# Fréchet-derivative-based global sensitivity analysis of the physical random function model of ground motions

*Dis Prev Res*2023;2:8.

## Abstract

Randomness in earthquake ground motions is prevalent in real engineering practices. Therefore, it is of paramount significance to utilize an appropriate model to simulate random ground motions. In this paper, a physical random function model of ground motions, which considers the source-path-site mechanisms of earthquakes, is employed for the seismic analysis. The probability density evolution method is adopted to quantify the extreme value distribution of structural responses. Then, the sensitivity analysis of the extreme value distribution with respect to basic model parameters is conducted via a newly developed Fréchet-derivative-based approach. A 10-story reinforced concrete frame structure, with nominal deterministic structural parameters and subjected to random ground motions, is studied. The results indicate that when the structure is still in a linear or weakly nonlinear stage in the situation of frequent earthquakes, the model parameter called the equivalent predominate circular frequency is of the most significance, with an importance measure (IM) greater than 0.8. Nonetheless, if the structure exhibits strong nonlinearity, such as in the case of a rare earthquake, the equivalent predominate circular frequency remains highly influential, but the Brune source parameter, which describes the decay process of the fault rupture, becomes important as well, with an IM increased from around 0.2 to around 0.4. These findings indicate that the IMs of basic model parameters are closely related to the embedded physical mechanisms of the structure, and the change in the physical state of the structure may provoke the change of IMs of basic inputs. Furthermore, some other issues are also outlined.

## Keywords

*,*sensitivity analysis

*,*seismic analysis

*,*probability density evolution method

## INTRODUCTION

In practical applications, quantification of various engineering uncertainties has become one of the most crucial concerns in the process of structural design and analysis. In general, considering the difference in sources of uncertainty, uncertainty can be categorized into two aspects^{[1–5]}: (1) the natural variability from the structural parameters, such as the uncertainty in mechanical properties of structural materials, geometric parameters, model discrepancies, etc., and (2) the randomness of external excitations, such as earthquake ground motions. Although considering both the randomness of ground motions and the structural parameters is the most objective^{[6,7]}, it may be more convenient for the preliminary structural design and evaluation to only take into account the randomness of ground motions^{[8–10]}. In this view, only the uncertainty of ground motions is focused on in this work.

For seismic analysis, the simulation of ground motions plays a significant role in the dynamical analysis of structures, especially when typical earthquake records of past strong ground motions may not be available for most engineering sites. To this end, numerous ground-motion models have been developed in the past decades, mainly by following three distinct pathways. The first path belongs to the category of seismology models^{[11–14]}, which predicts the earthquake ground motions by modeling the physical mechanisms, including the source, propagation medium, etc. The second and perhaps the most classical path in the engineering field is to adopt a given response spectrum or a power spectrum, on which a lot of fundamental and representative works have been done by Housner (1947)^{[15]}, Kanai (1957)^{[16]} and Tajimi (1960)^{[17]}, Hu and Zhou (1962)^{[18]}, Ou and Niu (1990)^{[19]}, Clough and Penzien (1995)^{[20]}, etc. This approach falls within the scope of engineering models, which aim at describing the second-order statistic characteristics of earthquakes from ground-motion records, and the non-stationarity of earthquakes is simulated by adopting a modulated function in the time domain^{[21,22]}. Different from seismology models, engineering models are more concerned with the influence of engineering sites on the ground motion. However, it has been confirmed that the physical mechanisms of the source and the complex propagation path also have critical impacts on the seismic response of engineering structures. For this reason, the third path is to combine the benefits of seismology models and engineering models, known as engineering seismic models, which focus on onshore earthquakes^{[23–26]} and offshore earthquakes^{[27–30]}. Considering the local site effect, Li and Ai (2006)^{[31]} proposed the idea of a physical random function model to reconstruct non-stationary stochastic ground motions. Building on this foundation, Wang and Li (2011)^{[23]} developed a physical random function model of ground motions (hereinafter referred to as "StoModel"). This model incorporates the randomness of the source and the site through four random variables that have specific physical interpretations. Then, the distributions of these four random variables can be identified based on actual earthquake records^{[32]}. Nevertheless, it is found that the distribution parameters of the random variables in StoModel may be significantly different^{[33]} when the statistical uncertainty from data of ground motions is involved. Therefore, there is a need to establish a more robust StoModel that holds the capability to reflect the randomness of earthquakes under various conditions of data. It should be emphasized that the adopted StoModel may not be the best ground-motion model currently available, but it is simple enough to help illustrate the present work in this paper.

In fact, the robustness of a stochastic model can be partially enhanced through the application of the global sensitivity analysis (GSA)^{[34]}. For instance, setting non-influential input variables of a stochastic model to nominal values would help decrease the statistical uncertainty arising from data, thereby enhancing the practical robustness. One of the quantitative measures in the GSA is the global sensitivity index (GSI). Among a variety of GSIs, the variance-based Sobol' index^{[35,36]} and the moment-independent index^{[37,38]} are two popular GSIs that have been applied in structural engineering^{[39]}, aerospace engineering^{[40]}, geotechnical engineering^{[41]}, and other fields. The variance-based GSI measures the contribution of each basic input (or the interaction effect of two or more inputs) on the variance of the quantity of interest (QoI), while the moment-independent GSI is defined on the stochastic distance between unconditional distribution and conditional distribution. Apparently, these two indices are always non-negative, indicating that they do not provide information about the direction of sensitivity. As a result, there is a risk of being misled into assuming that an increase in the uncertainty of inputs will invariably result in an increase in the uncertainty of outputs. In fact, the direction of sensitivity might be more essential than the importance measure (IM) when dealing with particular issues of engineering. For instance, the failure-probability-based GSI, defined as the derivative of failure probability with respect to basic distribution parameters, plays a crucial role in reliability-based design optimization^{[42]}. While GSA is supposed to provide adequate information for revealing the global physical features of stochastic systems, it is noticed that considering only second-order moments or failure probability may not be adequate. Therefore, it is reasonable to adopt a GSI that can effectively capture both the IM and the direction of sensitivity with respect to the probability distribution, rather than solely relying on second-order moments or failure probability.

In this paper, We conduct the GSA of StoModel using a Fréchet-derivative-based approach^{[43]}. The Fréchet-derivative-based GSI (Fre-GSI) is employed as the measure. To reduce computational costs associated with calculating the Fre-GSIs, we incorporate the probability density evolution method (PDEM) and the change of probability measure (COM)^{[4]}. To investigate the GSA of structural responses to basic random variables in StoModel, we use a typical high-rise reinforced concrete structure as the benchmark. The results of Fréchet-derivative-based GSA (Fre-GSA) provide insights into improving the robustness of the StoModel, and these improvements are discussed in detail.

## METHODS

### Physical random function model of ground motions

The StoModel studied in this research is based on the source-path-site mechanisms. Specifically, this model consists of two physical models: the amplitude spectrum model ^{[23]}

where the amplitude spectrum model

where

In the StoModel, ^{[24]}. The reader interested in the randomness of *et al*. (2018, 2022)^{[25,26]} for further information.

According to the site classification recommended in the Chinese code for seismic design of buildings (GB 50011-2010)^{[44]}, the marginal probability density functions (PDFs) of ^{[32]} using a database of 4438 seismic ground motions from the Pacific Earthquake Engineering Research Center (PEER). The assumption of Independence is made by

Probabilistic information of the physical random function model of ground motions^{[32]}

Random variable | Distribution type | Distribution parameters | ||

Lognormal | Site | |||

I | ||||

II | ||||

III | ||||

IV | ||||

Lognormal | Site | |||

I | ||||

II | ||||

III | ||||

IV | ||||

Gamma | Site | |||

I | ||||

II | ||||

III | ||||

IV | ||||

Gamma | Site | |||

I | ||||

II | ||||

III | ||||

IV |

Figure 1. PDFs of parameters of physical random function model of ground motions according to the site classification in the Chinese design code (GB 50011-2010)^{[44]}.

It is evident that the distribution parameters of basic random sources vary greatly for different site classes. This variation can be attributed to both the physical characteristics of different site classes and the statistical uncertainty originating from the earthquake ground motions. In other words, the distribution parameters derived from the analysis of ^{[32]} may possess epistemic uncertainty, as demonstrated in the study by Li & Liu (2015)^{[33]} where different distribution parameters are estimated from the records of 2008 Wenchuan earthquakes. To this regard, it is valuable to study how these uncertainties may affect the uncertainty of structural responses, such as maximum inter-story drift angle, top displacement, etc.

In Table 1, the PDFs of Lognormal distribution and Gamma distribution are given by

Besides, for the sake of simplicity, the distribution parameters in Table 1 are numbered in order as follows:

### Uncertainty propagation via the probability density evolution method

In this section, We introduce the basic theory and numerical algorithm of the PDEM^{[45]}, which is adopted to estimate the PDF of the QoI.

Without loss of generality, Let us consider a MDOF structure with the equation of motion given by:

where

Let

where ^{[46]}, the joint PDF of ^{[2]}

which is referred to as the generalized density evolution equation (GDEE). Finally, the PDF of QoI can be calculated by integrating

where

In general, the numerical algorithm of the PDEM consists of the following four steps:

**Step 1.1.** Generation of representative points. Denote

The way to partition the sample space can be referred to Chen *et al*. (2009)^{[47]}. Besides, to minimize the point discrepancy of representative points, the GF-discrepancy minimization strategy^{[48]} is adopted in this work.

**Step 1.2.** For each

**Step 1.3.** For each

with the initial condition ^{[45]}.

**Step 1.4.** Assemble the results in **Step 1.3**, i.e.,

### Uncertainty propagation via the change of probability measure

The aforementioned PDEM is available only if the input PDF is precisely determined. In other words, when the input PDF denoted by ^{[4]} is briefly introduced in this section. The combination of PDEM and COM is essential for a quick Fre-GSA in Section 2.4.

The backbone of the COM is based on the Radon-Nikodým theorem, which ensures that

where

where

For some simple stochastic systems, the analytical formula of Radon-Nikodým derivative can be found in Chen & Wan (2019)^{[4]}. Nevertheless, it is always impossible to obtain an exact expression of Radon-Nikodým derivative for complex and nonlinear stochastic systems, but the COM can be numerically accomplished with the aid of the PDEM.

The numerical algorithm of the PDEM-COM is summarized as follows:

**Step 2.1.** Complete one round of probability density evolution analysis via PDEM introduced in Section 2.2. Store the point set

**Step 2.2.** Considering the input PDF is changed from

where

**Step 2.3.** Solve the GDEE in Equation (10) with a new initial condition

It should be emphasized that the accuracy of the PDEM-COM depends on whether the support of *et al*. (2023)^{[49]}.

### Fréchet-derivative-based global sensitivity analysis

The Fre-GSA provides a quantitative approach to identify the most influential variables of input for stochastic systems. In this analysis, a Fre-GSI is calculated, and its parametric form is defined by^{[43]}

and the

where

For the ^{[50]}

which theoretically satisfies that

The

**Step 3.1.** Firstly, calculate

**Step 3.2.** Calculate

**Step 3.3.** Approximate the

where the norm term in the denominator can be numerically or analytically computed^{[4]}.

## ENGINEERING APPLICATION

The aim of this paper is to investigate how the distribution parameters of the StoModel may affect the stochastic responses of the structure by adopting the Fre-GSA. To achieve this goal, a 10-story reinforced concrete frame structure, as shown in Figure 2A, is considered. The finite element model of the structure is modeled via the OpenSees software. The constitutive model of concrete materials is described by the elastoplastic damage constitutive model^{[51]} ($$\textsf{ConcreteD}$$ command), which is consistent with the Chinese design code (GB 50010-2010)^{[52]}. The behavior of steel materials is characterized via the Giuffré-Menegotto-Pinto model^{[53]} ($$\textsf{Steel02}$$ command), which accounts for the effect of isotropic strengthening. The stress-strain curves of the concrete and steel materials are shown in Figure 2B and Figure 2C, respectively. The labels "Compressive" and "Tensile" stand for the compressive state and the tensile state of the concrete materials, respectively.

Assume the seismic fortification intensity is categorized as ^{[44]}, the peak ground accelerations (PGAs) of the transient dynamic analysis are assigned to

Comparisons of the dynamic amplification coefficients via the StoModel and the Chinese design code^{[44]} for four site classes are drawn in Figure 3. Note that in Figure 3, the Y-axis ^{[48]}. The results show that the mean response obtained from the StoModel is basically consistent with the one from the design code, and the range of the StoModel (mean value

Figure 3. Comparisons of the dynamic amplification coefficients via the physical random function model of ground motions (StoModel) and the Chinese design code (GB 50011-2010)^{[44]}

The results for the case of a frequent earthquake (PGA

In contrast to the case of a frequent earthquake, the PDF of the extreme top displacement becomes sharper as we move from Site I to Site IV, while the amplitudes of the Fre-GSIs turn out to be smaller. This difference may be attributed to the much stronger development of the structural nonlinearity in the case of a rare earthquake, as shown in Figure 6. The results for the rare earthquake (PGA^{[2]}.

## CONCLUSIONS

In this paper, We investigate the sensitivity of parameters in the StoModel by measuring the Fre-GSI. Numerical computation of the Fre-GSI is sharply accelerated by integrating the PDEM and the COM. As a benchmark model, we analyze a 10-story reinforced concrete frame structure while considering the consistency of the StoModel with the Chinese design code (GB 50011-2010)^{[44]}. The main conclusions of this study are as follows:

1. The StoModel is statistically consistent with the Chinese design code, in terms of the dynamic amplification coefficient.

2. Once the PDF of the Qol is estimated by the PDEM, the Fre-GSI can be obtained as a byproduct that can be rapidly computed via the COM.

3. For the case of a frequent earthquake, when the mechanical behavior of the structure is nearly linear, the parameter

4. For the case of a rare earthquake, when the structure enters a highly nonlinear stage, although

5. It is suggested that more information on the parameters

More research is needed to address certain issues. For instance, studies are still being conducted to better describe the randomness of ground motions using more realistic physical random functions and to take into account the inherent uncertainty in structural parameters. Additionally, the adopted ground-motion model in this paper also needs further improvements, particularly in aspects related to the physical mechanisms of the source model, path model, local-site effects, etc. Moreover, model uncertainty of the ground-motion model is a concerning factor that requires attention in future research efforts.

## DECLARATIONS

### Acknowledgments

The financial support provided by the National Natural Science Foundation of China (NSFC Grant No. 52208206) and the Fundamental Research Funds for the Central Universities (Grant Nos. G2022KY05103) is highly appreciated.

### Authors' contributions

Conceptualization, Investigation, Methodology, Formal analysis, Software, Writing - original draft, and Writing - review & editing: Wan Z

Resources, Funding acquisition, and Data curation: Tao W, Ding Y, Xin L

### Availability of data and materials

Some or all of the data, models, or code generated or used during the study are available from the author upon request.

### Financial support and sponsorship

The National Natural Science Foundation of China (NSFC Grant No. 52208206);

The Fundamental Research Funds for the Central Universities (Grant Nos. G2022KY05103).

### Conflicts of interest

All authors declared that there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2023.

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Wan Z, Tao W, Ding Y, Xin L. Fréchet-derivative-based global sensitivity analysis of the physical random function model of ground motions. *Dis Prev Res* 2023;2:8. http://dx.doi.org/10.20517/dpr.2023.13

**AMA Style**

Wan Z, Tao W, Ding Y, Xin L. Fréchet-derivative-based global sensitivity analysis of the physical random function model of ground motions. *Disaster Prevention and Resilience*. 2023; 2(2): 8. http://dx.doi.org/10.20517/dpr.2023.13

**Chicago/Turabian Style**

Zhiqiang Wan, Weifeng Tao, Yanqiong Ding, Lifeng Xin. 2023. "Fréchet-derivative-based global sensitivity analysis of the physical random function model of ground motions" *Disaster Prevention and Resilience*. 2, no.2: 8. http://dx.doi.org/10.20517/dpr.2023.13

**ACS Style**

Wan, Z.; Tao W.; Ding Y.; Xin L. Fréchet-derivative-based global sensitivity analysis of the physical random function model of ground motions. *Dis. Prev. Res.* **2023**, *2*, 8. http://dx.doi.org/10.20517/dpr.2023.13

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