1. Introduction
In general, an explosion is a process of rapid energy release. Previous studies and explosion damage data show that most of the explosive energy is converted primarily to thermal energy, ground motions and shock waves. With the development of anti-explosion design theory and the renewal of design concepts, the influence of explosion damage, especially explosive ground motions, on structures has been gradually recognized and paid more attention to in the engineering field. Actually, explosive ground motions can be partially analogous to natural ground motions since the source of an explosion is similar to that of the latter. However, compared with natural seismic ground motions, explosive ground motions have the characteristics of shorter duration, smaller amplitude and more abundant frequency components, which would perform a noticeable impact on typical engineering structures.
In the 1920s, Rockwell [
1] took the lead in studying the impact of the ground motions caused by mine blasting on nearby building structures, and initially proposed the concept of structural anti-explosion. As a matter of fact, the explosion has significant randomness, and applying the random vibration theory to identify and study the spectral characteristics of explosive ground motions is crucial for exploring the damage mechanism and the anti-explosion strategy of the engineering structures affected by such excitations. In this regard, Persson [
2] presented the bispectrum analysis of both natural earthquake ground motion signals and explosive ground motion signals. He found that the sample bispectrum of the signals with Gaussian distribution is lower than that of the signals with skewed distribution. Wu and Hao [
3] proposed empirical formulas for estimating the ground peak particle velocity and the dominant frequency of explosive ground motions based on data recorded in the field and found that explosive ground motions on the ground surface and in the free field are remarkably different due to the effect of surface reflection. In addition, with the rapid development of numerical analysis technologies such as finite element strategy, significant progress has been achieved in the investigations of the effects of explosive ground motions on structures. Hao and Wu [
4] studied the dynamic response of reinforced concrete frame structures under the action of the simulated explosive ground motions, and they compared the numerical results of structural damage with some test results obtained in previous studies and codes. Ye et al. [
5] analyzed the dynamic response and the damage state of a four-story masonry building utilizing measured explosive ground motion records, and they concluded that the structure suffered brittle failure due to tensile stress. With the help of the measured data, Zhang et al. [
6] investigated the possibility of evaluating the damaging effect of explosion sources on the ground by utilizing the actual explosive ground motion records through experimental tests, which showed that the treatment was feasible in the case of the same formation condition and within a certain distance. Obviously, explosive ground motion is a special form of ground motion that has attracted widespread attention [
7,
8,
9,
10,
11].
Dynamic response and damage analysis of structures induced by explosive ground motions are primarily based on measured recordings at present, according to the research described above, which undoubtedly belong to the deterministic analysis scheme. However, due to the limited number of measured explosive ground motion time-histories, as well as the limitations of site conditions and explosion environment, it is difficult for the existing measured records to completely satisfy the requirements of structural dynamic response analysis. Considering the nature of the random dynamic characteristics of explosive ground motions, the application of random vibration theory and method to conduct the artificial synthetic method for reasonable stochastic explosive ground motion model, as well as structural dynamic response and reliability analysis, plays a significantly important role in the anti-explosion analysis and design of engineering structures. In addition, the aforementioned studies generally focus on the dynamic response of structures under explosive seismic action, and rarely consider the dynamic reliability evaluation of structural systems, which requires further investigation for engineering purpose.
Due to the fact that the explosive ground motions are similar to the natural earthquake ground motions, the former can be simulated through the spectral representation method (SRM) [
12,
13,
14,
15,
16], which is widely applied in the generation of stochastic processes and is already extensively used for the simulation of explosive ground motions [
17,
18]. However, the conventional SRM is based on the Monte Carlo simulation method, which typically requires a large number of random variables and randomly generated sample functions to obtain a satisfactory level of simulation precision. In this end, it is difficult to use the Monte Carlo method for an accurate dynamic reliability evaluation and response analysis of complex civil engineering structures due to the large amount of computational costs and time expenses required to complete a dynamic time-history analysis of the investigated structure and the lack of assigned probability sampling [
19,
20,
21].
Overall, current research on explosive ground motions has entered a relatively mature stage by analogy with natural ground motions, where the Sadovsky formula is a typical example. However, there are still three limitations in the existing research that require further exploration.
The research of the relation of charge quantity and the distance to the explosion source with the seismic characteristics of explosive ground motions only staying on the seismic energy level in the present stage, where the influence of that on the seismic spectrum and seismic duration is neglected.
The classic Monte Carlo method for simulating explosive ground motions faces the problems of high-dimensional random variables and the incomplete probability information of the generated sample set, which leads to great difficulty for the efficient and refined dynamic response and reliability analysis of randomly excited structural systems.
The method of using measured records for structural dynamic response analysis has two shortcomings. On the one hand, the number of measured records is limited, and the records are susceptible to the environment and the noise of the measurement instrument, which is difficult to meet the strict requirements of engineering structures. On the other hand, it is essentially a deterministic method that ignores the impact of the randomness of explosive ground motions on structures, which is not in line with reality.
To overcome the challenges faced by the conventional Monte Carlo method, on the one hand, Chen et al. [
22,
23] proposed the stochastic harmonic function representation for simulating stationary and non-stationary stochastic processes in recent years. On the other hand, Liu et al. [
24,
25,
26,
27] recently offered a dimension-reduction technique wherein random functions are introduced as the constraints of random variables in order to generate stochastic (vector) processes for modeling the earthquake ground motions and the wind velocity fields. The computational costs associated with millions of random variables have been effectively reduced to the level of a few elementary random variables, demonstrating the commendable accuracy and efficiency of the dimension-reduction method. In addition, it is worth mentioning that each representative sample generated by the dimension-reduction method has an assigned probability and all the representative samples can constitute a complete probability set, making it possible to achieve the refined dynamic response analysis and reliability evaluation of randomly excited structural systems combining with the probability density evolution method (PDEM) [
28,
29]. Consequently, it effectively bypasses the deficiencies of the conventional Monte Carlo method. Actually, the PDEM is a new approach to solve the dynamic reliability of complex engineering structures under the action of dynamic disasters, which was initially proposed by Li and Chen and has been widely expanded and applied in both the academic and engineering communities [
30,
31,
32].
Based on the above research situation, the purpose of this investigation is to establish an efficient and accurate stochastic model for synthesizing explosive ground motions by introducing the dimension-reduction method and to realize the refined dynamic reliability evaluation of a nonlinear frame structure combining with the PDEM, which can provide a certain theoretical foundation and technical support for the anti-explosion research of engineering structures. The remaining sections are organized as follows.
Section 2 introduces the evolutionary power spectrum density (EPSD) function of explosive ground motions and proposes a method for identifying the EPSD parameters. In addition, an exponential model is suggested, which can essentially characterize the functioning relationship between the explosion mechanism and the EPSD parameters.
Section 3 expounds the SRM-based dimension-reduction method for simulating explosive ground motions, and the accuracy of the proposed dimension-reduction model is revealed through numerical examples.
Section 4 establishes a finite element model of a 10-story frame structure considering the material nonlinearity, and the dynamic response analysis is implemented combining with the PDEM.
Section 5 suggests an equivalent extreme value (EEV)-based strategy for evaluating the component and global dynamic reliability of the frame structure. Some conclusion remarks are summarized in
Section 6.