Probability Distribution of Elastic Response Spectrum with Actual Earthquake Data

Abstract

This study aimed to propose a probability-guaranteed spectrum method to enhance the reliability of seismic building designs, thereby addressing the inadequacy of the current code-specified response spectrum based on mean fortification levels. This study systematically evaluated the fitting performance of dynamic coefficient spectra under normal, log-normal, and gamma distribution assumptions based on 288 ground motion records from type II sites. MATLAB(2010) parameter fitting and the Kolmogorov–Smirnov test were used, revealing that the gamma distribution optimally characterized spectral characteristics across all period ranges (p < 0.05). This study innovatively established dynamic coefficient spectra curves for various probability guarantee levels (50–80%), quantitatively revealing the insufficient probability assurance of code spectra in the long-period range. Furthermore, this study proposed an evaluation framework for load safety levels of spectral values over the design service period, demonstrating that increasing probability guarantee levels significantly improved safety margins over a 50-year reference period. This method provides probabilistic foundations for the differentiated seismic design of important structures and offers valuable insights for revising current code provisions based on mean spectra.

1. Introduction

A significant area of China is located on the Eurasian Seismic Belt and the Circum-Pacific Seismic Belt, where numerous destructive earthquakes have occurred in recent decades such as the Tangshan Great Earthquake in 1976 and the Wenchuan Earthquake in 2008 [1,2]. These earthquakes have resulted in direct economic losses and casualties, while also causing various indirect damages and even social issues [3]. With the rapid development of high-rise and super high-rise buildings, their seismic safety has become a major focus in the engineering community [4,5]. Currently, various national codes commonly adopt the mean-based seismic response spectrum theory [6]. However, this approach lacks the systematic evaluation of probabilistic safety levels, which may lead to potentially unsafe designs for long-period structures.

Recent advancements in probabilistic distribution studies of seismic motion parameters have yielded significant progress. Ancheta et al. validated the variability of response spectra through the NGA-West2 database [7,8], while Bommer et al. highlighted that mean spectra may underestimate the seismic risk for long-period structures [9,10], and research indicates that the assurance level of current code spectra in long-period ranges could fall below 50%, a challenge also faced by China’s GB 50011 code. Multiple recent studies have demonstrated that the statistical characteristics of seismic parameters (e.g., spectral acceleration, duration) significantly influence structural seismic designs [11,12]. Rezaeian et al. further revealed the non-stationary nature of near-fault ground motions through stochastic modeling [13,14,15]. Al Atik et al. emphasized that different source mechanisms lead to notable variations in response spectrum shapes, yet current codes fail to adequately quantify such effects [16,17]. Additionally, while the conditional mean spectrum (CMS) method proposed by Kalkan et al. accounts for site-specific characteristics, it fails to address the engineering applicability of probability assurance levels [18,19]. Domestic scholars have conducted a series of studies on site-specific characteristics in China [20], but two major limitations persist: (1) Insufficient sample sizes or the lack of site classifications compromise statistical reliability. (2) The relationship between probability-assured spectra and design service periods remains unexplored. Luco et al.’s latest research shows that U.S. seismic codes have begun incorporating probabilistic adjustment factors [21,22,23], yet a systematic probability-assured spectrum framework is still lacking, underscoring the pioneering nature of this study.

To address these issues, this study employs a comprehensive dataset of site-specific ground motion records to systematically analyze the statistical properties of response spectra. A Gamma distribution-based method for constructing probability-assured spectra is proposed, followed by the quantification of the impact of different assurance levels on 50-year design service safety period. This paper is structured into four parts. Part 1 provides an analysis of response spectrum probability distribution characteristics using 288 ground motion records. Part 2 provides a comparison of three distribution hypotheses via K-S tests and the Akaike information criterion (AIC). Part 3 involves developing spectra at varying assurance levels (50–80%), evaluating code spectrum deficiencies, and proposing a quantitative safety-level assessment framework for design service periods, with a discussion of engineering cost–benefit implications. Part 4 summarizes the key findings and outlines future research directions.

2. Probability Distribution Assumption of Seismic Response Spectrum

A large number of earthquake records under certain conditions are required for conducting a statistical analysis of seismic response spectra. Considering that most areas in China are classified as type II sites, this study adopted the seismic data from K-NET and KiK-net seismic networks in Japan and selected 288 earthquake records from type II sites for further analysis. Figure 1 displays the dynamic coefficient spectra of 288 earthquake records at type II sites. The frequency distribution histograms of the response spectra of the statistical samples revealed that the response spectrum under each independent seismic motion was relatively extensively distributed near the average line and gradually scattered toward two sides. The probability of dynamic coefficient was assumed to follow a normal distribution, log-normal distribution, and gamma distribution based on the distribution characteristics and patterns of the response spectra in previous studies [24]. The selected normal, log-normal, and gamma distribution models characterize symmetric, right-skewed, and flexible-shaped probabilistic patterns, respectively, providing complete coverage of possible statistical properties inherent in response spectra. Next, the representative period points of the aforementioned three distribution patterns were selected for the fitting tests of distribution [25,26,27].

The dynamic coefficients at different period points (0.2, 0.4, 1, 2, 3, 4, 5, and 6 s) were selected for the aforementioned samples. First, the value range was divided into k nonoverlapping intervals, and the statistics of the sample distribution probabilities at various intervals were calculated [28,29,30]. Next, the probability distribution curves were fitted with normal, log-normal, and gamma distributions using the probability distribution fitting toolbox in MATLAB (Dfittool), as shown in Figure 2, Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9.

The distribution curves of the dynamic coefficient at different period points in actual earthquakes were obtained. Also, the optimal fitting according to the aforementioned three assumed probability distribution patterns were performed. Then, the assumed probability distribution patterns were tested to derive the optimal probability form that fits well with the distribution of dynamic coefficients in actual earthquakes.

3. Fitting Test of the Probability Distribution of Seismic Response Spectra

The value of ${\chi }^2$ was calculated for testing the fitted probability distribution curves at the period points. This value was determined using the deviation degree between the dynamic coefficient distributions at the actual period points and the theoretical values. A larger value of ${\chi }^2$ indicated greater deviation. According to Pearson’s theorem [31,32,33], if the assumed probability distribution holds, the statistic ${\chi }^2$ can be calculated as follows:

$${\chi }^2={\displaystyle \sum _{k=1}^r\frac{{(n_k-\mathrm{N}\cdot {\stackrel{\wedge }{p}}_k)}^2}{\mathrm{N}\cdot {\stackrel{\wedge }{p}}_k}}$$

(1)

The aforementioned distribution is the distribution of ${\chi }^2$ with (r − 1) degrees of freedom, in which $n_k$ denotes the distribution quantity of the actual dynamic coefficients in the ith interval at the selected period point, N denotes the total number of the samples for probability distribution fitting, and ${\stackrel{\wedge }{p}}_k$ denotes the theoretical frequency of the samples overall falling in the kth interval. The optimal probability distribution pattern can be determined by calculating the values of the ${\chi }^2$ under different assumed probability distributions.

The probability density functions corresponding to the normal distribution and log-normal distribution indicated that the mean value μ and the standard deviation σ were the distribution parameters to be fitted [34,35]. The parameters of the normal distribution and the log-normal distribution were determined based on the selected samples and the intervals among different period points, as listed in

Table 1. Parameters for the normal distribution fitting test.

	Distribution Parameter	μ	$\mathit{\sigma }$	Number of the Groups for the Assumed Test	Test Statistic χ2
Period Point (s)
0.2		2.1171	0.6727	5	2.8052
0.4		2.0381	1.0584	8	11.0621
1.0		1.4929	0.9758	7	9.8511
2.0		0.8411	0.7080	7	23.0924
3.0		0.5481	0.5157	6	10.4518
4.0		0.4379	0.4409	5	7.2075
5.0		0.3405	0.3490	6	11.3392
6.0		0.2685	0.2541	7	9.5330

and

Table 2. Parameters for the log-normal distribution fitting test.

	Distribution Parameter	μ	$\mathit{\sigma }$	Number of the Groups for the Assumed Test	Test Statistic χ2
Period Point (s)
0.2		0.7002	0.3225	5	3.8391
0.4		0.5632	0.5857	8	4.8379
1.0		0.1155	0.8664	7	12.5791
2.0		–0.6520	1.1223	7	19.6491
3.0		–1.1434	1.1552	5	8.1626
4.0		–1.4145	1.1991	5	10.8370
5.0		–1.6602	1.1758	5	6.7373
6.0		–1.8743	1.1704	7	16.0806

, respectively.

The distribution parameters $\alpha$ and $\beta$ should be fitted according to the probability density function corresponding to the gamma distribution.

Table 3. Parameters for the gamma distribution fitting test.

	Distribution Parameter	α	$\mathit{\beta }$	Number of the Groups for the Assumed Test	Test Statistic χ2
Period Point (s)
0.2		10.1999	0.2076	5	2.6048
0.4		3.5167	0.5795	7	1.5790
1.0		1.9032	0.7844	7	5.5817
2.0		1.1827	0.7111	7	9.8874
3.0		1.0579	0.5181	5	3.5116
4.0		0.9825	0.4456	5	2.9260
5.0		0.9912	0.3435	5	3.5318
6.0		1.0284	0.2611	7	5.9775

lists the gamma distribution parameters based on the selected samples and the intervals among different period points.

The K–S test results displayed in Table 1, Table 2 and Table 3 indicated that the distribution pattern of actual dynamic coefficients at the period points fitted well with the gamma distribution. The regions at different period points (0.2, 3.0, 4.0, and 5.0 s) with the theoretical frequency less than five were merged into five groups, and the values of ${\chi }^2$ were 2.6048, 3.5116, 2.9260, and 3.5318, respectively. At the significance level α = 0.05, the critical value of the rejection domain was calculated to be ${\chi }_{\alpha }^2\left(5-2-1\right)=5.991$. The gamma distributions of the dynamic coefficient at different period points (0.02, 3.0, 4.0, and 5.0 s) all passed the test at the significance level 0.05. The regions at two period points (0.4 and 1.0 s) with a theoretical frequency less than five were merged into seven groups, and the values of ${\chi }^2$ were 1.5970 and 5.5817, respectively. At the significance level α = 0.05, the critical value of the rejection domain was calculated to be ${\chi }_{\alpha }^2\left(7-2-1\right)=9.488$. The gamma distributions of the dynamic coefficient at two period points (0.4 and 1.0 s) all passed the test at the 0.05 significance level. The regions at two different period points (2.0 and 6.0 s) with the theoretical frequency less than five were merged into seven groups, and the values of ${\chi }^2$ were 9.8847 and 11.8542, respectively. At the significance level α = 0.01, the critical value of the rejection domain was calculated to be ${\chi }_{\alpha }^2\left(7-2-1\right)=15.086$. The gamma distributions of the dynamic coefficient at two different period points (2.0 and 6.0 s) all passed the test at the significance level of 0.01. The assumed test results of the dynamic periods at typical period points showed that the distribution pattern of dynamic coefficients fitted well with the gamma distribution.

The MATLAB Dfittool exhibited several limitations: (1) its preset algorithms demonstrated insufficient sensitivity to tail extreme values; (2) the binning process required manual intervention; and (3) it failed to account for period-dependent correlations among seismic parameters. We conducted supplementary verification using the Akaike information criterion test for all three distribution types to enhance the robustness of the hypothesis testing. The results consistently confirmed that the gamma distribution achieved the highest goodness of fit with the response spectra. A ${\chi }^2$ test at the 1.0 s period point without merging intervals was conducted, which confirmed that the gamma distribution maintained consistent significance levels across all analyses. Once the probabilistic distribution form of the seismic response spectrum was established, the corresponding probability assurance level spectra could be derived by specifying different confidence intervals.

4. Probability Guarantee of Standard Response Spectrum and Mean Elastic Spectrum

4.1. Fitted Spectra Based on the Actual Dynamic Coefficient Spectra at Different Probability Guarantee Levels

The analysis of the mean spectra of actual earthquakes indicated that the probability guarantee levels of the standard spectrum at some periodic intervals were low, much below the mean value of the actual earthquake spectrum. Therefore, it is unsafe in engineering practices. Seeking the spectra satisfying a certain probability level is of great significance for engineering practices. In this study, the dynamic coefficient spectra satisfying various probability levels were fitted based on the actual earthquake response analysis of the selected earthquake records. Also, the dynamic coefficients at the typical period points with varying probability levels were analyzed, as listed in

Table 4. Dynamic coefficients at different period points with different probability guarantee levels.

Typical Period Point T (s)	0.2	0.4	1.0	2.0	3.0	4.0	5.0	6.0
Dynamic coefficient β(T) at a probability guarantee level of 50%	2.05	1.85	1.24	0.62	0.39	0.30	0.24	0.19
Dynamic coefficient β(T) at a probability guarantee level of 60%	2.22	2.12	1.50	0.80	0.51	0.40	0.31	0.25
Dynamic coefficient β(T) at a probability guarantee level of 70%	2.41	2.44	1.82	1.02	0.66	0.53	0.41	0.32
Dynamic coefficient β(T) at a probability guarantee level of 80%	2.65	2.85	2.25	1.33	0.88	0.71	0.55	0.43

.

The fitting spectrum was divided into three segments according to the spectral pattern in China’s seismic design code: the linearly ascending segment, the horizontal segment, and the exponentially descending segment. Then, the dynamic coefficient spectra at different probability guarantee levels were fitted. The seismic data in this study were the records for type II sites. Therefore, T₀ and T_g were calibrated as 0.1 s and 0.4 s, respectively. For ${\beta }_{\mathrm{m}ax}$, the mean coefficients before 0.4 s were adopted as the fitted value. However, according to the principle, the fitted value of ${\beta }_{\mathrm{m}ax}$ at the probability guarantee levels of 50% and 60% was 1.95 and 2.17, respectively, both smaller than the value of 2.33 derived from the mean spectrum. Therefore, the values at the probability guarantee levels of 50% and 60% were fitted as 2.33 in this study. The value of $\gamma$ was calibrated with the data in Table 4 using the fitting toolbox in MATLAB cftool. The calibration results are presented in

Table 5. Calibration parameters of the dynamic coefficients at different probability guarantee levels.

Calibration Parameter	T0 (s)	Tg (s)	${\mathit{\beta }}_{\mathbf{m}\mathit{ax}}$	$\mathit{\gamma }$
Parameter of the dynamic coefficient spectrum at a probability guarantee level of 50%	0.1	0.4	2.33	0.8290
Parameter of the dynamic coefficient spectrum at a probability guarantee level of 60%	0.1	0.4	2.33	0.6945
Parameter of the dynamic coefficient spectrum at a probability guarantee level of 70%	0.1	0.4	2.42	0.5894
Parameter of the dynamic coefficient spectrum at a probability guarantee level of 80%	0.1	0.4	2.75	0.5238

.

The dynamic coefficient spectra were plotted based on the calibration parameters of the dynamic coefficients at different probability guarantee levels, as presented in Table 5 (Figure 10).

The dynamic coefficient spectrum curves fitted in accordance with actual earthquake records under specific conditions at various frequencies can be used in practical engineering. Given sufficient data in earthquake records, the fitted spectra may be of great significance in engineering practices. Therefore, the seismic dynamic coefficient spectra at different probability guarantee levels can be selected for various buildings in accordance with their importance degree or business requirements to achieve the seismic design.

4.2. Safety Level of Spectral Loads During the Designed Service Period

The probability safety level of spectral loads during the designed service life can be determined through analysis, in combination with the concept and the probability distribution pattern of regional seismic intensity as well as the response spectra at different probability guarantee levels. The safety level of the spectral load during the service life can be defined as the probability that the determined spectral load will not exceed the value during the service life. For example, for the peak acceleration of frequent earthquakes (from which the seismic coefficient $k_{usual}$ can be determined) and the dynamic coefficient spectrum at the probability guarantee level of 50% (from which the dynamic coefficient of the building with a specific load ${\beta }_{50\%}\left(\mathrm{T}\right)$ can be determined), the probability that an earthquake exceeding the seismic coefficient occurs in the region within 50 years can be calculated as $\alpha =k_{usual}\cdot {\beta }_{50\%}\left(\mathrm{T}\right)$, equaling to 63% × 50% = 31.5%. Accordingly, for the building designed with the peak acceleration of frequent earthquakes and the dynamic coefficient spectra at a probability guarantee level of 50% in this region, the spectral load level, that is, the probability when the load is below the spectral load, equals to 100–31.5% = 68.5%.

Table 6. Spectral loads under the dynamic spectra at different probability guarantee levels for frequent earthquakes.

Used Seismic Acceleration Intensity	Frequent Earthquake	Frequent Earthquake	Frequent Earthquake	Frequent Earthquake
Used dynamic Coefficient	Dynamic coefficient spectrum at a probability guarantee level of 50%	Dynamic coefficient spectrum at a probability guarantee level of 60%	Dynamic coefficient spectrum at a probability guarantee level of 70%	Dynamic coefficient spectrum at a probability guarantee level of 80%
Spectral load safety assurance rate (%)	68.5	74.8	81.1	87.4

,

Table 7. Spectral loads under the dynamic spectra at different probability guarantee levels for fortification earthquakes.

Used Seismic Acceleration Intensity	Fortification Earthquake	Fortification Earthquake	Fortification Earthquake	Fortification Earthquake
Used dynamic Coefficient	Dynamic coefficient spectrum at a probability guarantee level of 50%	Dynamic coefficient spectrum at a probability guarantee level of 60%	Dynamic coefficient spectrum at a probability guarantee level of 70%	Dynamic coefficient spectrum at a probability guarantee level of 80%
Spectral load safety assurance rate (%)	95	96	97	98

and

Table 8. Spectral loads under the dynamic spectra at different probability guarantee levels for rare earthquakes.

Used Seismic Acceleration Intensity	Rare Earthquake	Rare Earthquake	Rare Earthquake	Rare Earthquake
Used dynamic coefficient	Dynamic coefficient spectrum at a probability guarantee level of 50%	Dynamic coefficient spectrum at a probability guarantee level of 60%	Dynamic coefficient spectrum at a probability guarantee level of 70%	Dynamic coefficient spectrum at a probability guarantee level of 80%
Spectral load safety assurance rate (%)	99.0	99.2	99.4	99.6

list the safety levels calculated according to the earthquake intensities and the dynamic coefficients at different probability guarantee levels.

The Chinese structural design principle “no damage under minor earthquakes, repairable under moderate earthquakes, and non-collapse under major earthquakes” is implemented using code-specified methods. Typically, structural designs are based on the code spectra for minor earthquakes (frequent earthquakes), with seismic measures and structural details employed to achieve the design objectives. Accordingly, this study introduced probability assurance spectra for frequent, design, and rare earthquakes and provided multi-level probability assurance spectra options for engineering design to accommodate varying safety requirements.

The dynamic coefficient spectra with a 50% probability assurance level presented in the table align closely with the corresponding spectral values in China’s GB50011 seismic design code [36], with the maximum deviation across all period points remaining within 8.5%. In contrast, the spectral values for the dynamic coefficient spectra with the 60%, 70%, and 80% probability assurance levels consistently exceed the code-specified values at all period points. As shown in Table 6, the probability levels of the spectral load at different probability guarantee levels varied obviously when using the acceleration of frequent earthquakes. For example, the spectral load was only 68.5% when using the dynamic coefficient spectrum at the probability guarantee level of 50%, which increased to 87.4% when using the spectrum at the probability guarantee level of 80%. As illustrated in Table 7 and Table 8, the spectral loads under the dynamic coefficients at different probability guarantee levels displayed a slight change when using the accelerations of fortification and rear earthquakes.

The probability level of the spectral load changed more obviously when different seismic acceleration intensity levels were applied under the dynamic coefficient spectrum at the same probability guarantee level. For example, the probability level of the spectral load was 68.5% and 95%, respectively, when using frequent and fortification earthquake intensities under the dynamic coefficient spectrum at a probability guarantee level of 50%. It is thus concluded that the seismic force load level of the building can be enhanced by increasing the probability guarantee level of the dynamic coefficient, thereby improving structural safety. However, the degree of improvement achieved by enhancing the probability guarantee level is lower than the improvement achieved by increasing the seismic fortification intensity.

The probability assurance level response spectra provide an effective methodology for designing and constructing engineering structures with higher safety levels. The method is conceptually clear and offers multiple levels of choice; however, its impact on engineering costs is also significant. The comparative analysis performed by applying 50%, 60%, and 70% probability assurance level spectra under the design earthquake scenario to the same frame structure design revealed that (1) steel consumption increased by 16.5% and the overall cost rose by 14.2% when transitioning from the 50% to 60% assurance level; and (2) the cost increase became more pronounced at higher assurance levels, with steel consumption rising by 18.6% and the overall cost increasing by 16.1% when shifting from the 60% to 70% assurance level.

5. Discussion and Conclusions

This study investigated the probability distribution pattern of the seismic response spectrum based on actual earthquake records under specific conditions. Accordingly, the concepts of the spectra at different probability guarantee levels and the probability safety level of the spectral load of the building during the service life were derived. The probability assurance level response spectra provide an effective methodology for designing and constructing engineering structures with enhanced safety levels. The approach is conceptually clear and offers multiple assurance level options, although its impact on engineering costs is substantial.

The data adopted in this study were sourced from the Japanese seismic network. Although categorization was conducted based on site soil parameters and Chinese site classifications, significant differences remain in the seismic source mechanisms, propagation pathways, and other factors in Japan and China. Therefore, further in-depth analysis and research are required in areas such as seismic source mechanisms, design earthquake groupings, and additional site classifications to improve the applicability of this methodology in Chinese seismic design practices.

The main conclusions of this study are as follows:

(1) Quantifying the probabilistic safety level of response spectra addresses the underestimation risks inherent in mean-based spectra, thus providing a reference for future revisions to seismic design codes.

(2) Offering probabilistic assurance design spectra as optional tools enables differentiated seismic fortification choices in structural design, thereby accommodating individualized design requirements.

(3) Establishing a direct correlation between spectral value loads and design life safety levels for the first time provides a probabilistic foundation for performance-based designs.

The data used in this study were sourced from the Japanese seismic network. Although seismic station soil parameters and Chinese site classifications were considered in the analysis, disparities persist in terms of seismic source mechanisms, wave propagation pathways, and other factors compared with the seismic characteristics of China. Therefore, further in-depth exploration is required to explore seismic source mechanisms, design earthquake groups, and additional site classifications.