Failure Probability Of Structure Research Articles

System reliability analysis of Cable-stayed bridge using pair Copula and ICLHS with unequal weights under seismic loading

Considering the correlation between different components, quantifying the failure probability of a high-level hyper-static structure under non-stationary seismic loading is of great significance in guiding earthquake-resistant design and fortification of bridges. In this study, a high-pier and large-span railway cable-stayed bridge in the mountainous region were employed to evaluate the seismic performance. First, the basic theoretical methods of the non-stationary random seismic motion simulation, equivalent extreme value theory based on extreme value distribution (EVD), improved Latin hypercube sampling (ICLHS) with unequal weighted fractional moment estimation and Pair Copula technique are presented. Second, the finite element analysis model of the cable-stayed bridge was established based on the OpenSees platform, the seismic response samples of the bridge were calculated under three levels of seismic ground motion to determine the first passage failure probability of the main components of the bridge under the design failure modes. Finally, the Pair Copula function was introduced to estimate the seismic system reliability of the cable-stayed bridge. The optimal Copula type that can describe the correlation between the bridge components was selected through goodness-of-fit tests such as AIC, BIC, and the minimum distance method. The failure probability of the entire system was obtained by calculating the selected Copula function type under seismic loading.

Risk-based life cycle cost–benefit analysis for progressive collapse design of RC frame structures

Progressive structural collapse is a typical low-probability high-consequence event. Progressive collapse design can effectively improve the robustness of structures against accidental local damage and reduce the probability of collapse. However, progressive collapse design also increases the construction cost of buildings. Hence, it is necessary to conduct cost-benefit analyses for the progressive collapse design of building structures. This study first designs a reinforced concrete (RC) frame based on a progressive collapse design program that simultaneously optimizes material consumption and structural resistance. Then, the structural failure probability is quantified using an alternative path (AP) method-based fragility analysis that considers the structural uncertainties. Based on this, the life cycle costs and benefits of the structure designed with different design load factors are investigated. A four-story RC frame is used as an example. By considering the time value of money and threat probability, the break-even points and optimized design load factors for the progressive collapse design of the RC frame are analyzed. These research outcomes can help promote progressive collapse design in engineering practice and provide references for proposing scientific and reasonable design load factors.

A layer assigned probability space partition method for structural small failure probability problem

The Physical Synthesis Method (PSM) stands out as a robust framework for conducting structural reliability analyses due to its clear conceptual foundation. However, this approach often necessitates significant computational resources when addressing scenarios with small failure probabilities. In response to this challenge, this study introduces a layer assigned probability space partition method designed to identify pivotal points based on the ultimate bearing capacity failure criterion of structural components within the PSM framework. Drawing inspiration from Harbitz’s β-sphere, this method effectively utilizes the minimum reliability index of components to discern essential representative points within the probability space, thus streamlining computations. The efficacy of this approach is showcased through two case studies: a simply supported beam and a six-story reinforced concrete frame. The outcomes demonstrate that the proposed method, when integrated with PSM, exhibits a substantial enhancement in efficiency compared to the conventional Monte Carlo method. Besides, under equivalent computational resources, it achieves superior computational accuracy compared to the importance sampling method, particularly in scenarios with small failure probabilities. Furthermore, by introducing the notion of a common safe domain, this method addresses challenges in structural reliability analyses involving multiple failure surfaces.

The first-order time-variant reliability expansion method

Time-variant reliability problems are frequently encountered in engineering due to factors like material degradation or random loading modeled as random processes. The PHI2 method, which employs the First Order Reliability Method (FORM), is commonly used to solve such problems. However, it requires repeated searches for Most Probable Points (MPPs), making it computationally expensive. To improve efficiency with little sacrifice of accuracy, this study proposes a First Order Time-variant Reliability Expansion (FOTRE) method, which provides an efficient explicit formulation for MPP regarding time, in contrast to the expensive optimization approach of the PHI2 method. It requires only a single accurate search for the so-called “worst MPP” over the whole lifespan and offers the “adaptive accuracy of outcrossing rate”, which avoids the repeated search for MPPs ensuring computational accuracy. The inspiration behind the FOTRE method stems from the observation that the outcrossing rate tends to be small at time points with relatively large reliability indexes compared to the minimum reliability index βmin, which has a negligible impact on the subsequent structural failure probability over the entire lifespan. This innovative approach significantly improves the efficiency of solving time-variant reliability problems without compromising much of the numerical accuracy. The effectiveness and accuracy of the FOTRE method are demonstrated through several numerical examples.

Dynamic probability modeling-based inspection intervals optimization for civil aircraft composite structures using Bayesian updating

The evaluation of damage tolerance in composite materials is essential for ensuring the safety of aircraft structures. One of the most challenging aspects of applying probability modeling-based methods to evaluate damage tolerance is determining the actual damage size distributions for in-service aircraft structures. Although existing nondeterministic approaches have been used to optimize inspection intervals of composite structures, few studies have investigated the effects of updates on the actual damage size distribution and its impact on both the probability of structural failure and inspection intervals. This paper proposes a dynamic optimization method for inspection intervals of composite structures based on Bayesian updating. The damage size distribution of the composite structure is characterized by a general stochastic distribution. A Bayesian updating methodology is presented to iteratively update the actual damage size distribution whenever new data becomes available. Based on the constructed probability model, the inspection intervals of composite structures are determined under the objectives of optimal safety and economy for civil aircraft using a Monte Carlo approach. Compared to prior distribution models, the proposed method achieves higher safety for structures during a single inspection, reduces the failure probability of structures throughout their entire service life, and incurs lower maintenance costs. It also enables maintenance personnel to flexibly adjust inspection intervals while facilitating quantitative evaluation of both failure probabilities and maintenance costs associated with these intervals. These findings suggest that the proposed method holds great potential in enabling maintenance personnel to make informed decisions regarding inspection intervals for improved safety and economic performance.

Efficient Methods for Reliability Sensitivity Analysis of Distribution Parameters and Their Application

To efficiently evaluate the influence of the distribution parameters of the input variables on the failure probability of engineering structures and improve the reliability and safety of engineering structures in a targeted manner, new methods for the global reliability sensitivity analysis (RSA) of distribution parameters are proposed in this study based on the cubature formula (CF), surrogate sampling probability density function (SSPDF), and quasi-Monte Carlo (QMC) method. By introducing CF, the proposed methods can effectively improve the computational efficiency of the nested expectation and variance operators in the reliability sensitivity indices of the distribution parameters. Based on the concept of SSPDF, a surrogate importance sampling probability density function was developed. This not only overcomes the problem of the computational effort of propagating parameter uncertainty to the failure probability function (FPF), which depends on the dimensionality of the parameters; it also further improves the efficiency of the RSA of the parameters in the case of a small failure probability. Finally, by incorporating the idea of the QMC method, the process of calculating the reliability sensitivity indices of the parameters is reduced from a double-loop to a single-loop one. Three engineering examples are used in this study to demonstrate the efficiency and accuracy of the new algorithms.

Probabilistic fatigue life prediction of corroded PC beams considering spatial variability of pitting corrosion

Pitting corrosion intensifies local stress in materials, accelerating the fatigue crack initiation and propagation at the pit root. The randomly distributed corrosion pits can result in fatigue fractures in rebars and wires at non-critical sections, which further increases the fatigue failure probability of structures. However, previous studies have scarcely taken this into account. Here, we propose a novel model for fatigue life prediction of corroded post-tensioned prestressed concrete (PC) beams, considering the spatial variability of pitting corrosion. First, a Weibull distribution model of the pitting factor is developed based on the 3D laser scanning method. Then, the 3D non-Gaussian random field of the pitting factor in PC beams is generated using the inverse Nataf transformation. Next, the 3D pit-induced stress concentration is modeled using pit shape ratios. Following that, the initiation and propagation of fatigue cracks in pitting corroded wires and rebars are considered by the equivalent initial flaw size methodology. Finally, the fatigue analyses of the pitting corroded wires and rebars in all beam elements are conducted step by step until the load-bearing capacity of PC beams becomes lower than the applied load. The proposed prediction model is validated using the experimental data. The results reveal that neglecting the spatial variability of pitting corrosion overestimates the fatigue life of corroded PC beams, especially under high corrosion degrees.

A framework to demonstrate the utility and safety of the observational method

AbstractUnderground engineering projects are often characterised by significant uncertainties, leading to complex decision‐making when selecting appropriate design methods. Traditional approaches, such as the observational method (OM), have been widely used but rarely explicitly account for the integration of risk, costs, construction timeframes and engineering judgement. Successfully reaching a consensus to implement the OM on tunnelling projects can be challenging and requires ensuring stakeholders understand how the method impacts structural safety. Fostering such understanding can be an arduous process. A risk‐based decision framework is introduced using expected utility theory to aid the decision to choose the OM, or not. The framework integrates risk, cost, construction timelines and engineering judgement within an economic decision model using a probabilistic approach. It is tested on a Sydney, Australia case study. A Monte Carlo model analysis examines contingency actions and structural failure probability when applying the OM. The case study highlights the significance of considering indirect costs in complex decisions and harnesses experienced professionals’ insights for a comprehensive risk and cost assessment. We conclude that merging probabilistic‐based design with OM enhances the ability to demonstrate structural safety and economic factors within a unified model.

Investigation of seismic fragility curves of unbonded FREIs: Adaptive characteristics and modeling sensitivity

AbstractFiber‐reinforced elastomeric isolators (FREIs) are composed of layers of elastomer reinforced with either steel or fibers, and can be placed in a bonded or unbonded configuration between the upper and lower supports. The use of fiber reinforcement in FREIs was intended to reduce the production and installation costs compared to common steel‐reinforced elastomeric isolators (SREIs) and to develop an isolator suitable for widespread application, particularly in developing countries. The unique rollover deformation exhibited by unbonded FREIs (UFREIs), due to their flexible fiber reinforcement, enables them to adapt to multiple performance objectives at different hazard levels. In this study, the impact of different numerical models of UFREIs on the seismic response and failure probability of structures is investigated. The research employs incremental dynamic analysis and the development of fragility curves for different limit states, considering three sets of ground motions (including far‐field and pulse‐like). Moreover, the effect of full rollover was investigated by comparing the fragility curves of UFREIs when supported on modified support geometries, which involved three types of support: unmodified, accelerated, and delayed full rollover. The effectiveness of the adaptive characteristics to behave as a displacement restrain is demonstrated. The results emphasize the importance of employing an accurate model to simulate the behavior of UFREIs as an adaptive device for effectively utilizing their potential capacity, particularly at larger displacements where there is more dissipated energy due to full rollover.

Experiments of Lift-Bending Response on a Slender UAV Wing Structure with Control Surface under Extreme Flow Turbulence

The aeroelastic response of lightweight low-speed aircrafts with slender wings under extreme flow turbulence intensity is not well understood. Experiments on a commercial unmanned aerial vehicle (UAV) with a 3 m wingspan and aspect ratio of 13.6 were performed in a large open-return wind tunnel with extreme flow turbulence intensity of ≈10%. The wing bending displacement and the flow beneath the wing were measured by using laser-displacement sensors and tomographic particle image velocimetry (PIV) techniques, respectively. The unsteady lift produced by the wing was also measured by using a high-capacity load cell at an angle of attack of two degrees for three freestream velocities of 13.4 m/s, 17.9 m/s, and 26.8 m/s, representing the UAV’s stall speed, a speed approximately equal to the cruise speed, and a speed considerably higher than the cruise speed, respectively. It was found that a high flow turbulence intensity with large integral length scales relative to the wing chord plays a dominant role in the large unsteady lift and wing displacements measured. The power spectral density (PSD) of the wing structural vibration shows that flow shedding from the wing and the integral length scales have a significant impact on the overall power inherent in the bending vibration of the wing. Computations of the vorticity isosurfaces in the flow measurement volume surrounding the aileron reveal a striking observation: an aileron deflection of 10° becomes less effective in producing additional spanwise vorticity, which is proportional to circulation and lift at 26.8 m/s since the freestream already has elevated levels of vorticity. A paradigm shift in design is suggested for light aircraft structures with slender wings operating in highly turbulent flow, which is to employ multiple control surfaces in order to respond to this flow and mitigate large bending or torsion displacements and the probability of structural failure.

Spectrum-consistent [formula omitted]-based fragility curves

In the field of earthquake engineering, Fragility Curves (FCs) have become indispensable for regional risk assessments due to their efficacy in estimating the failure probabilities of diverse structures, including buildings and bridges, based on certain intensity measures. FCs are typically derived from either mechanical simulations or on-site observations, encapsulating crucial information about local stratigraphy and topography. The dependency of FCs on these local parameters, such as soil and topographic conditions, may vary depending on the selected intensity measure. When a spectral ordinate at a specific period is used as an intensity measure, FCs tend to be largely site-independent. On the other hand, if the intensity measure is chosen to be the acceleration ag at the bedrock substrate, as often occurs, then the FC is site-dependent. This distinction implies that identical structures at different locations, or even the same location with different soil and topographic conditions, have different FCs. This can significantly influence the results of regional risk studies, as using FCs for a specific building type at diverse locations with different hazards, stratigraphy, and topography can result in significantly different failure probabilities. The goal of this study is to develop a method so that ag-based FCs, developed at a certain location, can be used at any other location and on any other soil, preserving consistency in terms of spectral ordinate. This is done by a fully analytical approach, using a spectrum-consistent transformation. As an application, risk maps for Italy are generated and compared with an alternative method based on shifting the hazard curve.

Failure Probability and Economic Loss Assessment of a High-Rise Frame Structure under Synthetic Multi-Dimensional Long-Period Ground Motions

Multiple research studies and seismic data analyses have shown that multi-directional long-period ground motion affects crucial and intricate large-scale structures like oil storage containers, long-span bridges, and high-rise buildings. Seismic damage data show a 3–55% chance of long-period ground motion. To clarify, the chance of occurrence is 3% in hard soil and 83% in soft soil. Due of the above characteristics, the aseismic engineering field requires a realistic stochastic model that accounts for long-period multi-directional ground motion. A weighted average seismic amplification coefficient selected NGA database multi-directional long-period ground motion recordings for this study. Due to the significant low-frequency component in the long-period ground motion, this research uses empirical mode decomposition (EMD) to efficiently decompose it into a composite structure with high- and low-frequency components. Given the above, further investigation is needed on the evolutionary power spectrum density (EPSD) functions of high- and low-frequency components. Analyzing the recorded data will reveal these functions and their corresponding parameters. Proper orthogonal decomposition (POD) is needed to simulate samples of high- and low-frequency components in different directions. These samples can be combined to illustrate multi-directional long-period ground motion. Representative samples exhibit the seismic characteristics of long-period multi-directional ground motion, as shown by numerical examples. This proves the method’s engineering accuracy and usefulness. Moreover, this study used incremental dynamic analysis (IDA) to apply seismic vulnerability theory. This study investigated whether long-period ground motions in both x and multi-directional directions could enhance the seismic response of a high-rise frame structure. By using this method, a comprehensive seismic economic loss rate curve was created, making economic loss assessment clearer. This study shows that multi-directional impacts should be included when studying seismic events and calculating structure economic damages.

Stochastic critical buckling speed analysis of rim-driven rotating composite plate using NURBS-based isogeometric approach and HSDT

This article is the first attempt in the open literature to investigate the impact of uncertainty in material properties on the critical buckling speed of rim-driven rotating laminated plates. The formulations are constructed using the isogeometric approach (IGA) and higher-order shear deformation theory (HSDT). Coriolis and centrifugal forces resulting from rotation are taken into consideration. The results of the current formulation are compared to the previous results and validated. The accuracy and flexibility of the RBFN-based surrogate model, designed for uncertainty analysis, is verified by comparing the result with Monte Carlo simulation (MCS). Through a variety of parametric experiments, the critical buckling speed of rotating laminates driven by a rim is stochastically analyzed. Variance-based sensitivity analysis is employed to identify the influence of individual material properties on the critical buckling speed. Further, the impact of sensitive properties on the probability of structural failure is estimated. This research can be used to forecast the probability of failure of a structure at various levels of material uncertainty associated with it.

Stochastic Excitation-Incorporated Seismic Fragility Assessment Using Monte Carlo Simulations

Structures are susceptible to damage from natural disasters like tornadoes, floods, and seismic events. A resilient structure can withstand such damage and ensure cost-effectiveness. Therefore, accurately assessing the probability of structural failure is imperative for structural designers. This study employs an innovative fragility methodology to assess seismic damage in structures. In comparison to conventional fragility approaches, this method is cost-effective, universally applicable across various seismic sources, and highly precise. Specifically, this method involves the generation of artificial seismic records using the point-source stochastic seismological model and Monte Carlo simulation method. These synthetic seismic records offer distinct advantages when compared to the seismic data collected from real earthquakes at recording stations, such as the ability to adjust the epicentral distance of the synthetic waves as needed. Therefore, this approach enables a comprehensive analysis of structural responses under seismic waves characterized by different attributes. For example, the influence of different seismological parameters such as wave magnitude and epicentral distance are well studied and compared. The advanced finite element software, OpenSees, serves as the computational platform for examining how structures react to diverse seismic waves. The research findings suggest that, compared to traditional methods, utilizing OpenSees can significantly enhance computational efficiency. Furthermore, the methods employed introduce a novel technique for factoring in wave magnitudes and distances during the calculation of structural seismological responses/fragility. This innovation holds significance for the development of next-generation structural resilience designs.

Quantitative reliability measures determination of a steel-reinforced concrete beam taking into account fatigue damage accumulation in flexible pin stops

Reliability is one of the most important properties of road bridges due to their degree of responsibility and strategic importance for the transport system. For steel-reinforced concrete bridges, which are widespread nowadays, the question of their overall structure reliability dependence on the element’s condition of steel beam and reinforced concrete slab connection remains not fully studied. The authors have performed fatigue damage accumulation numerical modelling in the conjunction elements of steel-reinforced concrete beam from the cyclic temporary loads influence and analysed this process influence on the overall structure reliability. Based on the previous studies data, a relationship model between the conjunction element shear rigidity and the amount of damage accumulated by it is proposed. The nonlinear finite element calculation algorithm with regard to the conjunction element’s rigidity reduction is proposed and tested on the model of the steel-reinforced concrete superstructure. The selection method of the equivalent time load for modelling the accumulation of fatigue damage process from traffic flow is proposed. In order to optimise the computational resources used it is proposed to use loading process clusterization into enlarged cycles groups and the rational group size is determined. A numerical experiment on the simply supported steel reinforced concrete beam model with different loading cycles number (from 1 loading to 100 million loading cycles) to determine the reliability index β under the assumption of variability of two parameters — design steel resistance and the value of temporary load. FORM (First Order Reliability Method) was used for probabilistic analysis. The study results have shown that the accumulation of fatigue damage in the elements of steel-reinforced concrete superstructures has a significant effect both on the load-bearing capacity of the superstructure beams and on the quantitative indicators of structural reliability (increase in the probability of structural failure, decrease in the reliability index β).

Failure Probability Prediction for Offshore Floating Structures Using Machine Learning

Summary Accurately estimating the failure probability is crucial in designing civil infrastructure systems, such as floating offshore platforms for oil and gas processing/production, to ensure their safe operation throughout their service periods. However, as a system becomes complex, the evaluation of a limit state function may involve the use of an external computer solver, resulting in a significant computational burden to perform Monte Carlo simulations (MCS). Moreover, the high-dimensionality of the limit state function may limit efficient sampling of input variables due to the “curse of dimensionality.” To address these issues, an efficient machine learning framework is proposed, combining polynomial chaos expansion (PCE) and active subspace. This will enable the accurate and efficient evaluation of the failure probability of an offshore structure, which typically involves a large number of uncertain parameters. Unlike conventional PCE schemes that use the original random variable space or the auxiliary variable space for building a surrogate model, the proposed method utilizes a reduced-dimension space to circumvent the “curse of dimensionality.” An appropriate coordinate transformation is first sought so that most of the variability of a limit state function can be accounted for. Next, a PCE surrogate limit state function is constructed on the derived low-dimensional “active subspace.” The Gram-Schmidt orthogonalization process is used for making basis polynomial functions, which is particularly effective when input random parameters do not follow the Askey scheme and/or when a dependence structure between the input parameters exists. Therefore, a nonlinear iso-probabilistic transformation, which makes the convergence of a surrogate to the true model difficult, is not required, unlike traditional PCE. Numerical examples, including limit state functions related to structural dynamics problems, are presented to illustrate the advantages of the proposed method in estimating failure probabilities for complex structural systems. Specifically, the method exhibits significantly improved efficiency in estimating the failure probability of an offshore floating structure without compromising accuracy as compared to traditional PCE and MCS.

A loading contribution degree analysis-based strategy for time-variant reliability analysis of structures under multiple loading stochastic processes

This paper proposes a new single-loop method based on the loading contribution degree analysis (LCDA) and decoupling strategy for analyzing the time-variant reliability of structures considering multiple loading stochastic processes and degradation processes. In this method, the loading stochastic processes are first discretized using the expansion optimal linear estimation method. Then, the LCDA is carried out to obtain a total weighted loading stochastic process used to accurately reflects the structural response, thus constructing a single loading stochastic process analysis problem. Subsequently, a decoupling strategy is applied to decouple the degradation processes and the total weighted loading stochastic process to obtain the equivalent extreme-value limit state function (EEV-LSF) of the structure. Finally, an efficient probability distribution reconstruction tool is applied to derive the probability density function of EEV-LSF. Notably, the proposed LCDA can avoid the calculation of the response time series of a structure under the input loading stochastic processes, providing a novel perspective to simplify the calculation of structural extreme-value response. Additionally, the proposed method requires only one round of deterministic analysis to obtain the structural time-variant failure probability at the specified time, constructing an efficient single-loop analysis method. Three numerical examples are used to validate the proposed method.

Research on seismic fragility of subway station based on incremental dynamic analysis method

Seismic intensity measure is an significant parameter that affects the accuracy of structural seismic risk assessment, and plays an important role in the performance-based earthquake engineering research framework. Taking a two story, three span, and shallow buried subway station as the research object, a two-dimensional finite element analysis model of soil structure underground in a typical engineering site was established using ANSYS. Ten earthquake records were selected from PEER as input, and the structural seismic fragility analysis method based on incremental dynamic analysis (IDA) method was introduced into the subway station structure. The maximum story drift angle was used as the structural damage index, and three typical seismic intensity measures were selected, the exploration is conducted on the seismic intensity measures applicable to shallow buried subway stations and the variation pattern of seismic response of subway stations with increasing seismic amplitude. The seismic fragility curve of shallow buried subway station structures is established to obtain the failure probability of the structure at different seismic intensity levels. The analysis results indicate that for shallow buried subway stations, PGA is the most suitable seismic intensity measure, and the comparison with existing empirical fragility curves shows that the vulnerability analysis method is feasible and can quantitatively provide the failure probability of structures at different performance levels, providing reference for seismic design of underground structures.

Shakedown-reliability based fatigue strength prediction of parts fabricated by directed energy deposition considering the microstructural inhomogeneities

For mechanical components made by additive manufacturing (AM) techniques such as directed energy deposition (DED), micropores and other defects prevalently exist in the microstructure and they significantly reduce the reliability of these parts. To allow the influence of these microstructural features over the load bearing capacity to be considered in the design stage of AM structures, in the present paper we developed a multiscale numerical approach. The goal of this approach is to predict the failure probability of AM structures subjected to time-varied loadings, and it is realized through combining statistical homogenization, shakedown analyses, and reliability methods such as first-order reliability method and Monte Carlo simulation. Based on this strategy, we employed statistics of material parameters obtained from micromechanical models as inputs, implemented numerical tools and applied them to two exemplary structures, the plate with a hole and an aircraft bracket. Through a series of case studies carried out using different methods and under different assumptions of material randomness, the paper confirmed the robustness of the derived results and explained the mechanism of how micropores influence the structural reliability. The method developed in this paper can be a viable means for the design and optimization of metallic and metal matrix composite structures produced by AM techniques.

Study on the freeze–thaw effect on concrete arch dam using an improved response surface method

The damming materials will be deteriorated during the long-term service of concrete arch dam by weathering, seepage dissolution, freeze–thaw and other factors which will put dams at potential risk. However, there are few researches on the reliability assessment of arch dams considering the effect of material deterioration. In this paper, a prediction model of concrete mechanical properties under freeze–thaw cycles is established using regression analysis. Considering various uncertain factors, a time-varying structural reliability analysis model is established using finite element equivalent stress method and an optimized response surface method with improved sampling strategy and convergence criterion. A case study is later carried out. The results show that, at the early stage of freeze–thaw, the structural failure probability will increase steadily in a small range. However, as the tensile strength of concrete continues to decrease under long-time freeze–thaw action, the reliability of the arch dam structure will suddenly fall off a cliff. The proposed deterioration model and reliability analysis probability model are reasonable and practical and could be serve as a technical support for similar engineering projects.