Failure Probability Of Structure Research Articles

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.

Efficient estimation of structural probability of failure with adaptive kriging and multi-layered hyperball-based importance sampling

ABSTRACT The need for more resilient infrastructures entails an accurate and efficient structural reliability analysis. In recent years, Kriging-based meta-modeling has constantly been adopted to reduce the computational cost in estimating the probability of failure. Active learning models focus on most informative samples to increase computational efficiency. Nevertheless, the efficiency gained from the models may diminish when dealing with a rare event with a relatively small probability of failure. To resolve the difficulty, this study proposes a new method, Adaptive Kriging with Multi-layered Hyperball-based Importance Sampling (AK-MHIS), to estimate the structural probability of failure. AK-MHIS uses an adaptive multi-layered hyperball as its candidate sample pool with a new sample filtering mechanism and a more robust stopping condition for the active-learning phase. The performance of AK-MHIS method is validated in benchmark cases. The verification results confirm the superior performance of AK-MHIS to previous methods.

A conditional extreme value distribution method for dynamic reliability analysis of stochastic structures

An efficient post-processing simulation method is proposed to estimate small failure probabilities of stochastic dynamic structures involving the inherent randomness of structural physical-geometrical parameters and external excitations. To extract a small failure probability, the proposed method introduces an intermediate event to represent the realizations of extreme structural response in the tail of distribution. With the aid of this intermediate event, structural failure probability is reformulated as a product of the event’s occurrence probability and the conditional exceedance probability of extreme response when the event occurs. The latter corresponds to a distribution of extreme response under the condition of the intermediate event, referred to as the conditional extreme value distribution (CEVD). Accordingly, the proposed method is termed the CEVD method. To reconstruct the CEVD, a truncated shifted generalized lognormal distribution model is employed. Bayesian estimation method is utilized to determine the two shape parameters of this model based on the samples of both original extreme value distribution and CEVD, where the CEVD samples are generated by Markov chain Monte Carlo sampling. The efficiency and accuracy of the proposed method are demonstrated through two numerical examples considering seismic reliability analyses of a 10-story nonlinear frame and a soil-foundation-structure interaction system.

Novel evaluation methodology for mechanical behaviour and instability risk of roof structure using limited investigation data

ABSTRACT How to use limited investigation data to analyse the failure of roof engineering is a challenging problem. In this paper, a novel risk analysis method for the mechanical structure of a roof is proposed. Firstly, the Copula theory is presented and the construction method of multidimensional Gaussian Copula parameters is given. Secondly, a Copula method of the mechanical characteristics and instability risk of roof structure with E, ν, c and φ as uncertain variables is proposed. Thirdly, based on the investigation data of 192 roof groups in China, the influence of Copula mechanical parameters on the failure probability of the roof structure is analysed and discussed. This new evaluation methodology can use various Copula functions to simulate the positive and negative correlation structures, which provides an effective way to clarify the mechanical characteristics and instability risk of roof structure using limited investigation data. The results show that the mechanical failure of the roof structure is mainly at the bottom. As the correlation of mechanical parameters increases, the failure probability of the mechanical structure decreases significantly. In the simulation of positive and negative correlation parameters, Gaussian Copula and No.16 Copula, respectively, make the roof have the smallest failure probability.

Life-cycle thinking-based decision support framework for multispan simply supported bridges under typhoon-induced wave, current and surge conditions

Coastal bridges usually suffer from wave, current, and storm surge hazards simultaneously under typhoons. Comprehensive mitigation measures can reduce the structural failure probability under these hazards but usually lead to a conservative structural design and a significant increase in the project budget. With the recent development of sustainable structural design, the multicriteria decision-making method can assist engineers in incorporating the life-cycle investment and probability of hazards in the design selection. In this paper, a life-cycle thinking-based decision support framework that considers multicriteria trade-offs and sustainable design was proposed for the design of coastal bridges under typhoon-induced wave, current and surge conditions. The simulation of the typhoon-induced wave, current and surge conditions was developed using SWAN + ADCIRC model, joint probability and environmental contours. The multispan simply supported (MSSS) bridge was then used to illustrate the framework. The main findings include that the void slab girder is better than the other girder types in the equal-weighted, risk-averse and pro-economic scenarios for the example bridge, but the change in girder types has a small impact on the economic cost index score. The increase in the girder height can increase the failure probability when deck shifting dominates the superstructure's failure.

Reliability-based Fire Protection of Structural Cables due to Deck Fires on Cable Supported Bridges

Cable supported bridges are typically part of community-critical infrastructure and therefore the consequences arising from extensive structural damage due to a fire and the subsequent operational discontinuity could be significant beyond the damage to the bridge itself and the cost of repairs. Bridge design standards generally do not provide guidance and hence fire resistance requirements are often informed by first principles assessments of the thermal and structural response of the structure to proposed fire scenarios. A deterministic assessment can provide a reasonable worst-case scenario but it does not allow bridge stakeholders to make informed decisions with regards to uncertainty and the low likelihood of fires occurring to identify the benefits of investments in structural fire protection. This is because the target reliability required for each bridge may be different depending on its characteristics, and the expected fire severity has a probabilistic nature. This research aims to fill this gap by undertaking a reliability-based analysis to assess the probability of structural failure by considering the probability of a fire of a given size occurring and the uncertainty in the incident location and wind. The methodology is demonstrated through the case study of an operational 450 m long cable supported bridge section and a fire scenario based on a liquid fuel tanker. Liquid tankers can result into two type of fire scenario: first, a fire being contained within the tanker; and secondly, a fire due to a pool on the deck. Both fire scenarios have been characterised and it was determined that the contained tanker scenario presents a more onerous design case owing to its more prolonged duration. A Monte Carlo analysis was undertaken for a total of 10,000 independent trials of the combination of the location of the fire and the wind direction. The frequency of the fire was also examined to establish the probability of structural failure of a cable. It was established that the inclusion of wind can sometimes reduce the estimated height at which cables may experience temperatures with the potential to cause cable failure. However, for a number of the cases, the estimated height can remain broadly similar (i.e. within a couple metres difference) to the worst-case calculated value. Based on this analysis, the influence of the required level of reliability as well the fire protection duration specification agreed with bridge stakeholders can be used to determine the height at which cable fire protection is necessary to reduce the probability of structural failure to a level acceptable to the stakeholders. This approach allows bridge designers and stakeholders to make informed decisions with regards to provision and extent of fire preventive/protective measures for cable supported bridges.

Understanding of trabecular-cortical transition zone: Numerical and experimental assessment of multi-morphology scaffolds

Applications of additive manufacturing (AM) in tissue engineering develop rapidly. AM offers layer-by-layer creation of complex objects, developed to restore functionality of, or replace, damaged tissues. Porous 3D-printed functional gradient structures are of particular interest: their special architecture makes it possible to simulate the heterogeneity of the replaced tissue and, by continuously changing the mechanical properties, to avoid the concentration of stresses that can be caused by abrupt geometric changes. Such structures also allow combinations of different types of unit cells and a smooth transition between them, making design of personalised scaffolds with optimal parameters for the replacement of damaged host tissue at the interface between tissues possible. This paper presents the results of development of scaffold structures with gradients of porosity and multi-morphology using unit cells based on triply periodic minimal surfaces (TPMS). The mechanical behaviour of additively manufactured scaffold prototypes made of polylactide acid (PLA) was studied under compressive loading. Strain fields on their surface were captured using the Vic-3d Micro-DIC digital image correlation system and compared with those obtained with detailed numerical simulations, employing elastic-plastic properties of PLA, obtained in experiments. The effect of gradient parameters and unit-cell morphology on the stress distribution in scaffolds was analysed. A smooth gradient transition between cells with different morphologies was found to reduce the probability of structural failure under intense compressive loading. A good agreement between numerical results and experimental data was achieved, which justifies application of the developed approach to design of personalised bone scaffolds.

Adaptive stochastic configuration network ensemble for structural reliability analysis

This paper develops a new adaptive surrogate model method for structural reliability analysis, which is constructed by approximating the limit state function (LSF) using stochastic configuration network (SCN) ensemble model. The proposed ensemble model integrates three SCN base models and sets weights for each prediction result, reducing outliers and greatly improving the output accuracy and stability of the algorithm. In constructing a surrogate model for reliability analysis, a new adaptive sampling strategy is designed, including a novel learning function. The new strategy enables the most informative sample points are selected and significantly reduce the number of required sample point. Without further calls to the LSF, the structural failure probability is quickly evaluated using Monte Carlo simulation in terms of the constructed SCN ensemble surrogate model. Three benchmark problems, including a highly nonlinear problem with small failure probability and two multi-variable engineering structural problems, are used to verify the effectiveness of the proposed method.

AK-SEUR: An adaptive Kriging-based learning function for structural reliability analysis through sample-based expected uncertainty reduction

Reliability Analysis (RA) is a critical aspect of structural design and performance evaluation aiming to determine the probability of structural failure under given random input parameters. With modern development of modeling techniques, computational models have achieved higher fidelity but at the increased cost of computational time, which poses a significant challenge for RA. Consequently, surrogate model-assisted RA has been explored as a means of improved efficiency and accuracy. This study proposes a novel learning function, Sample-based Expected Uncertainty Reduction (SEUR), for surrogate model-assisted RA. The SEUR function uses statistical information from the metamodeling with fixed hyper-parameters to construct expected failure probability bounds to sequentially update the design of experiment (DoE). The joint probability densities of input variables are accounted for through simulation methods, including Monte Carlo (MC) and subset simulation (SS). Furthermore, the discrete simulated annealing algorithm is used to search for the optimal design point. The performance of proposed AK-SEUR function is systematically evaluated using six examples of different dimensions, failure probability levels and nonlinearities. The AK-SEUR function is demonstrated to be more effective and efficient than other popular active learning methods in dealing with nonlinear performance functions, small probabilities, and complex limit states. The proposed SEUR function has the potential to improve the efficiency and accuracy of RA, particularly in situations where computational models are time-consuming and the search for the optimal solution is challenging.

Multi-objective topology optimization design of truss structures based on evidence theory under limited information

Uncertainty information is often limited and has great discreteness, how to ensure the structure having good robustness under epistemic uncertainty becomes a technical problem for multi-objective topology optimization design. In this paper, the evidence theory is used to quantify the epistemic uncertainty; the plausibility measurement expressing upper limit of failure probability is applied to be a new evaluation of reliability. The total weight and the reliability of structures are taken as the optimization objectives, and a multi-objective robust topology optimization design model based on evidence theory is proposed. Parallelization technique based differential evolution for multi-objective optimization (DEMO) is preferable to search above robust Pareto front due to its merits of non-requirement of any gradient and superior mechanisms of non-dominate strategy. In order to verify the effectiveness of the proposed method, the multi-objective reliability optimization of a wood truss structure is implemented with considering elastic modulus and structural load as uncertain variables. According to the optimization results, six same truss specimens are made for the static random loading test. The failure probability of the truss structure is judged by the stress and node displacement obtained from the test, so as to verify the feasibility of the reliability optimization method based on evidence theory. The experimental results also indicate that the proposed method can avoid the deviation of the optimization results caused by the fluctuation of epistemic uncertainty, and provide a new method for designers to make the optimization results robust even when the data information is insufficient and the cognitive level is limited.

Seismic failure probability of adjacent frame structures retrofitted by flexible control measure

Due to the randomness of earthquake and the difference of dynamic characteristics of adjacent structures, the mechanical behavior of the adjacent structures under earthquake is very complex. A flexible control method using steel strands is proposed, which has little impact on the dynamic characteristics of the adjacent structures. A three-dimensional numerical calculation model of the adjacent frame structures is established by using LS-DYNA. The interstory drift ratios of the single structure, the adjacent uncontrolled structure and the adjacent controlled structure are analyzed by using incremental dynamic analysis (IDA) method. The failure probabilities of different limit states (LS) are studied based on the vulnerability, and the annual average and the design life failure probabilities are further discussed by the probabilistic seismic failure risk analysis. The results show that the maximum interstory drift ratio of the structure is reduced by the flexible control, and the failure probability of the adjacent controlled structure is less than that of the single structure and the adjacent uncontrolled structures. It should be noted that the influence of the flexible control on the failure probability is reduced when the PGA is close to 1.0 g. The four limit states LS1, LS2, LS3 and LS4 represent the slight damage, the moderate damage, the severe damage and the complete damage, respectively. The annual average failure probability from LS1 to LS4 decreases in turn, and the difference is about 10 times. Compared with the adjacent uncontrolled structures, the flexible control reduces the design life failure probabilities of the 10-storey structure and the 8-storey structure by 13.82% and 11.79%, respectively. The proposed control measure is helpful to reduce the failure probability of the adjacent frame structures under earthquake action.