the target ductility increased. The reduction of Nall and CDDmax was significant when the period was less than 2 s, while it slowed down when the period was in the range of 2∼6 s. CDDmax increased linearly as the target ductility increased, implying that with the increase in target ductility, the structures suffered more serious damage at the later loading stage with larger cycle amplitude.


## 4.3. Strength Reduction Coefficient and Damping Ratio

Figures10 and 11 present the total inelastic cycle number Nall and the maximum cumulative displacement ductility CDDmax of SDOF and MDOF systems with different strength reduction coefficients and damping ratios under LPGMs, where the target ductility is 4; the structural period is 2 s; ξ is the damping ratio.Figure 10
The influence of strength reduction coefficient and damping ratio on SDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)Figure 11
The influence of strength reduction coefficient and damping ratio on MDOF system. (a) The total inelastic cycle number. (b) The maximum cumulative displacement ductility.
(a)(b)It can be observed that SDOF and MDOF systems showed extremely distinctive characteristics ofNall and CDDmax varying the damping ratio. That might be attributable to the different calculation methods of structural damping in SDOF and MDOF systems. Linear viscous damping was adopted for the SDOF system, while the Rayleigh damping was adopted for the MDOF system.For SDOF systems,Nall and CDDmax had the similar ascending trends increasing the strength reduction coefficient except when the strength reduction coefficient was 6. While compared with Figure 8, the growth of Nall and CDDmax caused by the increase in the strength reduction coefficient was less than that caused by the increase in target ductility and the decrease in the structural period. For MDOF systems, Nall and CDDmax had stable changing trends varying the strength reduction coefficient with the same damping ratio.


## 5. Prediction of Cycle Number and Cycle Amplitude

### 5.1. Double Exponential Prediction Model

From the above analyses, it can be found that the structural period and target ductility have the most significant influence on the cycle number and cycle amplitude distribution under a certain type of ground motion. Therefore, regression analyses on cycle numbers and cycle amplitudes of SDOF and MDOF systems with different periods and target ductilities under LPGMs were conducted. The double exponential function (equation (10)) was adopted to predict the cycle amplitude on the basis of the simple exponential function adopted in the existing researches [22, 30]. The boundary conditions are D(1) = 1 and D(Nall) = μ.(10)DN=1eα−eβμ−eβeαN−1/Nall−1−μ−eαeβN−1/Nall−1,where D(N) is the cycle amplitude corresponding to the N-th cycle; N is the cycle number, which equals 1, 2, 3, … Nall in turn; α and β are the distribution parameters of the cycle amplitudes, which reflect the trend of cycle amplitude D(N) with the change of cycle number N; μ is the target ductility.


#### 5.1.1. The Total Inelastic Cycle Number:Nall

Quartic polynomial and quintic polynomial were utilized to fit the effect of target ductility (T) and structural period (μ) on the total inelastic cycle number Nall, respectively. For convenience, the fitting formula for Nall was exhibited in matrix form (equation (11)). All coefficients in quartic polynomial and quintic polynomial were integrated into a coefficient matrix C. CS shown in equation (12) and CM shown in equation (13) are the coefficient matrices obtained through the fitting process of Nall statistical results for the SDOF system and MDOF system, respectively. They can only be used in the cases studied in this paper. Corresponding variables related to T and μ are shown in equations (14) and (15), respectively.(11)NallT,μ=C.×LT×Rμ,(12)CS=−16.62−15.7534.16−6.3520.382943.47−32.81−10.941.914−0.1053−9.78324.550.1291−0.029590−5.83−4.5740.01212001.9820.2954000−0.1620000,(13)CM=−53.160026.490012.7200−2.12300.103552.5400−25.6100−5.80700.6520−0.0152−24.230014.84000.5777−0.055903.6100−2.85800.0078000.01020.1725000−0.02520000,(14)LT=1TT2T3T4T5T,(15)Rμ=1μμ2μ3μ4,where C is the coefficient matrix, in which CS is applied for SDOF systems and CM is applied for MDOF systems; LT is the column vector associated with the structural period T; Rμ is the row vector associated with the target ductility μ; symbol .× is the dot product operation of the matrix.


#### 5.1.2. The Distribution Parameters of Cycle Amplitude:α and β

The cumulative distribution function (CDF) of cycle amplitudes [21] was introduced to describe the trend of cycle amplitude distribution with the change of cycle number. It can be derived by equation (16). Figure 12 presents the CDF of structures with different periods and target ductilities under LPGMs.(16)CDF=∑NiNall,where ΣNi is the number sum of cycles whose amplitude is less than or equal to the amplitude corresponding to the i-th cycle.Figure 12
TheCDF of structures with different periods and target ductilities. (a) μ = 2, with different T. (b) μ = 6, with different T. (c) T = 0.5 s with different μ.
(a)(b)(c)It can be observed in Figures12(a) and 12(b) that, with the same target ductility, CDF of structures with different periods were almost the same. They exhibited a greater agreement with each other as the target ductility increased. However, CDF of structures with the same period showed a high dependence on target ductility, as shown in Figure 12(c). For the same value of CDF, the larger the target ductility, the larger the corresponding cycle amplitude was. Meanwhile, for structures with the same period, as the target ductility increased, the growth rate of cycle amplitude increased, while the difference of cycle amplitudes under two adjacent target ductility levels decreased.Accordingly, for structures with the same target ductility, the double exponential prediction model with the sameα and β should be adopted. The values of α and β should be the optimal fitting results of equation (10) according to the statistical cycle numbers and cycle amplitudes. The calculated optimal fitting parameters α and β for SDOF and MDOF systems with different target ductilities are listed in Table 1.Table 1
Calculated distribution parameters of cycle amplitudea and ß.
Structural systemParameterTarget ductilityμ23456SDOF systemα7.49512.1615.7820.8825.96β0.41990.68470.87651.0681.207MDOF systemα8.6859.86712.9416.3819.09β0.47590.70140.93531.1461.314Note. The values in the table can only be used in the cases studied in this paper.Moreover, there was a small difference between cycle amplitudes of structures with two high adjacent target ductility levels. The growth rate of the cycle amplitude of structures with higher-level target ductility was larger. Therefore, when the value of ductility is a non-integer, the cycle number and cycle amplitude could be predicted by the parameterα and β under the closest integer high-level target ductility, as a conservative estimation value.


### 5.2. Prediction of Cycle Number and Cycle Amplitude

The cycle number and cycle amplitude of loading protocols considering the action characteristics of LPGMs could be directly predicted by the proposed double exponential prediction model, instead of conducting the nonlinear time-history analyses. The specific prediction steps are shown as follows:Firstly, calculate the total inelastic cycle numberNall by equation (11) according to the structural period and target ductility. Then, obtain the values of distribution parameters of cycle amplitude α and β from Table 1 on the basis of the predetermined target ductility. Finally, the corresponding amplitudes of all cycles could be calculated by inputting the total inelastic cycle number Nall and distribution parameters α, β to equation (10).


### 5.3. Validation

In order to validate the accuracy and reliability of the proposed prediction model, three RC bridge pier columns (representing the SDOF system) and three RC frame structures (representing the MDOF system) were designed and analyzed. The top mass of all three pier columns is 500t, the height is 5 m, 10 m, and 15 m, respectively. The uniaxial compressive strength of the concrete is 40 MPa. The elastic modulus of reinforced concrete is 31.62 GPa. The structural period is 1.14 s, 2.26 s, and 2.77 s, respectively. The equivalent stiffness of the columns is 15188.7 kN/m, 3864.7 kN/m, and 2572.6 kN/m, respectively. Three RC frame structures with 4, 8, and 12 stories [4] were designed according to the Chinese seismic design code [36]. The seismic precautionary intensity is 8-degree. The site condition is site-class 3. The structural fundamental period is 0.54 s, 1.17 s, and 2.18 s, respectively. For all systems, the damping ratio is 0.05 and the target ductility is 4.The nonlinear time-history analyses and rainflow cycle counting were carried out first, obtaining the counting values of cycle numbers and cycle amplitudes. Then, the predicted values of cycle numbers and cycle amplitudes were obtained by the proposed prediction model. Figures13–15 present the comparison of the counting value and predicted value in cycle number, cycle amplitude, and cumulative displacement ductility.Figure 13
Validation of the cycle number and cycle amplitude for SDOF system. (a)μ = 4; T = 1.14 s. (b) μ = 4; T = 2.26 s. (c) μ = 4; T = 2.77 s.
(a)(b)(c)Figure 14
Validation of the cycle number and cycle amplitude for MDOF system. (a)μ = 4; T = 0.54 s. (b) μ = 4; T = 1.17 s. (c) μ = 4; T = 2.18 s.
(a)(b)(c)Figure 15
Validation of the cumulative displacement ductility. (a) SDOF system. (b) MDOF system.
(a)(b)It can be observed that the predicted values of cycle numbers were in good agreement with the counting values. The predicted values of cycle amplitude basically coincided with counting values in the earlier stage, while those were slightly different from counting values in the later stage. This difference had a relatively small impact on the whole prediction model due to the large cycle amplitude in the later stage.It also can be seen that the predicted values and counting values of cumulative displacement ductility were almost the same. This implied that, from the perspective of cumulative damage, the difference of cycle amplitude values in the later stage had almost no influence. It further validated the accuracy and reliability of the proposed double exponential prediction model.


## 5.1. Double Exponential Prediction Model

From the above analyses, it can be found that the structural period and target ductility have the most significant influence on the cycle number and cycle amplitude distribution under a certain type of ground motion. Therefore, regression analyses on cycle numbers and cycle amplitudes of SDOF and MDOF systems with different periods and target ductilities under LPGMs were conducted. The double exponential function (equation (10)) was adopted to predict the cycle amplitude on the basis of the simple exponential function adopted in the existing researches [22, 30]. The boundary conditions are D(1) = 1 and D(Nall) = μ.(10)DN=1eα−eβμ−eβeαN−1/Nall−1−μ−eαeβN−1/Nall−1,where D(N) is the cycle amplitude corresponding to the N-th cycle; N is the cycle number, which equals 1, 2, 3, … Nall in turn; α and β are the distribution parameters of the cycle amplitudes, which reflect the trend of cycle amplitude D(N) with the change of cycle number N; μ is the target ductility.


### 5.1.1. The Total Inelastic Cycle Number:Nall

Quartic polynomial and quintic polynomial were utilized to fit the effect of target ductility (T) and structural period (μ) on the total inelastic cycle number Nall, respectively. For convenience, the fitting formula for Nall was exhibited in matrix form (equation (11)). All coefficients in quartic polynomial and quintic polynomial were integrated into a coefficient matrix C. CS shown in equation (12) and CM shown in equation (13) are the coefficient matrices obtained through the fitting process of Nall statistical results for the SDOF system and MDOF system, respectively. They can only be used in the cases studied in this paper. Corresponding variables related to T and μ are shown in equations (14) and (15), respectively.(11)NallT,μ=C.×LT×Rμ,(12)CS=−16.62−15.7534.16−6.3520.382943.47−32.81−10.941.914−0.1053−9.78324.550.1291−0.029590−5.83−4.5740.01212001.9820.2954000−0.1620000,(13)CM=−53.160026.490012.7200−2.12300.103552.5400−25.6100−5.80700.6520−0.0152−24.230014.84000.5777−0.055903.6100−2.85800.0078000.01020.1725000−0.02520000,(14)LT=1TT2T3T4T5T,(15)Rμ=1μμ2μ3μ4,where C is the coefficient matrix, in which CS is applied for SDOF systems and CM is applied for MDOF systems; LT is the column vector associated with the structural period T; Rμ is the row vector associated with the target ductility μ; symbol .× is the dot product operation of the matrix.


### 5.1.2. The Distribution Parameters of Cycle Amplitude:α and β

The cumulative distribution function (CDF) of cycle amplitudes [21] was introduced to describe the trend of cycle amplitude distribution with the change of cycle number. It can be derived by equation (16). Figure 12 presents the CDF of structures with different periods and target ductilities under LPGMs.(16)CDF=∑NiNall,where ΣNi is the number sum of cycles whose amplitude is less than or equal to the amplitude corresponding to the i-th cycle.Figure 12
TheCDF of structures with different periods and target ductilities. (a) μ = 2, with different T. (b) μ = 6, with different T. (c) T = 0.5 s with different μ.
(a)(b)(c)It can be observed in Figures12(a) and 12(b) that, with the same target ductility, CDF of structures with different periods were almost the same. They exhibited a greater agreement with each other as the target ductility increased. However, CDF of structures with the same period showed a high dependence on target ductility, as shown in Figure 12(c). For the same value of CDF, the larger the target ductility, the larger the corresponding cycle amplitude was. Meanwhile, for structures with the same period, as the target ductility increased, the growth rate of cycle amplitude increased, while the difference of cycle amplitudes under two adjacent target ductility levels decreased.Accordingly, for structures with the same target ductility, the double exponential prediction model with the sameα and β should be adopted. The values of α and β should be the optimal fitting results of equation (10) according to the statistical cycle numbers and cycle amplitudes. The calculated optimal fitting parameters α and β for SDOF and MDOF systems with different target ductilities are listed in Table 1.Table 1
Calculated distribution parameters of cycle amplitudea and ß.
Structural systemParameterTarget ductilityμ23456SDOF systemα7.49512.1615.7820.8825.96β0.41990.68470.87651.0681.207MDOF systemα8.6859.86712.9416.3819.09β0.47590.70140.93531.1461.314Note. The values in the table can only be used in the cases studied in this paper.Moreover, there was a small difference between cycle amplitudes of structures with two high adjacent target ductility levels. The growth rate of the cycle amplitude of structures with higher-level target ductility was larger. Therefore, when the value of ductility is a non-integer, the cycle number and cycle amplitude could be predicted by the parameterα and β under the closest integer high-level target ductility, as a conservative estimation value.


## 5.1.1. The Total Inelastic Cycle Number:Nall

Quartic polynomial and quintic polynomial were utilized to fit the effect of target ductility (T) and structural period (μ) on the total inelastic cycle number Nall, respectively. For convenience, the fitting formula for Nall was exhibited in matrix form (equation (11)). All coefficients in quartic polynomial and quintic polynomial were integrated into a coefficient matrix C. CS shown in equation (12) and CM shown in equation (13) are the coefficient matrices obtained through the fitting process of Nall statistical results for the SDOF system and MDOF system, respectively. They can only be used in the cases studied in this paper. Corresponding variables related to T and μ are shown in equations (14) and (15), respectively.(11)NallT,μ=C.×LT×Rμ,(12)CS=−16.62−15.7534.16−6.3520.382943.47−32.81−10.941.914−0.1053−9.78324.550.1291−0.029590−5.83−4.5740.01212001.9820.2954000−0.1620000,(13)CM=−53.160026.490012.7200−2.12300.103552.5400−25.6100−5.80700.6520−0.0152−24.230014.84000.5777−0.055903.6100−2.85800.0078000.01020.1725000−0.02520000,(14)LT=1TT2T3T4T5T,(15)Rμ=1μμ2μ3μ4,where C is the coefficient matrix, in which CS is applied for SDOF systems and CM is applied for MDOF systems; LT is the column vector associated with the structural period T; Rμ is the row vector associated with the target ductility μ; symbol .× is the dot product operation of the matrix.


## 5.1.2. The Distribution Parameters of Cycle Amplitude:α and β

The cumulative distribution function (CDF) of cycle amplitudes [21] was introduced to describe the trend of cycle amplitude distribution with the change of cycle number. It can be derived by equation (16). Figure 12 presents the CDF of structures with different periods and target ductilities under LPGMs.(16)CDF=∑NiNall,where ΣNi is the number sum of cycles whose amplitude is less than or equal to the amplitude corresponding to the i-th cycle.Figure 12
TheCDF of structures with different periods and target ductilities. (a) μ = 2, with different T. (b) μ = 6, with different T. (c) T = 0.5 s with different μ.
(a)(b)(c)It can be observed in Figures12(a) and 12(b) that, with the same target ductility, CDF of structures with different periods were almost the same. They exhibited a greater agreement with each other as the target ductility increased. However, CDF of structures with the same period showed a high dependence on target ductility, as shown in Figure 12(c). For the same value of CDF, the larger the target ductility, the larger the corresponding cycle amplitude was. Meanwhile, for structures with the same period, as the target ductility increased, the growth rate of cycle amplitude increased, while the difference of cycle amplitudes under two adjacent target ductility levels decreased.Accordingly, for structures with the same target ductility, the double exponential prediction model with the sameα and β should be adopted. The values of α and β should be the optimal fitting results of equation (10) according to the statistical cycle numbers and cycle amplitudes. The calculated optimal fitting parameters α and β for SDOF and MDOF systems with different target ductilities are listed in Table 1.Table 1
Calculated distribution parameters of cycle amplitudea and ß.
Structural systemParameterTarget ductilityμ23456SDOF systemα7.49512.1615.7820.8825.96β0.41990.68470.87651.0681.207MDOF systemα8.6859.86712.9416.3819.09β0.47590.70140.93531.1461.314Note. The values in the table can only be used in the cases studied in this paper.Moreover, there was a small difference between cycle amplitudes of structures with two high adjacent target ductility levels. The growth rate of the cycle amplitude of structures with higher-level target ductility was larger. Therefore, when the value of ductility is a non-integer, the cycle number and cycle amplitude could be predicted by the parameterα and β under the closest integer high-level target ductility, as a conservative estimation value.


## 5.2. Prediction of Cycle Number and Cycle Amplitude

The cycle number and cycle amplitude of loading protocols considering the action characteristics of LPGMs could be directly predicted by the proposed double exponential prediction model, instead of conducting the nonlinear time-history analyses. The specific prediction steps are shown as follows:Firstly, calculate the total inelastic cycle numberNall by equation (11) according to the structural period and target ductility. Then, obtain the values of distribution parameters of cycle amplitude α and β from Table 1 on the basis of the predetermined target ductility. Finally, the corresponding amplitudes of all cycles could be calculated by inputting the total inelastic cycle number Nall and distribution parameters α, β to equation (10).


## 5.3. Validation

In order to validate the accuracy and reliability of the proposed prediction model, three RC bridge pier columns (representing the SDOF system) and three RC frame structures (representing the MDOF system) were designed and analyzed. The top mass of all three pier columns is 500t, the height is 5 m, 10 m, and 15 m, respectively. The uniaxial compressive strength of the concrete is 40 MPa. The elastic modulus of reinforced concrete is 31.62 GPa. The structural period is 1.14 s, 2.26 s, and 2.77 s, respectively. The equivalent stiffness of the columns is 15188.7 kN/m, 3864.7 kN/m, and 2572.6 kN/m, respectively. Three RC frame structures with 4, 8, and 12 stories [4] were designed according to the Chinese seismic design code [36]. The seismic precautionary intensity is 8-degree. The site condition is site-class 3. The structural fundamental period is 0.54 s, 1.17 s, and 2.18 s, respectively. For all systems, the damping ratio is 0.05 and the target ductility is 4.The nonlinear time-history analyses and rainflow cycle counting were carried out first, obtaining the counting values of cycle numbers and cycle amplitudes. Then, the predicted values of cycle numbers and cycle amplitudes were obtained by the proposed prediction model. Figures13–15 present the comparison of the counting value and predicted value in cycle number, cycle amplitude, and cumulative displacement ductility.Figure 13
Validation of the cycle number and cycle amplitude for SDOF system. (a)μ = 4; T = 1.14 s. (b) μ = 4; T = 2.26 s. (c) μ = 4; T = 2.77 s.
(a)(b)(c)Figure 14
Validation of the cycle number and cycle amplitude for MDOF system. (a)μ = 4; T = 0.54 s. (b) μ = 4; T = 1.17 s. (c) μ = 4; T = 2.18 s.
(a)(b)(c)Figure 15
Validation of the cumulative displacement ductility. (a) SDOF system. (b) MDOF system.
(a)(b)It can be observed that the predicted values of cycle numbers were in good agreement with the counting values. The predicted values of cycle amplitude basically coincided with counting values in the earlier stage, while those were slightly different from counting values in the later stage. This difference had a relatively small impact on the whole prediction model due to the large cycle amplitude in the later stage.It also can be seen that the predicted values and counting values of cumulative displacement ductility were almost the same. This implied that, from the perspective of cumulative damage, the difference of cycle amplitude values in the later stage had almost no influence. It further validated the accuracy and reliability of the proposed double exponential prediction model.


## 6. Developed Quasi-Static Loading Protocols

Quasi-static loading protocols considering the action characteristics of LPGMs were developed based on the proposed double exponential prediction model. The protocols consist of two stages: force-control loading stage and displacement-control loading stage. The RC pier column (μ = 4, T = 2.26 s) and the column on weak story of the RC frame structure (μ = 4, T = 2.18 s) were taken as examples. Developed quasi-static loading protocols are shown in Figure 16.Figure 16
Developed quasi-static loading protocols for SDOF and MDOF systems under LPGMs. (a) SDOF system (T = 2.26 s). (b) MDOF system (T = 2.18 s).
(a)(b)


### 6.1. The First Loading Stage (Force-Control Loading Stage)

When the cycle amplitude was less than the yield displacementδy, that is, before the yielding of structural components, there was almost no damage. Hence, loading protocols in different cases could adopt the same loading method in this stage. Force-control loading was employed in this paper. There are two cycles, as shown in Figure 16. The first cycle is applied with the amplitude of 0.75 Fy (Fy is the theoretical yield load of structural components). The second cycle is applied with the amplitude of Fy. The yielding of the column depends on the tension strain of longitudinal rebars on the control section. The lateral displacement of the horizontal actuator could be regarded as the yield displacement (δy) when the tension strain of longitudinal rebars on the control section reached the yield strain.


### 6.2. The Second Loading Stage (Displacement-Control Loading Stage)

This stage aimed to reflect the influence of LPGMs action characteristics on structural components, which is the critical part of the developed loading protocols. Cycle numbers and cycle amplitudes in this stage were predicted by the proposed prediction model, as shown in Figures13(b) and 14(c).In order to regularize the loading protocols, it is necessary to grade the predicted cycle amplitude and recount corresponding cycle numbers. The target displacement amplitude, i.e., the target ductility (μ) multiplied the yield displacement (δy), was divided to 10 grades with an approximately equal interval. The cycle numbers corresponding to the graded cycle amplitudes were recounted based on the predicted results, as shown in Table 2.Table 2
The graded cycle amplitudes and corresponding cycle numbers.
Grade①②③④⑤⑥⑦⑧⑨⑩Graded cycle amplitude (×δy)11.31.61.92.22.52.83.13.44Cycle numberSDOF system5864311011MDOF system71297422121It can be found in Table2 that, in the second loading stage, RC columns may be at the state of long-term damage accumulation instead of failure. In this case, the conventional quasi-static loading protocols with a monotone increase of cycle amplitude should be applied after the second loading stage until the failure occurred. Thereby, damage accumulation and residual bearing capacity of structural components considering the action characteristics of LPGMs could be comprehensively analyzed through the quasi-static tests.The developed quasi-static protocol (Figure16(b)) was applied on an RC column, and the structural response was compared with that under the time-history response subjected to an LPGM record, in order to verify the accuracy and efficiency of the developed quasi-static loading protocol. Figure 17 shows the time-history displacement response of the weak story in the 12-story RC frame structure (T = 2.18 s) subjected to an LPGM record (AIC003EW from East Japan Earthquake). Comparison of the structural response is presented in Figure 18.Figure 17
Time-history displacement response subjected to AIC003EW.Figure 18
Comparison of the structural response.It can be found that the structural response under the developed protocol showed a good agreement with that under the seismic time-history response. This indicated that the proposed quasi-static protocols are reasonable to consider and reflect the seismic response characteristics of structural systems subjected to LPGMs.In fact, the target ductility of the specific structural component is difficult to directly determine in practical application. For structural components in the MDOF system, the inter-story drift ratio could be used as the control index instead of ductility. That is, the target ductility could be determined by the target drift ratio, as shown in the following equation:(17)μ=θhδy,where θ is the target drift ratio; h is the storey height; δy is the yield displacement, which can be obtained through conventional quasi-static tests.In this case, developed quasi-static loading protocols could be applied to experimental investigations of structural components in MDOF systems.


## 6.1. The First Loading Stage (Force-Control Loading Stage)

When the cycle amplitude was less than the yield displacementδy, that is, before the yielding of structural components, there was almost no damage. Hence, loading protocols in different cases could adopt the same loading method in this stage. Force-control loading was employed in this paper. There are two cycles, as shown in Figure 16. The first cycle is applied with the amplitude of 0.75 Fy (Fy is the theoretical yield load of structural components). The second cycle is applied with the amplitude of Fy. The yielding of the column depends on the tension strain of longitudinal rebars on the control section. The lateral displacement of the horizontal actuator could be regarded as the yield displacement (δy) when the tension strain of longitudinal rebars on the control section reached the yield strain.


## 6.2. The Second Loading Stage (Displacement-Control Loading Stage)

This stage aimed to reflect the influence of LPGMs action characteristics on structural components, which is the critical part of the developed loading protocols. Cycle numbers and cycle amplitudes in this stage were predicted by the proposed prediction model, as shown in Figures13(b) and 14(c).In order to regularize the loading protocols, it is necessary to grade the predicted cycle amplitude and recount corresponding cycle numbers. The target displacement amplitude, i.e., the target ductility (μ) multiplied the yield displacement (δy), was divided to 10 grades with an approximately equal interval. The cycle numbers corresponding to the graded cycle amplitudes were recounted based on the predicted results, as shown in Table 2.Table 2
The graded cycle amplitudes and corresponding cycle numbers.
Grade①②③④⑤⑥⑦⑧⑨⑩Graded cycle amplitude (×δy)11.31.61.92.22.52.83.13.44Cycle numberSDOF system5864311011MDOF system71297422121It can be found in Table2 that, in the second loading stage, RC columns may be at the state of long-term damage accumulation instead of failure. In this case, the conventional quasi-static loading protocols with a monotone increase of cycle amplitude should be applied after the second loading stage until the failure occurred. Thereby, damage accumulation and residual bearing capacity of structural components considering the action characteristics of LPGMs could be comprehensively analyzed through the quasi-static tests.The developed quasi-static protocol (Figure16(b)) was applied on an RC column, and the structural response was compared with that under the time-history response subjected to an LPGM record, in order to verify the accuracy and efficiency of the developed quasi-static loading protocol. Figure 17 shows the time-history displacement response of the weak story in the 12-story RC frame structure (T = 2.18 s) subjected to an LPGM record (AIC003EW from East Japan Earthquake). Comparison of the structural response is presented in Figure 18.Figure 17
Time-history displacement response subjected to AIC003EW.Figure 18
Comparison of the structural response.It can be found that the structural response under the developed protocol showed a good agreement with that under the seismic time-history response. This indicated that the proposed quasi-static protocols are reasonable to consider and reflect the seismic response characteristics of structural systems subjected to LPGMs.In fact, the target ductility of the specific structural component is difficult to directly determine in practical application. For structural components in the MDOF system, the inter-story drift ratio could be used as the control index instead of ductility. That is, the target ductility could be determined by the target drift ratio, as shown in the following equation:(17)μ=θhδy,where θ is the target drift ratio; h is the storey height; δy is the yield displacement, which can be obtained through conventional quasi-static tests.In this case, developed quasi-static loading protocols could be applied to experimental investigations of structural components in MDOF systems.


## 7. Conclusions

Quasi-static loading protocols considering the action characteristics of LPGMs were developed in this paper. They were developed by statistical and regression analysis based on the results of nonlinear time-history analyses considering LPGMs imposed on different structural systems with various periods and target ductilities. The structural response under developed protocols was in accordance with that under the time-history response subjected to LPGMs. Proposed loading protocols can be directly applied to the experimental study on the seismic performance of RC pier columns and columns on the weak story of the high-rise RC frame structure subjected to LPGMs. It provides a new method investigating the influence of LPGMs on the seismic performance of structural components.



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*Source: 8863335-2021-02-08.xml*