Introduction

Earthquake waves are transformed as they diverge from the epicenter and go through the alluvial layers. Factors related to the distance of points from the epicenter are known as “path impacts,” and factors associated with alluvial layers located on the bedrock are known as “site effects.” Site effects manifest in the form of the intensification of earthquake waves and alterations in seismic waves’ characteristics such as their amplitude, frequency, and the duration of strong movements. Today, earthquakes are one of the most devastating natural disasters leading to the destruction of man-made structures. Several destructive earthquakes in recent decades, including those in Mexico City (1985), North Ridge (1994), and Turkey (1999), as well as severe earthquakes in Iran, such as that which affected Bam (2003), have clearly revealed that the geological conditions and surface topography play influential roles in the extent of vibrations. As a case in point, consider the Mexico City earthquake. On September 19, 1985, an earthquake with an 8.1 magnitude on the Richter scale struck Mexico, brought about moderate damage to its epicenter (Pacific coast); however, it produced severe damage 350 km away in Mexico City. Another case is the earthquake that occurred on November 13, 2017, at the Sare Pole Zahab in Kermanshah province, Iran. Expert groups who were sent to the area to assess various aspects of the earthquake detected that the damage distribution was not uniform; there were significant differences in distinct parts of the city, even between comparable types of structures. There were two types of structural damage. In the northern sections of the city’s main boulevard, the damage was insignificant even though most of the other areas were severely damaged.

In 1996, Kaiser et al. (1996) proposed that two parameters, peak ground velocity (PGV) and peak ground acceleration (PGA), could be used to estimate the dynamic vibration load. Ground motions can be quantified by these two parameters. During an earthquake, the accumulated energy in the focal area is released in the form of seismic waves. The focal mechanism is an important feature of the seismic source that influences the propagation of seismic waves. In other words, the ground motions generated by earthquakes are closely related to the focal mechanism. Even if the distance to the source is the same in different directions, the dynamic load can vary significantly (Ma et al. 2018, 2019).

Douglas et al. (2007) conducted research examining focal mechanism effects on the earthquake ground motions. They reported a considerable influence of the focal mechanism on ground motion produced by earthquakes. Mining studies also indicated the importance of focal mechanisms (Ma et al. 2019). According to extant theories, the vibrations produced by the explosion are reduced in all directions away from the source of the explosion. However, Ma et al.’s (2019) results showed that the ground vibrations did not decrease as they moved away from the source of the explosion; this phenomenon was due to the focal mechanism parameter.

Additionally, ground motions are distance-related and azimuth-related in spatial distribution. Alluviums with diverse structures show different responses due to the arrival of earthquake waves. Researchers have also revealed that responses to a particular earthquake vary from station to station due to the characteristics of various sites (Pavel et al. 2020; Ottonelli et al. 2021; Alam et al. 2021). Other researchers have reported similar results who used a similar approach to the present study, including Zheng et al. (2021) and Alimohammadi et al. (2020, 2019a, b), Alimohammadi and Yaghin 2019). The geological properties of the site are a fundamental factor influencing seismic responses (Diaz-Fanas et al. 2020). For example, when the environment structure has horizontal layers, only volumetric waves moving up and down in the surface layers are trapped, whereas when the environment structure is two- or three-dimensional (lateral heterogeneity), the surface waves created by this inconsistency are also trapped.

Soil stratification close to the surface can significantly intensify or weaken seismic forces (Garcia-Suarez and Asimaki 2020). Resonance in the soil is of great significance for structural design, as structures vulnerable to resonance can be destroyed easily (Mase et al. 2021). So far, numerous studies have been conducted on soil resonance from various perspectives (Gosar et al. 2010; Kaurkin et al. 2021; Guangyin et al. 2021; Trevisani et al. 2021; Xie et al. 2020; Moschetti and Hatzell 2021; Sgattoni and Castellaro 2020; Gallipoli et al. 2020). The term resonance relates to the increase in amplitude when the frequency of an applied alternating force to a system is equal to or close to the system’s natural frequency. By applying an oscillating force to a dynamic system resonant frequency, the system oscillates at a higher amplitude than in cases where an equal force is applied to other non-resonant frequencies (Halliday et al. 2005).

Studies conducted by the Roads, Housing and Urban Development Research Center to present the structure design regulations against earthquakes in Iran (Regulations 2800 (2014)) suggest that replacing the hard soil layer close to the earth’s surface with soft soil (sand to gravel) intensified the earthquake motion on the ground surface (thus intensifying the ground motion response curve). Moreover, by doubling the soil layer thickness (the soil close to the ground surface was assumed to be coarse-grained), the movement of the ground surface was weakened. Thus, the possibility of soil resonance was significantly reduced as a direct consequence of the increase in the thickness of the damper soil layers (sand and gravel).

Additionally, a reduction in the arrival acceleration into the soil intensified the ground response. This effect can be explained by the fact that moderate earthquakes produce more soil resonance than strong earthquakes due to damping. Small earthquakes produce minute strains, and in tiny strains, damping is slightly activated. Conversely, large earthquakes generate considerable strains, providing an enormous amount of damping, thereby reducing the severity of ground movements.

The importance of studying the phenomenon of resonance in the soil, as well as the factors affecting ground motions and the phenomenon of resonance in soil, was mentioned above. Due to the importance of the resonance in the soil for structural design, this phenomenon should be studied from different perspectives.

The present study is of relevance because it examines factors affecting resonance in soil from different perspectives. However, the effect of dynamic properties and repetitive patterns in earthquake signals on resonance in soil has not been studied, which is addressed in this study. This study contributes to the understanding of the effect of dynamic properties and repetitive patterns in earthquake acceleration time series on resonance in soil. It provides the impetus for future research on the effect of earthquake signals on the resonance phenomenon in soil. This study demonstrates whether, in addition to properties of soil, focal mechanisms, distance, and nonlinear dynamic properties of ground motions, among other factors, affect the resonance in soil. It also suggests which ground motions should be considered in the case of structural design to examine resonance in the soil. This research is beneficial to structural design and soil engineers and researchers who study the effect of earthquake signals on soil behavior.

The main objective of the present study is achieved by generating a soil column with sixteen horizontal layers. Subsequently, ten well-known earthquake accelerations (motions) were applied to the soil column to study soil resonance and soil failure. Ultimately, the multifractal detrended fluctuation analysis (MF-DFA) technique was employed to estimate the nonlinear dynamic properties of motions, and subsequently, the association between the earthquake signal’s nonlinear dynamic properties and soil resonance was investigated.

In the next section, the soil column and earthquake details used for analysis will be provided. In the “Results and discussion” section, the results will be presented and discussed. Finally, the “Conclusion” section will provide a summary and conclusions.

Materials and methods

In order to investigate the effect of earthquakes on generated soil profile, ten well-known motions (earthquakes) were considered, and acceleration data were collected from PEER Ground Motion Database (https://ngawest2.berkeley.edu/). Details concerning the ten utilized motions (earthquakes) are sorted in Table 1.

Table 1 The details of the ten employed motions (earthquakes)
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Earthquakes that occurred between 1940 and 1995 and ranged in magnitude from 6.8 to 7.3 Mw were examined in this study. The lowest (5 km) and highest (19 km) depths at which the earthquakes occurred were the earthquakes of SPITAK and LOMA, respectively. The ten investigated earthquakes occurred in different countries. The location of the Kobe earthquake is shown in Fig. 1. This earthquake was recorded in 1995 and had a depth of 17.6 km; approximately 6434 people lost their lives.

Fig. 1
figure 1

The location of the Kobe earthquake (marked with an asterisk) recorded in 1995

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Generating soil profile and plotting shear strain percentage and 5% damped spectral acceleration

After collecting earthquake data, a soil profile was generated using DEEPSOIL software version 7. The soil profile was generated by employing the techniques of generalized quadratic/hyperbolic (GQ/H) model with shear strength control, proposed by Groholski et al. (2016), and non-masing unload-reload rules, proposed by Phillips and Hashash (2009). This soil model was selected because it requires few input data requirements (the curve fitting parameters θ1 through θ5 values and maximum stress τmax), and DEEPSOIL software calculates the curve-fitting parameters of the generated soil profile.

The minimum thickness of the layers in the study soil profile was 1.9 m, and the total soil profile thickness was 30 m. The maximum frequency in the soil layers was 50 HZ, and the mean layer properties (including width, unit weight, and effective vertical stress) were 30 m, 20 KN/m3, and 152.85 KPA, respectively. The shear wave velocity varied according to the depth; specifically, it was 70 for the ground’s surface and 700 for a depth of 30 m. The shear strength and soil type were 100 KPa and sand, respectively. The soil properties of each layer are provided in Table 2.

Table 2 Details concerning the generated soil layers through DEEPSOIL software version 7
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There are two general methods used to analyze soil response based on the ingress of an earthquake wave into the soil profile: the frequency-domain method and the time-domain method. The nonlinear time-domain method was employed to solve motion equations to estimate soil responses. This method was selected because it produces more reliable results than other methods.

Likewise, there are two available time integration methods used to solve motions equations: the Newmark β method (implicit) and the Heun Method (explicit). In the present study, the first method was adopted. Usually, to solve a second-order differential equation, one must first transform that equation into two first-order differential equations. The advantage of the Newmark method (Newmark, 1959) is that it enables us to solve the second-order differential equation without the transformation (for further details, see Lindfield and Penny 2019).

DEEPSOIL software was utilized to solve the equations of the motion and extract shear strain percentage and 5% damped spectral acceleration graphs. According to the valid regulations (such as Regulations 2800 (2014)), when the shear strain percentage touches a value greater than 1%, the soil is ruptured (soil failure phenomenon). The shear strain percentage curves were related to the investigation of the soil failure phenomenon in layers of the study soil profile. Depicting 5% damped spectral acceleration versus period graphs supports the study of the resonance in soil by examining the response spectra.

The phenomenon of resonance in soil occurs when the response spectra curve in a layer of soil exceeds the response spectra curve of the input motion into the soil profile. The significant duration of the ground motions was extracted prior to the nonlinear dynamic properties’ estimation of ground motions. The significant duration is the time span (in seconds) between the occurrences of 5% and 95% of the total Arias Intensity. Arias Intensity can be used to represent the intensity of a motion as a function of acceleration (Eq. 1). The significant duration was computed using DEEPSOIL software. Ultimately, the nonlinear dynamic characteristics of the motions’ significant durations were estimated.

$${I}_a(t)=\frac{\pi }{2g}\int_0^t{\left[a(t)\right]}^2 dt$$
(1)

Multifractal detrended fluctuation analysis (MF-DFA)

The application of detrended fluctuation analysis (DFA) is commonly adopted for the measurement of mono-fractal scaling characteristics and exploring long-range correlations in noisy, non-sub-basinary time series (Kantelhardt et al. 2002). Multifractal detrended fluctuation analysis (MF-DFA) technique can be used for multifractality assessment in the time series, and its accurate results for non-sub-basinary series (Adarsh et al. 2020; Miloş et al. 2020) prove the proper functionality of this tool for finding multifractal patterns in the hydrological time series. This technique also can be used for generalized Hurst exponent estimation. Accordingly, the MF-DFA technique was adopted to examine generalized Hurst exponent. The singularity spectrum f(α) can be employed for representing the multifractal patterns in the time series. The scaling exponent τ(q) and first-order Legendre transforms provide a basis for singularity spectrum evaluation (Eq. (2)), demonstrating the time series segment dimensions identified by α (see Eq. (3)). If f(α) is a single point, it can be concluded that the time series is mono-fractal (for further details, see (Rahmani and Fattahi 2021a)).

$$\tau (q)=q\times H(q)-1$$
(2)
$$\alpha =\frac{\partial \tau (q)}{\partial q},f\left(\alpha \right)=\alpha q-\tau (q)$$
(3)

where α and q are the Hölder exponent (singularity index) and the local variations’ order weighting, respectively. H(q) points to the generalized Hurst exponent and can be assessed by Eq. (4).

$${F}_q(s)\propto {s}^{H(q)}$$
(4)

Equation 4 is known as the power-law association between variation segment Fq(s) and time scale s. The H(q) is valued by gradient of the fitted Fq(s) and s (for further details concerning MF-DFA, see Mandelbrot 1989, Rahmani and Fattahi 2021b).

A narrow singularity spectrum depicts relative systematic fluctuations in the tike series that occur over similar amplitudes over time; this is a sign for a mono-scaling structure. A wide singularity spectrum illustrates more periodic fluctuations in the time series that occur over a greater range of amplitudes in different time lengths, resembling the multifractal processes. The range of the singularity spectrum represents the dynamics of the system (Zhang et al. 2019; Munro et al. 2018).

The large fluctuations in a signal hold small singularity exponent α and are located in the left tail of the spectrum, whereas α for the small fluctuations is large and located in the right tail of the spectrum. Accordingly, the multifractality strength is characterized by the large deflection of the local singularity exponent α from the central orientation α (0). When α is almost constant, the signal is monofractal, while α variations indicate a multi-scaling process (Ihlen and Vereijken 2013; Cruz and Sampaio 2020).

The range α illustrates the variety of singularity exponents, resembling the dynamics of the system, and the value of Δf(α) represents the probability of an identified α. The singularity spectrum may be categorized in three, regarding the shape. Accordingly, a singularity spectrum with a left long tail, right long tail, and symmetric spectrums can be identified. This subdivision comes from the range of a singularity spectrum which includes a left limb (L) and a right limb (R), representing the dynamics of the highest and lowest fluctuations, respectively. When the mean values of Δf (α) > 0, the minimum fluctuation rate in the time series happens with a higher probability than the maximal fluctuation rate, and the signal structure is not sensitive to local fluctuations with a large magnitude. Likewise, when the mean values of Δf (α) < 0, then the maximal fluctuation rate occurs with a higher probability comparing the minimal rate and the signal structure is not sensitive to small magnitude local fluctuations (Cruz and Sampaio 2020). It has to be mentioned that the MATLAB software was used for multifractal analysis, and MATLAB, Microsoft Office, and DEEPSOIL softwares were employed to generate the graphs.

Results and discussion

The analysis was performed in two phases. In the first phase, shear strain percentage and 5% damped spectral acceleration diagrams were plotted and interpreted. In the second phase, the dynamic properties of the motions’ significant duration were estimated, and the possible relationship between the nonlinear dynamic properties of motions’ significant duration and resonance phenomena in soil and soil failure was investigated.

Phase 1 results

After generating the soil profile described in the “Materials and methods” section, ten ground motions were applied to the produced soil profile, and the shear strain percentage and 5% damped spectral acceleration diagrams were generated. An examination of the 5% damped spectral acceleration graphs revealed that all ground motions, except those in El Alamo and Trinidad, produced resonance in the soil. In cases where ground motions resonated, they did so within a period of less than 0.1 s and from 1 to 8 s. However, in the period of less than 0.1 seconds, the rate of intensification was considerable, and in the period from 1 to 8 seconds, it was negligible. A striking example of resonance in soil was produced in Imperial Valley (II) (Fig. 2).

Fig. 2
figure 2

5% damped spectral acceleration versus period graph concerning Imperial Valley (II) input ground motion and the sixteen layers of generated soil profile

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Figure 2a clarifies that between the period of 0.01 and 0.08 s, considerable resonance was produced by Imperial Valley (II) ground motion (layer 16’s response spectra (RS) curve (blue line) exceeded response spectra of input motion (black line)). This demonstrates that during this period, the soil profile is considerably prone to resonance and buildings that have a natural frequency from 0.01 to 0.08 s will be severely damaged. In the period between 0.5 and 1 s, intensification (resonance in soil) was detected, although it was slight.

Figure 2b illustrates that the generated soil profile experienced resonance in all layers. The highest resonance occurred at the period of 0.06 s, which was equal to 1.26 g. In cases where resonance was produced in the soil profile, layers 15 and 16 were the most sensitive layers and experienced the highest resonance. Layers close to the surface witnessed insignificant resonance. However, for some ground motions, the resonance recorded in layers 15 and 16 was insignificant, while it was noticeable in others. According to \(F=\frac{1}{T}\) (F = frequency, T = period), the earthquake frequency is high during low periods and vice versa. Figure 2b shows that the resonance occurred in the first part of the diagram, verifying that the soil resonated at high frequencies. Hence, although high-rise buildings will be damaged, short-rise structures will be safe. In the second part of the diagram (Fig. 2b), there was an insignificant resonance, illustrating that the soil profile has low resonance at low frequencies and high periods, meaning that short-rise buildings are safe.

During the period from 0.08 to 0.4 s, the RS lines of the different layers (the colored lines in Fig. 2b) were significantly lower than the RS line of the input motion (the black line in Fig. 2b). This indicated that the soil damped the earthquake motions well during this period and prevented resonance in the soil profile. The most significant resonances occurred in layers 15 and 16. For this soil sample, the occurrence of earthquakes poses the greatest risk to underground structures (such as basements, underground tunnels, and underground parking lots).

The shear strain percentage graphs (Fig. 3) indicate that few ground motions brought about the soil failure phenomenon (Cape Mendocino, Kobe, Loma Prieta, and Gazli). Also, an examination of shear strain percentage and 5% damped spectral acceleration graphs did not confirm a significant relationship between the intensity of resonance in the soil and soil rupture (soil failure phenomenon). As a case in point, consider Imperial Valley (II). Even though Imperial Valley (II) produced the most significant resonance in soil, Loma Prieta’s ground motion led to the most soil rupture in the generated soil profile. All ground motions that caused the soil failure produced resonance in the soil; however, the opposite effect was not valid. The Loma Prieta shear strain percentage graph is presented in Fig. 3 as an illustration.

Fig. 3
figure 3

The Loma Prieta’s shear strain percentage graph

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Between 11 and 12 s, the percentage of shear strain exceeded 1% (touched 8%), indicating soil failure (experienced in layer 16). All soil failures occurred in layers 15 and 16, with layer 16 experiencing more extensive ruptures than layer 15.

Phase 2 results

In this phase, the nonlinear dynamic properties of ground motions were investigated. The significant duration was calculated before dynamic analysis. If the entire earthquake signal is analyzed, there is a high probability that the multifractal or mono-fractal behavior of the signal will not be detected. For this reason, only the significant duration part of the earthquake signal was analyzed. An example is the significant duration graph of Loma Prieta motion (Fig. 4).

Fig. 4
figure 4

The significant duration of Loma Prieta motion marked in gray on the graphs

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As shown in Fig. 4, the significant duration of Loma Prieta motion was from 6.56 to 14.375 s.

Figure 5 shows the multifractal singularity spectrum diagrams.

Fig. 5
figure 5

The multifractal singularity spectrum of ground motions

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Figure 5 shows that the singularity spectrum curves of the Cape Mendocino, Trinidad, Kobe, and El Alamo motions were of the right-tailed type. This indicates that the probability of small fluctuations (minute accelerations) in the acceleration time series of these motions was higher than large fluctuations—likewise, their signals were not sensitive to large local fluctuations. Smart (Taiwan), Loma Prieta, and Spitak (Armenia) motions showed left-tail singularity spectrum curves, designating more large fluctuations (acceleration magnitude) and less sensitivity to small fluctuations in their earthquake signals. The singularity spectrum curves of Gazli, Imperial Valley (II), and Borah Peake motions were symmetric, which indicates the equal extent of low and high fluctuations in their motion signals. Regulations 2800 (2014) suggest that accelerations in earthquake waves with small magnitudes, compared to those with large magnitudes, intensify the ground’s response. Therefore, they produce minute strains in the soil structure, leading to an escalation in the resonance probability in the soil. Accordingly, the resonances caused by Imperial Valley (II), Gazli, Borah Peake, Kobe, and Cape Mendocino motions were justifiable and expected.

The mean generalized Holder exponent α (0) values of Cape Mendocino, Borah Peake, Imperial Valley (II), and Gazli motions were less than 0.5, indicating long memory with anti-correlated conduct. Regarding Smart (Taiwan), Loma Prieta, El Alamo, Kobe, Trinidad, and Spitak (Armenia) motions, the mean generalized Holder exponent α (0) values were higher than 0.5, demonstrating the correlated attitude of ground motions (earthquake signals). The results reveal that all the ground motions with a mean generalized Holder exponent α (0) greater than 0.5 triggered resonance in the soil. The results also suggest that the anti-correlated behavior of fluctuations in earthquake waves (ground motions) enhanced the resonance likelihood in soil.

By computing ∆α values (Figs. 5 and 6) diagrams, the multifractal singularity spectrum width was measured and interpreted. The results show that the Kobe motion possessed the highest ∆α value, meaning that this motion had the most multifractal behavior, whereas the Spitak motion manifested the most inconsiderable multifractal behavior.

Fig. 6
figure 6

The ∆α values concerning the ten study ground motions

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The multifractal singularity spectrum width revealed that all ground motions held mono-scale structures, except for those of El Alamo, Kobe, and Loma Prieta, as these ground motions were multifractal. The results indicated that ground motions with a single-scale structure considerably increased the likelihood of resonance in the soil.

An examination of the results of phases 1 and 2 reveal that ground motions with either mono-scale or multi-scale and with symmetrical- or right-tail singularity spectrum augment the likelihood of resonance in soil. Even though dynamic properties and repetitive patterns in ground motions (earthquakes) had no effect on resonance frequency and period, they amplified the likelihood of resonance in soil. Since considerable resonances in soil bring about soil structure failure, one can conclude that ground motions escalate the probability of soil failure if ground motions have (1) right-tail or symmetrical singularity spectrum curves and (2) mono-scale structures.

The earthquake signals (ground motions) applied for the analysis in phases one and two were in the X direction. Comparable computations and analyses were performed for the earthquake signals in the Y and Z directions presented similar results. In other words, the dynamic properties of earthquake acceleration in all three axes (X, Y, and Z) had similar effects on resonance in the soil.

Studies on soil resonance conducted by the Roads, Housing, and Urban Development Research Center suggested that in addition to the soil structural properties (materials and thickness), the frequency and duration of the arrival of earthquake waves into the soil are significant. The present study broadened the general understanding of the impacts of the nonlinear dynamic properties of earthquake signals (ground motions, earthquake waves) on resonance in soil. The present study showed that the dynamic properties of ground motions play a role in resonance in soil.

To the best of the authors’ knowledge, no studies to date have examined the relationship between the nonlinear dynamic characteristics of earthquake waves and resonance in soil and soil failure.

Conclusion

This study incorporated the influence of nonlinear dynamic properties of earthquake acceleration (ground motions) on resonance in soil. The study is relevant due to the importance of the resonance in the soil for structural design. A soil profile was generated, and the effects of nonlinear dynamic properties of ground motions on soil resonance were investigated. The results indicated that all earthquakes, except those in El Alamo and Trinidad, produced resonance in the generated soil profile. Also, the critical periods during which the ground motions resonated were less than 0.1 s and from 1 to 8 s. Moreover, layers close to the bedrock (15 and 16) experienced the highest resonance, while layers close to the surface experienced inconsiderable resonance. If the earthquake signals (ground motions) meet at least one of the following conditions, the resonance probability in the soil increases significantly: (1) The earthquake signal has a right-tail or symmetrical multifractal singularity spectrum curve, (2) the earthquake signal holds a mono-scale structure, or (3) the earthquake signal exhibits long memory with anti-correlated behavior.

An analysis of the effect of earthquake acceleration in the Y and Z directions on soil resonance provided similar results to that in the X direction. Even though dynamic properties and repetitive patterns in ground motions (earthquakes) had no effect on resonance frequency and period, they amplified the likelihood of resonance in soil. This study revealed the influence of dynamic properties of earth movements on resonance in soil. Specifically, the results suggest that, in the case of structural design, when assessing the risk of resonance in the soil, only ground motions that include at least one of the above conditions should be considered. In other words, ground motions with the mentioned characteristics increase the possibility of creating a significant resonance in the soil, bringing about serious damage to structures. This study proposed a new perspective on resonance in soil, and future studies can examine the effect of intrinsic characteristics of ground motions on resonance in soil by considering statistical parameters that were not addressed in this study.