Since double sources of uncertainties are usually involved in practical engineering – namely, the epistemic and aleatory uncertainties – it is becoming increasingly crucial to incorporate these double sources of uncertainties into structural reliability analysis in order to ensure the safety of structures.

In this paper, a two-level Active-learning Kriging (AK) meta-model approach is put forward to address the challenges of imprecise structural reliability analysis, where the epistemic and aleatory uncertainties are characterized by using a parameterized probability-box (p-box) model. At the inner loop, a new learning function called the Relative Entropy Function (REF) is proposed to enhance the active learning process. The proposed REF facilitates the selection of informative points efficiently, accelerating the overall AK with Monte Carlo Simulation (AK-MCS) process. In that regard, the proposed AK-MCS with REF is effective in estimating failure probabilities with high accuracy in a precise probability sense. Moving to the outer loop, another Kriging meta-model is established to relate the distribution parameters within the p-boxes to the conditional failure probabilities. This outer-loop model allows for efficient estimation of the failure probability bounds via the efficient global optimization.

The efficacy of the proposed method is verified through four numerical examples, which include a finite-element model. A pertinent double-loop MCS is employed to obtain comparative results. Furthermore, the proposed method is applied to structural progressive collapse analysis, serving as a guide for robustness-based design.

The computational results demonstrate that the proposed method is effective in dealing with structural reliability problems involving double sources of uncertainties.

Uncertainty is a common occurrence in various aspects of practical engineering, including structure design, manufacturing, operation, and maintenance. While individual uncertainties may have minimal impacts on structural response, the combination of multiple uncertainties can lead to significant deviations in the structural behavior^{[1]}. As a result, it is crucial to perform structural reliability analysis, particularly for critical structures such as nuclear power plants, hospitals, and monuments, where structural failure can pose significant safety risks and have monumental consequences.

In practical engineering, uncertainties can generally be classified into two categories: aleatory and epistemic uncertainties^{[2–4]}. On the one hand, aleatory uncertainty, also known as probability uncertainty, is inherent in nature and beyond our control. It is characterized by a level of uncertainty that cannot be reduced or eliminated. On the other hand, epistemic uncertainty, also referred to as subjective uncertainty, arises from insufficient knowledge of variables and a lack of comprehensive experimental data. As knowledge and information increase, epistemic uncertainty tends to decrease^{[5, 6]}. In practical engineering problems, it is crucial to recognize that our knowledge is inherently imperfect. As a result, accurately assessing the probability of a structure becomes challenging due to the presence of epistemic uncertainty. This means that even the reliability index, which is used to measure the safety and performance of a structure, is fundamentally uncertain.

Traditional reliability analysis methods may not be suitable for handling the problem at hand because they are designed to handle precise probability problems that assume complete probabilistic information of variables without accounting for aleatory uncertainty. To address this issue, non-probabilistic models and imprecise probability models have emerged as potential solutions. Non-probabilistic models, such as the interval model^{[7, 8]}, convex model^{[9–11]}, and fuzzy set theory^{[12]}, have been proposed. However, these models may not effectively distinguish between epistemic and aleatory uncertainties, which can result in sub-optimal decision-making. Imprecise probability models, on the other hand, have gained attention as a more effective way to handle uncertainties. These models include probability-boxes (p-boxes)^{[3, 13, 14]}, Dempster-Shafer evidence theory^{[15–17]}, interval probabilities^{[18]}, fuzzy probabilities^{[19]}. By utilizing imprecise probability models, it is possible to more accurately capture and express the uncertainties associated with the variables. However, using imprecise probability models comes with a significant challenge—computational cost. The large number of Limit State Function (LSF) evaluations, especially for time-consuming finite element analyses, can increase the computational effort significantly. Some other methods, such as nested Monte Carlo Simulation (MCS)^{[20]}, random sets^{[21]}, and advanced line sampling^{[22]}, have been developed to address the imprecise problems, but their computational efficiency remains unsatisfactory. To overcome the challenge of computational complexity, active learning-based meta-models, also known as surrogate models, have been introduced. These surrogate models aim to approximate the LSF with reduced computational effort while maintaining an acceptable level of accuracy. By using surrogate models, the computational burden associated with imprecise probability models can be significantly reduced, making them more manageable and efficient in practice.

In recent decades, significant strides have been made in the development and widespread application of various meta-models aimed at enhancing the precision of structural reliability analysis. Notably, the Kriging meta-model [also known as the Gaussian Process Regression (GPR) model] has garnered substantial attention and utilization in this field^{[23–25]}. Other prominent meta-models include response surfaces^{[26]}, support vector machines^{[27, 28]}, neural networks^{[29–31]}, and high-dimensional model representations^{[32]}. Among these, the Kriging meta-model stands out due to its exceptional qualities in precise interpolation and local uncertainty quantification. As a result, the active Kriging meta-model technology has found applications in addressing imprecise probability problems. Noteworthy contributions from researchers involve the fusion of Active-learning Kriging (AK) meta-models with advanced sampling techniques to evaluate failure probabilities^{[33–38]}. Nevertheless, efficiently estimating failure probability bounds for imprecise reliability problems remains a challenging endeavor, particularly when employing the AK model. This paper addresses this challenge by employing parametric p-boxes^{[14]} to accurately characterize the imprecise random variables involved.

This paper introduces a novel methodology to address imprecise reliability problems employing parametric p-boxes. The approach introduces an efficient learning function termed the "Relative Entropy Function (REF)" to enhance the AK meta-model. By optimizing the point selection strategy, the REF significantly accelerates the speed of active learning. The organizational structure of this paper is outlined as follows: In Section 2, the problem formulation is meticulously presented. Moving on to Section 3, a succinct review of the Kriging meta-model is offered, alongside its fusion with the REF, giving rise to an AK meta-model with MCS (AK-MCS) in the precise probability sense. Furthermore, the paper applies the devised AK-MCS with REF to tackle imprecise probability problems, facilitating the estimation of failure probability bounds. Section 4 demonstrates the effectiveness of the proposed method through the exposition of four numerical examples. To delve deeper into the applicability, Section 5 explores a practical implementation of the proposed method. Some concluding remarks are contained in the final section.

In structural reliability analysis, the LSF of a stochastic system mapping inputs to outputs is typically expressed as:

where

In parametric p-boxes, the PDF or CDF of

PDFs and CDFs of the parametric p-boxes.

In the precise reliability sense, the condition

where

In this case, the range of failure probability

where both

In this section, the classical Kriging meta-model is briefly revisited. Subsequently, a new learning function named the " Relative Entropy Function (REF)" is introduced. This REF is then combined with the Kriging meta-model to establish a novel active-learning algorithm termed AK-MCS-REF. This algorithm is further extended to deal with the problems of imprecise structural reliability analysis^{[14]}, where both epistemic and aleatory uncertainties are considered. The objective of this extension is to effectively determine the bounds of failure probability.

The Kriging meta-model, often categorized as a GPR, serves as a regression algorithm designed for spatial modeling and prediction (interpolation) of stochastic processes or random fields, relying on covariance functions^{[39]}. As outlined in^{[40]}, the Gaussian process is frequently applied to achieve Kriging predictions. This involves the assumption of a relationship between the real response of the system and the input variables, characterized by:

where the deterministic function ^{[41, 42]}, is chosen to model

The Kriging meta-model can be conceptualized as a process in which a Gaussian prior model is combined with observations to derive a posterior model. Given a training set ^{[43]}:

where ^{[44]}.

Consider a collection of unobserved points denoted as

where

Kriging meta-model prediction.

In the context of the Kriging meta-model integrated with active learning, the selection of the next best point is significantly influenced by the learning function^{[38]}. An effective learning function has the capacity to accelerate the refinement process of the Kriging meta-model while also ensuring result accuracy. This is particularly advantageous for intricate finite element models that demand substantial computational resources, as it can markedly enhance computational efficiency. Recent years have witnessed the emergence of several learning functions tailored specifically for reliability analysis.

Based on the Effcient Global Optimization (EGO)^{[43]}, the Effcient Global Optimization (EFF) was proposed for efficient global reliability analysis^{[45]}. The EFF elucidates the degree to which the actual value of the performance function at a specific point

The general formulation of EFF reads:

where

In the context of reliability analysis, significant attention is directed toward the limit state of the performance function. As a consequence, the value of ^{[24]}:

In recent advancements, a novel learning function rooted in information entropy has been introduced, as shown in^{[46]}. It is worth noting that information entropy serves as a gauge of the uncertainty level of the data: higher entropy values signify greater uncertainty, while lower values indicate reduced uncertainty.

The information entropy is defined as^{[47]}

where ^{[46]}.

The learning function ^{[46]}.

The detailed derivation of Equation (10) {can be found} in Appendix A.

The learning criteria and stopping criteria of the above three learning functions are compared in

The learning criterion and stopping condition of EFF, U, and H

EFF | ||

U | ||

H |

As previously mentioned, information entropy serves as a means to quantify the uncertainty associated with a random variable, rendering it a suitable tool for gauging the confidence in the model response value during Kriging prediction. In the active learning procedure, the realizations of random variables that affect the system with large uncertainty are used to construct a GPR, and then the Kriging prediction will be close to the true response of the system. This ensures that the Kriging prediction closely approximates the real system response, particularly within critical regions such as limit state areas. Subsequently, based on the stability criterion, a new learning function rooted in the Relative Entropy (RE) will be put forward.

RE finds its application in quantifying the disparity between two probability distributions. Within active learning algorithms, the probability distribution of the system response

The RE serves as a metric for assessing the disparity between

where

The REF learning function plays a pivotal role in gauging the stability of the estimated system response

To summarize, the proposed AK-MCS-REF framework entails a comprehensive set of seven steps for conducting structural reliability analysis in a precise probability sense:

Evaluate

If

In the context of parametric p-boxes, the distributions of the random vector

where ^{[48]}, can be employed in imprecise structural reliability analysis to address the complexities posed by parametric p-boxes. Within this hierarchical framework, the parametric p-boxes can be segregated into two distinct components. The AK-MCS with REF learning function, as detailed above, is well-suited for addressing the task of evaluating the conditional failure probability ^{[43]} can be utilized to optimize

Flowchart of AK-MCS-REF-EGO.

^{[43]}. Specifically, for

For

where

^{[43]} to estimate the bounds of failure probability.

where

In this section, the efficiency and accuracy of the proposed method are demonstrated through four numerical examples. The double-loop MCS^{[49]} and the proposed AK-MCS-REF approach are implemented using MATLAB. The Kriging meta-models are established using the GPR model, with the explicit basis and kernel function defined as Constant and ardsquaredexponential, respectively. Notably, the application of the ARD kernel proves beneficial for addressing problems with inputs of varying dimensions, as it allows for the consideration of diverse length scales across input dimensions^{[50]}. A detailed discussion for effects of using different kernels can be found in^{[51]}.

The LHS technique is utilized to generate the distributed parameter sample pool, while the Sobol sampling technique is applied to generate the sample pool of random variables. In the final numerical example, a comparison is made between the proposed learning function and the previously mentioned ones, namely, EFF^{[45]}, U^{[24]}, and H^{[46]}, all at the precise probability level. This comparison serves to further establish the superiority of the proposed learning function in effectively balancing calculation accuracy and efficiency, particularly for computationally intensive finite element models. The accuracy is evaluated using the relative error of the reliability index, given by:

where

A classic series system with four branches is first employed to validate the effectiveness of the proposed method^{[24]}. The LSF is defined as follows:

To validate the efficacy of the proposed learning function, a precise probability level analysis is initially conducted, where the variables

Experimental design in precise probability sense.

Results in precise probability sense

Note: _{call} |
||||

MCS | 3.1778 | - | ||

AK-MCS-REF | 3.1829 | 0.16 | 85 |

In the context of imprecise probability, the variables

The LHS technique is employed to generate the parameter sample pool within the domains of

Initial distribution parameters and their corresponding failure probabilities

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 |

Evolution of the bounds of reliability index (Example 1).

Bounds of reliability index for Example 1

Double-loop MCS | 2.9633 | - | 3.4090 | - | ||

Proposed method | 2.9794 | 0.55 | 3.4051 | 0.11 | 88+10+10 |

The second example employs a nonlinear LSF characterized by independent random variables, as detailed in^{[52]}. This LSF involves ten imprecise random variables and is represented as follows:

Likewise, the variables

Statistical information of random variables in Example 2

Normal | 0 |

As depicted in

Evolution of the bounds of reliability index (Example 2).

Bounds of reliability index for Example 2

Double MCS | 2.1139 | - | 3.2437 | - | ||

proposed method | 2.1274 | 0.64 | 3.2760 | 0.99 | 111+11+1 |

The applicability of the proposed method for non-monotonic functions is demonstrated through the utilization of a highly nonlinear undamped single-degree-of-freedom system [^{[24]}. The LSF for this case is expressed as follows:

Single-degree-of-freedom system.

The statistical characteristics of the input random variables are presented in

Statistical information of random variables in Example 3

Normal | 1 | 0.1 | |

Normal | 0.1 | 0.01 | |

Normal | 1 | 0.05 | |

Normal | 0.5 | 0.05 | |

Normal | |||

Normal |

As depicted in

Evolution of the reliability index boundary (Example 3).

Bounds of reliability index for Example 3

Double MCS | 2.0303 | - | 2.6614 | - | ||

Proposed method | 2.0546 | 1.2 | 2.6569 | 0.17 | 60+4+22 |

A practical validation is performed using a two-bay, four-story spatial concrete frame structure, aimed at demonstrating the practical engineering applicability of the proposed method. The structure takes into account the intricate nonlinear behaviors inherent to both concrete and rebar materials. To accurately capture the behavior of the system, a nonlinear beam-column finite element representation of each member is implemented within the OpenSees software^{[53]}. In accordance with

Two-bay four-story spatial concrete frame.

where

Statistical information of random variables in Example 4

Concrete compressive strength | Lognormal | 26.8 | 2.68 | |

Concrete strain at maximum strength | Lognormal | 0.0001 | 0.05 | |

Concrete crushing strength | Lognormal | 10.0 | 1 | |

Concrete strain at crushing strength | Lognormal | 0.0035 | 0.000175 | |

Yield strength of rebar | Lognormal | 355 | 355 | |

Initial elastic modulus of rebar | Lognormal | 200 | 20 | |

Strain-hardening ratio of rebar | Lognormal | 0.001 | 0.00005 | |

External load | Lognormal | 54 | 10.8 | |

External load | Lognormal | 54 | 10.8 | |

External load | Lognormal | 42 | 8.4 | |

External load | Lognormal | 42 | 8.4 | |

External load | Lognormal | 30 | 6 | |

External load | Lognormal | 30 | 6 | |

External load | Lognormal | 18 | 3.6 | |

External load | Lognormal | 18 | 3.6 |

^{[24]}, EFF^{[45]}, and H^{[46]}, to facilitate a comprehensive evaluation. Upon comparison, it becomes evident that the proposed learning function REF enhances computational efficiency without compromising accuracy. Furthermore, the AK-MCS-REF methodology exhibits superior performance compared to classical approaches such as IS and SS, demonstrating improved accuracy and efficiency in the context of reliability analysis in a precise probability sense.

Reliability results in precise probability sense

MCS | - | 3.0334 | - | ||

IS | 6.83 | 3.0134 | 0.66 | 1106 | |

SS | 18.28 | 2.9824 | 1.68 | 2800 | |

AK-MCS-U | 6.53 | 3.0538 | 0.67 | 135 | |

AK-MCS-EFF | 1.32 | 3.0294 | 0.13 | 175 | |

AK-MCS-H | 13.40 | 2.9953 | 1.26 | 76 | |

AK-MCS-REF | 1.24 | 3.0297 | 0.12 | 67 |

Next, the proposed method is applied to conduct imprecise reliability analysis. In this context, the loads are treated as imprecise random variables modeled using parametric p-boxes, with interval variables specifying their standard deviations, as shown in

Statistical information of imprecise random variables

Lognormal | 54 | ||

Lognormal | 54 | ||

Lognormal | 42 | ||

Lognormal | 42 | ||

Lognormal | 30 | ||

Lognormal | 30 | ||

Lognormal | 18 | ||

Lognormal | 18 |

Bounds of reliability index for Example 4

LHS-IS | 2.8208 | - | 3.0694 | - | 11204350 | |

Proposed method | 2.8638 | 1.52 | 3.0240 | 1.48 | 515+309+358 |

Evolution of the bounds of reliability index (Example 4).

This section delves into the imprecise reliability analysis of progressive collapse in a practical frame structure, showcasing the engineering application of the proposed method. The analysis takes into account both epistemic and aleatory uncertainties, providing a comprehensive assessment of the reliability of the structure under progressive collapse conditions.

The investigation involves a planar four-bay eight-story reinforced concrete frame structure, as depicted in ^{[54]}, the analysis focuses on key members. Eight distinct columns, denoted as

Four-bay eight-story reinforced concrete plane frame.

The nonlinear static pushdown method is utilized for analyzing the progressive collapse of structures that have been damaged due to column removal. The code DoD2013 specifies the combination of load effects to be employed in the pushdown method in the following manner:

where

where ^{[55]}.

The deformation criterion serves as the basis for assessing the progressive collapse resistance of damaged structures. When reinforcement rupture takes place, the vertical displacement of the failed joint node at the column removal typically ranges between ^{[56–58]}. For this study, the median value of

The equation represents the vertical displacement of the failed joint node, denoted as

Statistical information of random variables for progressive collapse analysis

Peak concrete stresses in non-core areas | Lognormal | 0.172 | ||

Peak concrete strain in non-core areas | Lognormal | 0.0018 | 0.15 | |

Ultimate concrete strain in non-core areas | Lognormal | 0.005 | 0.15 | |

Peak concrete stresses in core areas | Lognormal | 0.172 | ||

Peak concrete strain in core areas | Lognormal | 0.005 | 0.14 | |

Ultimate concrete stresses in core areas | Lognormal | 14.7 | 0.15 | |

Ultimate concrete strain in non-core areas | Lognormal | 0.02 | 0.15 | |

Yield strength of rebar | Lognormal | 0.072 | ||

Initial elastic modulus of rebar | Lognormal | 200 | 0.034 | |

Strain-hardening ratio of rebar | Lognormal | 0.02 | 0.1 | |

Dead load | Normal | 0.1 | ||

Live load | Normal | 0.47 |

The proposed method yields reliability index intervals for various working conditions, as presented in

Bounds of reliability index under various working conditions

C1 | 2.0103 | 2.5427 | |

C2 | 1.9954 | 2.5690 | |

C3 | 2.0047 | 2.5690 | |

C4 | 2.0084 | 2.6045 | |

E1 | 1.8250 | 2.3867 | |

E2 | 1.8184 | 2.3867 | |

E3 | 1.8055 | 2.3781 | |

E4 | 1.7829 | 2.3378 |

Histogram of reliability index interval under various working conditions.

Typically, robustness pertains to the ability of a structure to maintain satisfactory load-bearing capacity despite the random perturbation of specific parameters, ensuring overall performance meets the required standards. The assessment of structural robustness after a progressive collapse event employs the logarithmic ratio of anteroposterior failure probabilities. The robustness index is formally defined as follows^{[55]}:

where

For the sake of simplicity, the failure probability of the undamaged structure is assumed to range between ^{[59]}. Consequently,

Bounds of robustness index under different working conditions

0.563 | 0.558 | 0.561 | 0.563 | 0.500 | 0.498 | 0.494 | 0.487 | |

0.770 | 0.781 | 0.781 | 0.796 | 0.705 | 0.705 | 0.702 | 0.686 |

This paper presents a novel approach for conducting imprecise structural reliability analysis, addressing both epistemic and aleatory uncertainties through a hierarchical model represented by p-boxes. The proposed method introduces a new learning function based on RE within an AK-MCS. This new learning function, referred to as the REF, efficiently and accurately evaluates failure probabilities in the context of precise probability analysis.

The REF learning function effectively guides the selection of optimal sampling points, contributing to improved accuracy and efficiency when compared to existing learning functions. Notably, the ability of REF to determine optimal points enhances reliability assessment in precise probability analysis. Additionally, when tackling the challenge of calculating failure probabilities associated with distributed parameters within p-boxes, the proposed approach capitalizes on the information acquired from prior AK-MCS-REF iterations. This allows for leveraging the LSF evaluations of previous AK-MCS-REF as an initial DoE for subsequent iterations. Consequently, the establishment of a second-level Kriging meta-model becomes straightforward, centered on distribution parameters and their corresponding failure probabilities. The application of an EI function facilitates the identification of optimal distribution parameter points for further exploration. Ultimately, through an EGO algorithm, the method culminates in obtaining bounds for failure probabilities or reliability indexes.

The effectiveness and accuracy of the proposed method are demonstrated through four numerical examples, validating its performance. Furthermore, the method is practically applied to analyze the imprecise reliability of a frame structure under progressive collapse. Key findings include (1) the superiority of the REF learning function over existing alternatives, leading to enhanced accuracy and efficiency in precise probability analysis; (2) the substantial reduction in computational efforts compared to traditional double-loop MCS in imprecise reliability analysis; and (3) the utility of the method in providing practical insights for robustness design in scenarios such as a progressive collapse of engineering structures.

It is crucial to acknowledge that the proposed method comes with certain limitations that warrant consideration and potential future developments. Firstly, in scenarios where distribution parameters encompass large intervals or involve small-scale failure problems, the method may necessitate a substantial number of LSF evaluations to ensure desired accuracy. Consequently, this could result in time-intensive and resource-demanding computations. Secondly, the method's foundation on the Kriging metamodel implies that the challenge of dimensionality, commonly referred to as the curse of dimensionality, might still pose obstacles. This becomes especially pertinent as the number of interval and random variables exceeds 20. As such, accurately capturing intricate relationships between variables and the LSF could become more challenging in high-dimensional spaces. To address these limitations and enhance the method's applicability, future research endeavors may focus on refining strategies to mitigate the computational burden associated with large parameter intervals or small failure scenarios. Additionally, exploring advanced techniques for managing high-dimensional spaces, such as dimension reduction methods, could prove beneficial for more effectively handling complex relationships and improving efficiency.

Since

Actually, Equation (12) can be derived as follows

Let

In conclusion, the analytical expression of the learning function REF is

Made substantial contributions to the conception and design of the study and performed data analysis and interpretation: Du Y

Performed data acquisition and provided administrative, technical, and material support: Xu J

Some or all data, models, or codes generated or used during the study are available from the corresponding author by request.

The National Natural Science Foundation of China (Nos. 52278178, 51978253), Natural Science Foundation of Hunan Province, China (No. 2022JJ20012), the Open Fund of State Key Laboratory of Disaster Prevention in Civil Engineering, China (No. SLDRCE21-04) and Fundamental Research Funds for the Central Universities, China (No.531107040224) are gratefully appreciated for the financial support of this research.

All authors declared that there are no conflicts of interest.

Not applicable.

Not applicable.

© The Author(s) 2023.

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