An Iterative PSD-Based Procedure for the Gaussian Stochastic Earthquake Model with Combined Intensity and Frequency Nonstationarities: Its Application into Precast Concrete Structures

Abstract

Earthquakes cause severe damage to human beings and financial development, and they are commonly associated with a lot of uncertainties and stochastic factors regarding their frequency, intensity and duration. Thus, how to accurately select an earthquake record and determine an earthquake’s influence on structures are important questions that deserve further investigation. In this paper, the author developed an iterative power spectral density (PSD)-based procedure for the Gaussian stochastic earthquake model with combined intensity and frequency nonstationarities. In addition, they applied this procedure to five precast concrete structures for dynamic analysis and verification. The research proved the effectiveness of the iterative procedure for matching the target response spectra and for generating the required seismic records. The application examples verified the accuracy of the seismic design for the precast concrete structures and indicated the reliable dynamic demands of the precast concrete structures under the stochastic excitation of nonstationary earthquakes. In general, the research provided a meaningful reference for further stochastic earthquake selections, and it could play an effective role in further assessments of precast structures.

1. Introduction

An earthquake is a sudden and rapid shaking of the ground caused by crustal movement and plate compression. Earthquakes can cause fires, tsunamis, landslides or avalanches and they can lead to severe damage to human societies. During the past few decades, the influence of earthquakes has been gradually recognized by researchers, and lots of solutions (e.g., retrofitting, upgrading or strengthening strategies) have been proposed by researchers all over the world [1,2,3,4,5,6,7].

Commonly, an earthquake is associated with a lot of uncertainties in terms of its source, attenuation and site amplification, and an earthquake occurance generally includes a series of stochastic factors in terms of its frequency, intensity and duration [8,9,10,11,12]. Taking the intensity as an example, the peak ground acceleration of an unknown earthquake is commonly stochastic in a probabilistic way, which means that it may exceed the fortification earthquake level and may lead to severe postdisaster damage [13,14,15,16,17]. Thus, how to accurately select an earthquake record and determine an earthquake’s influence on structures are important questions that deserve further investigation [18,19,20,21,22]. At this stage, a commonly adopted method is to select earthquake records from the existing database and match them with the target response spectra in the corresponding seismic regulations. It has been noted that this approach can be effective in choosing the required records to a certain extent, but, commonly, the generated average response spectra are different to the target response spectra with respect to their statistical details [23,24,25,26,27]. Meanwhile, as earthquake theory develops, researchers hope to focus more on the stochastic parameters of earthquakes, but the traditional selection strategy from the database shows limitations and cannot reflect so many parameters. Thus, the stochastic earthquake model was developed by researchers to characterize these stochastic parameters and to capture specific information for structural dynamic analysis [28,29,30,31,32].

Shinozuka and Deodatis [33] gave a state-of-the-art review of the stochastic process models for earthquake simulations, and the research proved the effectiveness and importance of the stochastic earthquake model, which laid a critical foundation for further research. Loh and Yeh [34] carried out spatial variation research and performed the stochastic modelling of seismic differential ground motions. It was observed that the corner frequency and phase velocity were the controlling parameters in estimating multiple differential grep deformations. Chen and Ahmadi [35] analyzed the seismic response of secondary substructures in base-isolated systems via the stochastic earthquake model, and the data for the Mexico City earthquake indicated the sensitivity of the base-isolated structures in terms of long periods of excitation. Grigoriu [36] proposed two models (i.e., $X_n\left(t\right)$ and $Y_n\left(t\right)$) to generate the stochastic band-limited samples of Gaussian stationary processes in light of the spectral representation, as well as the harmonic superposition, which gave an important basis for further stochastic earthquake simulation. Rietbrock et al. [37] proposed a stochastic earthquake model for the UK based on peak ground acceleration, peak ground velocity and pseudospectral acceleration, and the model indicated an ideal accuracy and predicting effect. Yamamoto and Baker [38] adopted wavelet packets to simulate the stochastic earthquake model, and the proposed stochastic earthquake model showed great consistency with existing established models for earthquake prediction in terms of the variabilities and means. Huang [39] proposed the orthogonal decomposition algorithm in the simulation of the multivariate nonstationary stochastic process, and the fast Fourier transform operation was introduced during the process, which distinctively enhanced the generating efficiency of the stochastic process. Bhattacharyya et al. [40] proposed a novel time-frequency representation approach based on the enhanced wavelet transform, and Fourier-to-Bessel expansions were further adopted to generate stochastic nonstationary signals, which provided some references for the efficient modelling of stochastic earthquakes. Cao et al. [41] performed the probabilistic seismic fragility comparison of different approaches under the stochastic nonstationary earthquake, and the research proved the importance of the stochastic earthquake model in the probabilistic performance assessment and fragility method selection. Feng et al. [42] proposed a nonparametric probabilistic density evolution method (PDEM)-based approach for seismic fragility evaluation of frame structures under the stochastic earthquake model, and the results showed that the stochastic earthquake model was directly related to the accuracy of the fragility calculation.

On the other hand, precast concrete structures are a type of construction method that involves the use of prefabricated concrete components, such as beams, columns, slabs, walls and foundations, that are manufactured off-site in a workshop setting and then transported to the construction site for assembly [43,44,45,46]. Utilizing precast concrete offers many advantages over traditional construction methods in terms of speed, efficiency, cost and quality. The prefabricated components are manufactured off-site in a controlled environment, meaning they are produced in a much shorter timeframe than traditional construction methods. This also eliminates the need for weather delays and other issues that can slow down the construction process [47,48,49,50]. The cost of the materials is often much lower than traditional construction methods, and there is a wide range of prefabricated components available for use. Additionally, as the components are produced off-site and can be delivered to the project site, the labor costs associated with traditional construction methods are eliminated. The quality of precast concrete structures is also higher than that of traditional construction methods [51,52,53,54]. The components are manufactured in a controlled environment, meaning they can be designed to exact specifications and with high-quality materials, resulting in a structure that is much stronger and more durable than one constructed using traditional methods. The environmental impact of precast concrete structures is also less than that of traditional construction methods. The materials used in precast concrete components are generally easier to recycle than traditional construction materials, and, as the components are manufactured off-site, there is less waste produced on-site. Overall, precast concrete structures offer many advantages over traditional construction methods in terms of speed, cost, quality and environmental impact. By utilizing prefabricated components, the construction process can be completed much faster, at a lower cost and environmental impact and, most importantly, with a higher construction quality [55,56,57,58,59,60].

Elliott [61] summarized the superiorities of earthquake-resistant precast concrete structures in terms of the precast concept, materials in the precast structures, precast frame systems, precast floor systems, precast beam systems, precast column systems, precast shear wall systems, precast joint systems, etc., which provided an overall literature review of precast concrete structures for further research. Kurama et al. [62] gave a state-of-the-art review of earthquake-resistant precast concrete structures and divided the structural forms into moment frames, structural walls, floor diaphragms and bridges, which contributed greatly to the literature review and laid an important foundation for further analysis. Polat [63] conclusively defined the parameters that affected the application of precast concrete structures in the United States and gave the development tendency of the precast-concrete industry through the past few decades, which provided some important references for future explorations of precast concrete structures. Koskisto and Ellingwood [64] gave an optimization strategy for precast concrete structures in view of reliability theory, and they formulated the design limitations of flexural failure probability, shear failure probability, cracking probability and excessive deflection probability. Ozden et al. [65] analyzed the seismic performance of precast concrete structures affected by the Van earthquakes in Turkey, and reported the importance of proper design, as well as seismic details of precast concrete joints, in the construction stages for structures in high-seismicity areas. Belleri et al. [66] assessed the seismic damage of a three-story precast concrete structure through large-scale experiments and structural identifications, which indicated the effectiveness of the structural form in seismic repairing regions. Kataoka et al. [67] performed nonlinear numerical analysis of a precast concrete slab–beam–column joint for its seismic behaviors, and they gave suggestions for the influence of critical parameters in the seismic design. Lacerda et al. [68] experimentally investigated the influence of vertical groutings in precast concrete structures, and the result implied an increase in the rotational flexural stiffness, as well as the flexural strength capacity, of the precast concrete structures via the vertical filling. Cao et al. [69] proposed innovative external precast concrete structural forms for seismic retrofitting without any inner disturbance, and their experiments validated the effectiveness of the novel precast structures in practical application. Lago et al. [70] investigated the effectiveness of diaphragms in precast concrete structures with cladding panels, and they proposed a novel fastening system to increase the total energy dissipation capacity of the structural system. Feng et al. [71] performed a comparative study of numerical approaches for precast concrete structures based on damage mechanics, and three practical strategies were proposed for efficient performance prediction in static cyclic modelling. Ye et al. [72] proposed a novel hybrid beam–column joint for precast concrete structures and carried out experimental and numerical studies of its seismic performance, which proved the effectiveness of the precast form. Xu et al. [73] used spectrum-compatible stochastic earthquakes with near-field nonstationarities to conduct a comparative study of precast steel-reinforced concrete substructures for seismic retrofitting, and the precast substructures potentially indicated a superior upgrading capacity and effectively alleviated the premature damage of existing buildings.

From the above literature review, it can be found that the stochastic earthquake model is developing rapidly and needs a better record-selection strategy and a more realistic uncertainty-quantification method in terms of seismic excitation. In the meantime, precast concrete structures have indicated a lot of superiorities, and they are promising for practical engineering in high-seismicity regions [74,75,76,77,78,79]. At this stage, the research of the nonstationary stochastic dynamic performance of precast concrete structures is scarce and deserves further exploration. It is meaningful research work and possesses value in terms of practical applications, which require the consideration of more stochastic input factors and reflect more realistic seismic capacities in the life cycle period. Thus, in this paper, the author has developed an iterative power spectral density (PSD)-based procedure for the Gaussian stochastic earthquake model with combined intensity and frequency nonstationarities, and they have applied this model to five precast concrete structures for dynamic analysis and verification. During the analysis, four iterative calculations were performed for the stochastically generated nonstationary earthquakes, and it was observed that the deviations with the target spectrum distinctively dropped along with the iterative procedure. Meanwhile, two system-level indexes were adopted for the assessment of the precast structures (i.e., the maximum and residual interstory drift ratio) to obtain a more comprehensive evaluation. In general, the research proved the effectiveness of the iterative procedure for matching the target response spectra and for generating the required seismic records. The application examples verified the accuracy of the seismic design for the precast concrete structures and indicated the reliable dynamic demands of the precast concrete structures under the stochastic excitation of nonstationary earthquakes. The detailed principles and applications of the iterative PSD-based procedure for the Gaussian stochastic earthquake model with combined intensity and frequency nonstationarities are illustrated in the following sections.

2. The Principles of the Iterative PSD-Based Procedure for the Gaussian Stochastic Earthquake Model

Generally, structures have many stochastic variables, including geometry variables (${\mathsf{\Theta }}_{\mathit{s}}={({\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2,\dots ,{\mathsf{\Theta }}_x)}^T$) and force variables (${\mathsf{\Theta }}_{\mathit{f}}={({\mathsf{\Theta }}_{x+1},{\mathsf{\Theta }}_{x+2},\dots ,{\mathsf{\Theta }}_n)}^T$). The variable $\mathsf{\Theta }$ is then adopted to reflect the structural uncertainty (i.e., $\mathsf{\Theta }=({\mathsf{\Theta }}_s,{\mathsf{\Theta }}_f)$). The variable $\mathsf{\Theta }$ has n groups of matrices, and the dynamic equation for balance is denoted in Equation (1), where $\mathit{M}$, $\mathit{D}$ and $\mathit{K}$ are the mass matrix, damping matrix and stiffness matrix of the structure, respectively. The values $\ddot{\mathit{G}}(\mathsf{\Theta },t)$, $\dot{\mathit{G}}(\mathsf{\Theta },t)$ and $\mathit{G}(\mathsf{\Theta },t)$ denote the acceleration matrix, velocity matrix and displacement matrix of the structure, respectively. For any concerned stochastic response of a structure, $\mathit{G}(\mathsf{\Theta },t)$ can be assigned for analysis, which depends on the structural stochastic variable $\mathsf{\Theta }$. The value $\ddot{g_{inp}}(\mathsf{\Theta },t)$ denotes the stochastic earthquake model with combined intensity and frequency nonstationarities [16].

$$\mathit{M}·\ddot{\mathit{G}}(\mathsf{\Theta },t)+\mathit{D}·\dot{\mathit{G}}(\mathsf{\Theta },t)+\mathit{K}·\mathit{G}(\mathsf{\Theta },t)=-\mathit{M}·\ddot{g_{inp}}(\mathsf{\Theta },t)$$

(1)

There are quite a few approaches to characterizing the stochastic earthquake model. In this paper, the spectral representation theory is utilized for the stochastic earthquake model, and the function of the bilateral evolutionary power spectral density (PSD) is introduced, as shown in Equation (2). The equation ${\beta }_k=k·{\beta }_{if}$ holds, and ${\beta }_{if}$ denotes the interval frequency. The set $\left\{\mathsf{\Theta }{1}_k,\mathsf{\Theta }{2}_k\right\}$$(k=1,2,\dots ,N_{tr})$ denotes the orthogonal stochastic variable in standard form, and it is obtained from a stochastic mapping from $\left\{\mathsf{\Psi }{1}_n,\mathsf{\Psi }{2}_n\right\}(n=1,2,\dots ,N_{tr})$.

$$\ddot{g_{inp}}(\mathsf{\Theta },t)=\sum _{k=1}^{N_{tr}}\sqrt{2S_{\ddot{g_{inp}}}(t,{\beta }_k)·{\beta }_{if}}·\left[cos\left({\beta }_{k}t\right)·\mathsf{\Theta }{1}_k+sin\left({\beta }_{k}t\right)·\mathsf{\Theta }{2}_k\right]$$

(2)

As for the Gaussian-based stochastic earthquake model, the Gaussian-based orthogonal forms of $\left\{\mathsf{\Psi }{1}_n,\mathsf{\Psi }{2}_n\right\}$ are incorporated. Equations (3) and (4) show the principles, where one or two phase angles are introduced as stochastic variables (i.e., $P{A}_1$ and $P{A}_2$). By this mapping strategy, the parameter dimensions are obviously lowered from $2N_{tr}$ to 2, and the systematic efficiency in the calculation is, consequently, increased [80]. In this approach, both $P{A}_1$ and $P{A}_2$ are uniformly distributed.

$$\mathsf{\Theta }{1}_k=\mathsf{\Psi }{1}_n={\mathsf{\Phi }}^{-1}\left[\frac{1}{\pi }arcsin(\frac{sin(n·P{A}_1)+cos(n·P{A}_1)}{\sqrt{2}})+\frac{1}{2}\right],korn=1,2,\dots ,N_{tr}$$

(3)

$$\mathsf{\Theta }{2}_k=\mathsf{\Psi }{2}_n={\mathsf{\Phi }}^{-1}\left[\frac{1}{\pi }arcsin(\frac{sin(n·P{A}_2)+cos(n·P{A}_2)}{\sqrt{2}})+\frac{1}{2}\right],korn=1,2,\dots ,N_{tr}$$

(4)

To incorporate and consider both the intensity and frequency nonstationarities, the classic Clough–Penzien model is introduced, and Equation (5) shows the Clough–Penzien model for the evolutionary power spectral density (i.e., $S_{\ddot{g_{inp}}}(t,\beta )$), which incorporates both the nonstationary intensity parameter and the nonstationary frequency parameter, as explained in Equations (6) and (7). In Equation (6), ${\xi }_g\left(t\right)$, ${\xi }_f\left(t\right)$, ${\beta }_g\left(t\right)$ and ${\beta }_f\left(t\right)$ reflect the stochastic characteristics of the earthquake model for the frequency nonstationarities, and in Equation (7), $A_{amp}\left(t\right)$ and $S_{amp}\left(t\right)$ reflect the stochastic characteristics of the earthquake model for intensity nonstationarities.

Table 1. The symbols and definitions of the stochastic earthquake model with frequency and intensity nonstationaries.

Number	Symbol	Definition
1	${\xi }_0$	Soil damping.
2	${\beta }_0$	Primary angular frequency.
3	${\mu }_1$	Field classification.
4	${\mu }_2$	Seismic group.
5	${\mu }_3$	Peak acceleration arrival time.
6	${\mu }_4$	Shape factor.
7	${\gamma }_e$	Equivalent peak parameter.
8	$T_{inp}$	Total duration.
9	${\overline{a}}_{max}$	Average peak acceleration.

gives the symbols and definitions of the stochastic earthquake model with frequency and intensity nonstationaries. Based on the different soil sites, the detailed values can be found in [81].

$$S_{\ddot{g_{inp}}}(t,\beta )=A_{amp}^{{}^2}\left(t\right)·S_{amp}\left(t\right)·\frac{{\beta }_g^4\left(t\right)+4{\xi }_g^2\left(t\right){\beta }_g^2\left(t\right){\beta }^2}{{\left[{\beta }^2-{\beta }_g^2\left(t\right)\right]}^2+4{\xi }_g^2\left(t\right){\beta }_g^2\left(t\right){\beta }^2}·\frac{{\beta }^4}{{\left[{\beta }^2-{\beta }_f^2\left(t\right)\right]}^2+4{\xi }_f^2\left(t\right){\beta }_f^2\left(t\right){\beta }^2}$$

(5)

$${\xi }_g\left(t\right)={\xi }_0+{\mu }_2\frac{t}{T_{inp}},{\xi }_f\left(t\right)={\xi }_g\left(t\right),{\beta }_g\left(t\right)={\beta }_0-{\mu }_1\frac{t}{T_{inp}},{\beta }_f\left(t\right)=0.1{\beta }_g\left(t\right)$$

(6)

$$A_{amp}\left(t\right)={\left[\frac{t}{{\mu }_3}·exp(1-\frac{t}{{\mu }_3})\right]}^{{\mu }_4},S_{amp}\left(t\right)=\frac{{\overline{a}}_{max}^2}{{\gamma }_e^2\pi {\beta }_g\left(t\right)·\left[2{\xi }_g\left(t\right)+1/\left(2{\xi }_g\left(t\right)\right)\right]}$$

(7)

In order to improve the accuracy of the stochastic nonstationary earthquake model with combined intensity and frequency nonstationarities, and to fit it with the target spectral acceleration, an iterative procedure was further introduced to adjust the evolutionary power spectral density. The corresponding iterative procedure is listed in Equation (8), and it includes the following four steps: (1) first, spectral representation theory is utilized to generate the initial stochastic earthquakes and to obtain the initial spectral acceleration, on average; (2) second, the gaps between the target spectral acceleration and the average spectral acceleration are analyzed, as indicated in Equation (8); (3) third, a new group of stochastic earthquakes with intensity and frequency nonstationarities is generated by the aforementioned equations; (4) fourth, the above PSD-based steps are repeated 4–5 times, and the target-spectrum-compatible stochastic earthquakes with required nonstationarities and less deviations are obtained. Figure 1 displays the flow diagram of the PSD-based iterative procedure of the stochastically generated nonstationary earthquakes in the practical application.

$$S_{\ddot{g_{inp}}}{(t,\beta )|}_{i+1}=\left\{\begin{array}{c}{S}_{\ddot{g_{inp}}}(t,\beta ),0<\beta \le {\beta }_c\\ S_{\ddot{g_{inp}}}{(t,\beta )|}_i·\frac{Q^T{(\beta ,D)}^2}{Q^S{(\beta ,D)}^2{|}_i},\beta >{\beta }_c\end{array}\right.$$

(8)

where $\beta =2\pi /T_0$, $\beta$ represents the structural frequency and $T_0$ represents the basic period. The values $S_{\ddot{g_{inp}}}{(t,\beta )|}_{i+1}$ and $S_{\ddot{g_{inp}}}{(t,\beta )|}_i$ represent the $(i+1)$th and ith iterative procedure of the evolutionary power spectral density, respectively. The value $Q^T(\beta ,D)$ represents the target response spectra in the code requirement with a damping ratio of D, and $Q^S{(\beta ,D)|}_i$ represents the average response spectra of the generated stochastic nonstationary earthquakes with a damping ratio of D after the ith iteration. The value ${\beta }_c$ represents the truncated frequency and it reflects the adjusting range of the evolutionary power spectral density in the PSD-based iterative procedure.

3. Application of the Iterative PSD-Based Procedure for the Gaussian Stochastic Earthquake Model

In this section, an application of the iterative PSD-based procedure for the Gaussian stochastic earthquake model with combined intensity and frequency nonstationarities is described, based on the precast concrete structures. In total, five precast concrete frames (PCFs) were well designed according to the principle of ‘equivalence to monolithic behavior’. That is, the reinforcements and dimensions were the same as the conventional cast in the place of the concrete structures, but the constructional details corresponded to the precast assembly operations. The five PCFs were 5 spans wide and 10 stories high; 5 spans wide and 8 stories high; 4 spans wide and 8 stories high; 4 spans wide and 6 stories high; and 3 spans wide and 6 stories high, respectively, and the dimensional details are displayed in Figure 2. To perform a dynamic assessment, an appropriate software is commonly required, and, in this analysis, the OpenSees software was utilized [82,83], which is a well-known and efficient platform in the earthquake community. The simulation approach of a PCF is given in Figure 3, in which the precast beams and precast columns are modelled via the nonlinear beam–column elements. To reflect the precast characteristics in the PCF, the Joint2D element was introduced, which can reflect the joint flexural moment–rotation relationship via the central spring and reflect the interfacial moment–rotation relationship via the four interfacial springs. For the central spring, Pinching4 material was used to reflect the pinching effects and joint shear deformation. For the interfacial spring, hysteresis material was used to reflect the degradation effects and interfacial bond–slip phenomenon. Figure 3 gives a verification example with experimental data in terms of the cyclic hysteresis curves [84], which, in a sense, indicate the accuracy and effectiveness of the numerical strategy. More details of the modelling details of the PCF can be found in Cao et al. [85,86,87].

According to the principle of the iterative PSD-based procedure for the Gaussian stochastic earthquake model with combined intensity and frequency nonstationarities, in total, four iterative scenarios were performed in this example. Meanwhile, 10 sets of values for $P{A}_1$ and $P{A}_2$ in Equations (3) and (4) were sampled (ranging from 0 to $2\pi$, as indicated in

Table 2. The information of stochastic variables for the Gaussian stochastic earthquake model.

Number	Random Variables	Symbol	Distribution	Mean (Unit)	COV
1	Earthquake phase angle 1.	$P{A}_1$	Uniform.	3.142 (1)	0.577
2	Earthquake phase angle 2.	$P{A}_2$	Uniform.	3.142 (1)	0.577

), and 10 stochastic nonstationary earthquakes were generated in this example for the evaluation of their seismic behavior. Additionally, the PCF in this example was located on a fortification site of 8 degrees, which means the corresponding peak ground acceleration (PGA) for the design was 0.2 g, followed by a probability of 10% in fifty years. Figure 4 displays the individual stochastic spectral acceleration (gray lines), the average spectral acceleration (blue lines) and the target spectral acceleration (red lines) of all four iterative scenarios, and the corresponding deviations are also given in Figure 4 and

Table 3. The deviations between the target and average spectral acceleration for all four iterative scenarios.

Iterative Scenario	Iteration 0	Iteration 1	Iteration 2	Iteration 3	Iteration 4
Deviation 1.	0.2231	0.0518	0.0209	0.0139	0.0114
Deviation 2.	0.0236	0.0114	0.0072	0.0059	0.0053

. It was found that with an increase in iteration times, the matching degree between the average spectral acceleration and the target spectral acceleration was enhanced. In the meantime, the deviations obviously dropped between Iteration 0 and Iteration 4. In this analysis, two deviation parameters were adopted, and the expressions are given in Equations (9) and (10), where $Ta{r}_i$ and $Av{e}_i$ present the target spectral acceleration and the average spectral acceleration at the ith period, respectively, and n represents the total number of structural periods in the matching range of the spectral acceleration.

$$\mathrm{Deviation}1=\sum _i^n{(Ta{r}_i-Av{e}_i)}^2$$

(9)

$$\mathrm{Deviation}2=\sqrt{\sum _i^n{(Ta{r}_i-Av{e}_i)}^2/(n-1)}$$

(10)

For Deviation 1, the result before the iteration was 0.2231, and the result after the fourth iteration was 0.0114, with a dropping ratio of 94.89%. For Deviation 2, the result before iteration was 0.0236, and the result after the fourth iteration was 0.0053, with a dropping ratio of 77.54%. The dropping ratios proved the accuracy and effectiveness of the iterative PSD-based procedure for the Gaussian stochastic earthquake model with combined intensity and frequency nonstationarities, and, meanwhile, they provided a critical basis for the subsequent dynamic assessment of the PCF. Figure 5 presents the relationship between the evolutional PSD, the earthquake’s frequency and the earthquake’s duration in the iterative PSD-based stochastic model. Figure 5a–c presents the results of Iteration 0, and Figure 5d–f presents the results of Iteration 4. It was shown that after the fourth iteration, the peak values and general densities of the evolutional PSD were elevated. Meanwhile, Figure 6 presents the means and standard deviations of the iterative PSD-based stochastic model. Figure 6a,b presents the means and standard deviations of Iteration 0, and Figure 6c,d presents the means and standard deviations of Iteration 4. The blue lines represent the target values, and the gray lines represent the average values. In general, the results of the 10 stochastic nonstationary earthquakes satisfied the target values and showed the same varying trends. Figure 7 presents a typical stochastic earthquake and iteration comparison, in which the gray lines represent the earthquakes before an iteration and the red lines represent the earthquakes after an iteration. Figure 7a,b shows the comparison of two typical earthquakes after the first iteration, Figure 7c,d shows the comparison of two typical earthquakes after the second iteration, Figure 7e,f shows the comparison of two typical earthquakes after the third iteration and Figure 7g,h shows the comparison of two typical earthquakes after the fourth iteration.

For the performance assessment of the PCF in this example, two engineering demand parameters were utilized, which were the maximum interstory drift ratio (MIDR) and residual interstory drift ratio (RIDR), respectively. These two indexes are broadly accepted in the fields of building engineering and dynamic assessment [69,88,89]. Two intensity levels were defined, which were the frequent earthquake level (FEL) with a probability exceeding 63% in 50 years and the rare earthquake level (REL) with a probability exceeding 2% in 50 years.According to GB50011 [90] and FEMA-356 [91], the thresholds for the MIDR and RIDR were adopted as 0.0018 and 0.00036 for the FEL, respectively, and the corresponding thresholds for the MIDR and RIDR were adopted as 0.02 and 0.004 for the REL, respectively. It is worth mentioning that the generated 10 stochastic nonstationary earthquakes in Figure 4 were for the fortification earthquake level (with a probability exceeding 10% in 50 years), and the corresponding accelerations were adjusted to the FEL and REL linearly in this analysis according to the generating principle of the stochastic earthquake model. Figure 8 presents the average roof accelerations of the five PCFs under three typical stochastic nonstationary earthquakes. Figure 9 presents the MIDR development along the structural height under the stochastic nonstationary earthquakes for both the FEL and REL, and Figure 10 presents the RIDR development along the structural height under the stochastic nonstationary earthquakes for both the FEL and REL. For the MIDR, the average results for all the conditions were calculated using the blue lines, and for the RIDR, the average results for all the conditions were calculated using the pink lines. The thresholds for both the MIDR and RIDR are shown by the red lines. Generally, the seismic performances of the five PCFs were satisfied within the thresholds in terms of the MIDR and RIDR. Especially for the average lines, they were all obviously lower than the red threshold lines for all the dynamic scenarios, which verified the accuracy of the seismic design for the precast concrete structures and indicated the reliable dynamic demands of the precast concrete structures under the stochastic excitation of nonstationary earthquakes. In general, the research provided a meaningful reference for further stochastic earthquake selections and could play an effective role in further precast structure assessments.

4. Conclusions

In this paper, an iterative PSD-based procedure for the Gaussian stochastic earthquake model was developed with combined intensity and frequency nonstationarities, and it was applied to five precast concrete structures for dynamic analysis and validation, from which the following findings can be drawn:

(1) An earthquake contains a lot of uncertainties and stochastic factors in terms of its frequency, intensity and duration, and accurately defining an earthquake’s influence on structures is an important task that deserves further exploration. In order to improve the accuracy of the stochastic nonstationary earthquake model and to fit it with target spectral acceleration, an iterative procedure was developed to adjust the evolutionary power spectral density. The iterative procedure included four steps, and a series of stochastic parameters associated with the earthquake can be considered during the procedure. An example of four iterative scenarios was performed to verify the procedure. It was found that, with an increase in iteration times, the matching degree between the average spectral acceleration and the target spectral acceleration was enhanced. The deviations obviously dropped between Iteration 0 and Iteration 4. For Deviation 1, the result before iteration was 0.2231, and the result after the fourth iteration was 0.0114, with a dropping ratio of 94.89%. For Deviation 2, the result before iteration was 0.0236, and the result after the fourth iteration was 0.0053, with a dropping ratio of 77.54%. The dropping ratios proved the accuracy and effectiveness of the iterative PSD-based procedure for the Gaussian stochastic earthquake model with combined intensity and frequency nonstationarities, and they provided an important basis for the dynamic assessment of engineering structures in the future.

(2) An application of the iterative PSD-based procedure for the Gaussian stochastic earthquake model was performed, based on five PCFs designed according to the principle of ‘equivalence to monolithic behavior’. Two engineering demand parameters were utilized, which were the MIDR and RIDR, and two intensity levels were defined, which were the FEL with a probability exceeding 63% in 50 years, and the REL with a probability exceeding 2% in 50 years. The thresholds for the MIDR and RIDR were adopted as 0.0018 and 0.00036 for the FEL, respectively, and the corresponding thresholds for the MIDR and RIDR were adopted as 0.02 and 0.004 for the REL, respectively. It is worth noting that the accelerations were linearly adjusted to the FEL and REL from the fortification level in this analysis, according to the generating principle of the stochastic earthquake model. Generally, the seismic performances of the five PCFs were within the thresholds, in terms of the MIDR and the RIDR. Especially for the average lines, they were all obviously lower than the red threshold lines for all the dynamic scenarios, which verified the accuracy of the seismic design for the precast concrete structures and indicated the reliable dynamic demands of the precast concrete structures under the stochastic excitation of nonstationary earthquakes. In general, the research provided some meaningful references for further stochastic earthquake selections and precast-structure assessments.