Probability Density Evolution Method Research Articles

An enhanced fatigue damage model based on Weibull strength distribution

AbstractConsidering the inherent characteristics of strength degradation, a residual strength model is improved reasonably through the incorporation of the two‐parameter Weibull distribution. Building upon this foundation, an enhanced nonlinear fatigue damage accumulation model that is closer to the actual situation is proposed in view of the randomness of fatigue damage. The model takes into account the effect of load action on fatigue damage. To ascertain the feasibility of the proposed model, the time‐varying reliability of the model is evaluated by applying the probability density evolution method. The model's precision in predicting residual life is corroborated through five experimental data sets under bilevel and multilevel loading conditions. The results demonstrate that the proposed model exhibits excellent applicability and remarkable precision, thereby confirming its potential value in practical applications.

Random dynamic response analysis of an asphalt concrete core wall dam on deep overburden with double random factors

The spatial variability of material parameters and the randomness of ground motion are factors that influence the seismic response of a dam. To obtain more reliable seismic response results, it is crucial to study the influence of various uncertain factors on the response to earthquakes. In this paper, the K-L series expansion method and LH sampling technique are combined to characterize the spatial correlation of material parameters using a Gaussian autocorrelation function. This allows for the simulation of random fields of material parameters. By improved the Clough-Penzien power spectrum model and incorporating the concept of a random function, it becomes possible to generate nonstationary random ground motion that accurately reflects the site conditions. Taking an asphalt concrete core dam on deep overburden as an example, a random dynamic response analysis was conducted, considering various random factors. Statistical analysis and distribution testing were conducted on the horizontal acceleration and vertical permanent deformation responses. The process of the evolution of the probability density of the horizontal acceleration at the top of the dam was determined using the generalized probability density evolution method. This method helps to reveal the influence of various random factors on the response of the dam body. The research results show that among the spatial variability of material parameters and the randomness of ground motions, the randomness of ground motions is the main influencing factor of the random response results of the dam body. The mean response of two random factors is significantly larger than that of a single random factor. When considering dual random factors simultaneously, the complexity of the random factors increases, resulting in statistical properties that differ from those of single random factors. Therefore, to obtain more reliable random seismic response results, it is necessary to consider both the spatial variability of material parameters and the randomness of ground motions.

A numerical critical shear crack model and its application to post‐peak behavior assessment of RC and SFRC beams

AbstractIn this paper, a numerical critical shear crack model, in which the shear‐flexural coupling effect is considered and the full process of stress and strain is captured, is proposed to evaluate the post‐peak behavior of reinforced concrete (RC) and steel fiber reinforced concrete (SFRC) beams. The shape of critical shear crack is identified by combining the modified compression field theory considering the bridge effect of steel fibers and the section analysis method. Based on the shape of critical shear crack, the shear capacity of beam members is provided by the shear tension zone and the shear compression zone. The shear capacity of the shear tension zone is calculated by the modified compression field theory and that of the shear compression zone is determined by multiaxial strength criterion of concrete. The vertical displacements caused by the flexure deformation and shear deformation are deduced by the moment area method and integration of shear strain. To verify the proposed numerical approach, a test database of 486 RC beams and 313 SFRC beams was established to predict the shear strength, and the force‐displacement relationships of twelve beam members are used to validate the feasibility for full process analysis. The stochastic analysis of beams with different failure modes is conducted via GF‐discrepancy‐based point selection method and probability density evolution method. The limitation between different failure modes is defined according to the degradation percent of shear capacity and it is taken as threshold value of failure domain, and the failure probability analysis indicates that the designed flexure beam member suffered severe degradation of shear capacity, resulting in a significant decline in safety probability.

Global sensitivity analysis of the maximum live load and its applications

The design live loads are determined by the probability distribution of the maximum live load, which is influenced by the amplitudes and time intervals of various sustained and extraordinary loads. If the relative impact of different input variables on the maximum can be clarified, more targeted load surveys and modeling can be achieved. However, there is currently no global sensitivity analysis that simultaneously considers all input variables. In this study, the probability density function of the maximum live load is determined using the load coincidence principle and probability density evolution method. The relative entropy is employed as a measure for conducting a global sensitivity analysis across five common building occupancy types. The results indicate a significant imbalance in the impact of different input variables. The load amplitudes have a much greater effect than the time intervals. Among various load amplitudes, those related to the extraordinary loads often have the most significant impact. Regarding the time intervals, the occurrence intervals corresponding to the extraordinary loads caused by furniture stacking and normal crowding consistently have the least influence. For the time intervals with minimal impact, it is suggested to treat them as deterministic values in the live load modeling. This treatment has a negligible impact (not exceeding 10%) on the mean and upper fractile of the maximum, which are generally used for design in load codes.

Probability density evolution method based stochastic simulation of near-fault pulse-like ground motions

Quantifying the near-fault effect and establishing a reasonable model of near-fault pulse-like ground motions are particularly important for seismic design of structures in near-fault regions. Given the pronounced randomness associated with earthquakes, this study first proposes a novel stochastic model of near-fault pulse-like ground motions by combining the improved finite-fault model (IFFM) and the multivariate copula-based velocity-pulse model (CVPM). Further, a probability density evolution method (PDEM) based stochastic simulation method is developed, by which the model parameters can be determined in a unified probability space so as to ensure the consistency of two independent models. For illustrative purposes, the observed records collected from the 1999 Chi-Chi earthquake are used to generate new stochastic ground motions set. Two ground motions sets based on classical stochastic simulation methods are also presented for comparison. Numerical results show that the proposed method for stochastic simulation of near-fault pulse-like ground motions is reliable; the statistics of peak ground accelerations and spectral characteristics of simulated samples are consistent with station records. Besides, the proposed method accommodates the noteworthy randomness and proportion consistency of components associated with near-fault pulse-like ground motions, making it suitable for the stochastic response and reliability analysis of seismic structures in near-fault regions. This superiority is challenging to classical stochastic simulation methods that lack reasonable consideration of randomness and correlation associated with model parameters.

Revaluation of occupancy duration for live load using big data of enterprise credit information

The occupancy duration is an essential variable in establishing the time-varying model of the live load. The manual questioning and inquiries, which are the common survey methods for occupancy duration, suffer from being expensive and time-consuming. This has hindered the updating of the data basis and progress in live load research. This study proposes a novel big data survey method, which utilizes 210 million enterprise credit information to collect occupancy duration samples. Along with the big data survey method, an address-oriented mining method is presented to efficiently extract duration samples of the target type from big data. Based on the actual distribution of the extracted duration data, a more general time-varying model, called the compound renewal process, is suggested to describe the live load. The stochastic harmonic function representation method and probability density evolution method are then adopted to access the maximum load distribution of the live load process. The maximum load distributions of the sustained and combined loads are calculated and compared with previous studies to evaluate the impact of the new survey results. Comparison shows that the occupancy duration of office buildings is significantly shorter, resulting in an increase in the maximum load. The proposed data survey method enables continuous updating of duration samples, and the compound renewal process allows better compatibility for new survey data. The up-to-date duration data and a realistic time-varying model provide new ideas for determining the design live load.

Seismic analyses of single-layer dome structures with random geometrical imperfections under stochastic ground motions

In this paper, seismic analyses of single-layer dome structures are carried out, with a particular emphasis on the incorporation of randomness in both geometrical imperfections and ground motions. First, the paper introduces models for random geometrical imperfections and stochastic ground motions, employing the Stochastic Imperfection Mode Superposition Method (SIMSM) for imperfections and utilizing the Spectral Representation Method (SRM) for ground motions. These models are designed to comprehensively account for the inherent uncertainties in real-world structural behavior. Subsequently, the Probability Density Evolution Method (PDEM) is applied to investigate the probabilistic response and seismic reliability of dome structures, which allows for a detailed examination of key performance indicators, such as displacement, plastic ratio, and strength damage, under the influence of random factors. Both deterministic and stochastic analyses are conducted to gain insights into the impact of randomness in geometrical imperfections and ground motions. The computational results unveil a crucial finding: an unfavorable distribution of imperfections can significantly compromise the dynamic performance and safety of single-layer dome structures. Furthermore, the paper delves into parametric analyses, with a specific focus on imperfection amplitude, the rise-span ratio and roof load as pivotal parameters. These analyses provide valuable insights into the sensitivity of the structural response to variations in these critical design factors.

Seismic safety assessment of dam slopes considering rockfill softening characteristics, shear strength uncertainties, and stochastic ground motion

The stability of the dam slope is one of the crucial indicators for maintaining the overall stability and reliability of the dam. The seismic stability response analysis of the dam slope involves a significant number of factors that introduce uncertainty, primarily including the stochastic ground motion and the uncertainty of the shear strength parameters of rockfill. Furthermore, the rockfill material is susceptible to strain-softening behavior at low confining pressures when subjected to ground motion, which exacerbates its progressive failure. Therefore, the assessment of seismic safety for dam slopes should give due consideration to this characteristic of rockfill material. This study extensively explores the influence of rockfill softening, stochastic ground motion, and the variability of rockfill shear strength parameters on the stability of the dam slope. Conducting probabilistic analysis in conjunction with the generalized probability density evolution method (GPDEM), the findings suggest that it is imperative to evaluate the safety characteristics of the dam slope by considering the coupled effects of stochastic ground motion, the variability of shear strength parameters, and the effect of rockfill material softening. Based on this, a preliminary dam slope failure classification was proposed, and a framework for dam slope seismic safety assessment was established.

Wind-resistant reliability assessment of a super-high tower crane considering the randomness and correlation of turbulent spectral parameters

The need for wind-sensitive super-high tower cranes during the construction of high-rise structures has been growing, which makes it critical to assess the wind-resistant reliability of cranes. The accuracy of assessment is dominated by the refined simulation of real stochastic wind fields, which is usually based on the deterministic turbulent spectra including fixed spectral parameters. However, it might cause significant errors to estimate extreme responses of structures. Therefore, a simulation method for stochastic wind fields considering the randomness and correlation of turbulent spectral parameters is proposed in this study. The random functions of correlated spectral parameters are constructed based on several independent elementary random variables. A novel simulation method for stochastic wind fields involving the established random turbulent spectral functions is utilized to generate fluctuating wind samples at the Ma'anshan Yangtze River (MYR) Bridge site. In combination with the probability density evolution method, the stochastic response analysis and reliability assessment of wind-induced vibration of the crane are addressed. Results prove the effectiveness of the proposed approach in depicting probabilistic properties of real wind fields. Meanwhile, the wind-resistant comfort reliability of the tower crane is higher when considering the randomness and correlation of turbulent spectral parameters.

Multi-hazard stochastic response analysis and reliability evaluation of a 5 MW monopile offshore wind turbine based on extended platform FAST-S

This study investigates the random dynamic response of a 5 MW monopile offshore wind turbine (MOWT) subjected to wind, wave and earthquake loads. Based on the aero-hydro-servo-elastic code FAST, a multi-field coupled simulation module considering earthquake, FAST-S, is developed for the fully coupled multi-hazard analysis of OWTs. Compared to the original FAST code, a seismic analysis module (SeismoDyn) and a soil-structure interaction (SSI) module (SoilDyn) have been supplemented in FAST-S. SeismoDyn is developed by modifying the Kane’s kinetic equation in the FAST, and the SoilDyn is developed through Winkler foundation model in order to consider the nonlinear SSI. Moreover, FAST-S has been validated by other recognized codes. Based on FAST-S, a fully coupled numerical analysis model of the NREL 5 MW MOWT is established, the random responses and reliability evaluation of the offshore wind turbine under wind, wave and earthquake loads are analyzed by probability density evolution method. The results show that the aerodynamic and hydrodynamic actions increase the damping of the wind turbine system, which has a certain effect on inhibiting the tower’s vibration caused by the earthquake. Therefore, compared with the coupled wind-wave-earthquake case, the sole earthquake case may be more unfavorable, and should be paid more attention to in high risk earthquake regions. The results show that the probability density functions of offshore wind turbine response under strong earthquakes obviously deviates from the typical regular distribution models. Based on the theory of equivalent extreme value event, the dynamic reliability of the offshore wind turbine is calculated. Seismic actions maybe the major factor in the design of the support structure. Nevertheless, owing to the aerodynamic and hydrodynamic damping effects of wind and wave loadings, the structural responses under wind-wave-earthquake are more moderate than ones subjected to solely earthquake. The strong seismic loads would cause significant complex variations of time-dependent probability density function of the OWT response.

A modified Chebyshev collocation method for the generalized probability density evolution equation

Solving the generalized density evolution equation (GDEE) is essential to obtain the structural random response by the probability density evolution method (PDEM). To improve the collocation method, this study proposes a modified Chebyshev collocation method (MCCM) scheme for the solution of the GDEE based on the pseudo-spectral method, and derives a refine time integration format for GDEE. The GDEE is numerically solved by combining the fine time integration method and MCCM, and the accuracy, efficiency and numerical stability of the algorithm are investigated. Two numerical examples indicate that the results for the proposed method are in good agreement with analytical solutions and Monte Carlo simulation results. The proposed four numerical collocation schemes could effectively suppress numerical dispersion and numerical dissipation. In addition, the modified Chebyshev collocation method based fourth-order Runge-Kutta scheme presents excellent computational accuracy, efficiency and numerical stability, especially highlighting significant advantages in the computational efficiency.

Dynamic reliability analysis of Aerial Building Machine under extreme wind loads using improved QBDC-based active learning

Aerial building machine (ABM) is a new comprehensive construction equipment used for the construction of high-rise buildings, and the research of its wind-induced response is essential for structural safety. However, the traditional reliability estimation for complex structures is generally computationally expensive. This study proposes a novel and time-saving active learning (AL) approach to predict the structural time-variant response and calculate the reliability under extreme wind loads based on improved Query-by-Dropout-Committee (QBDC), Deep Learning (DL) and Probability Density Evolution Method (PDEM). To establish an effective training set, the load sample pool from the turbulent wind field is constructed by the number theory method (NTM). Then a DL surrogate model, aimed to simulate the excitation-response relationship of the structure, is well-trained from data calculated by the Finite Element Method (FEM) under the workflow of the improved QBDC-based AL approach. Lastly, the structural responses predicted by the surrogate model are used to solve the Generalized Density Evolution Equation (GDEE) to obtain the Probability Density Function (PDF) of the structural response. The dynamic reliability can be gained using absorbing boundary condition. To demonstrate the feasibility of the proposed approach, an ABM in China is taken as a case study. It is found that (1) The external trusses of the platform are weak positions under the horizontal wind loads. It is suggested to append the longitudinal (windward direction) trusses to reduce the local stress and deformation. (2) Under turbulent wind loads with the average speed of 40 m/s, the maximum structural response of the ABM will exceed the threshold value and the reliability is 0.932. (3) The reliability calculation error derived by the proposed approach and the traditional PDEM is within 0.5 % in this study, while the computational efficiency of the AL approach is 77.08 % higher than that of the traditional PDEM. The novelty lies in the proposed active learning approach that can easily obtain the stochastic response and dynamic reliability of the ABM, where the analysis workflow not only has a low computational cost but also maintains a high calculation accuracy.

Structural reliability analysis based on probability density evolution method and stepwise truncated variance reduction

To address the substantial computational burden associated with probability density evolution method (PDEM) in structural reliability analysis, this study proposes a novel look-ahead learning function named stepwise truncated variance reduction (STVR), integrating polynomial chaos Kriging (PCK) and PDEM. Three key features of STVR are highlighted. First, it enables quantifying the maximum reduction in predictive errors of PCK within the regions of interest (ROI) when adding a new point. Second, closed-form expression for STVR is derived through Kriging update formulas, eliminating the need for computationally intensive Gauss–Hermite quadrature or extensive conditional simulations of PCK. Third, a dynamic adjustment procedure is proposed for the probability level-related parameter in STVR, with the aim of achieving a good balance between the exploitation and exploration of ROI during the sequential experimental design process. The performance of STVR is demonstrated through two benchmark analytical functions and three numerical examples of varying complexity. Results indicate that the dynamic adjustment procedure for the probability level-related parameter in STVR outperforms the empirical setting of a minor value. Then, STVR proves more advantageous than existing pointwise and look-ahead learning functions, particularly in addressing complex dynamic reliability problems.

Time‐varying reliability analysis based on hybrid Kalman filtering and probability density evolution

AbstractThe study introduces a novel approach for time‐varying reliability analysis of structures called “hybrid UKF‐PDEM” by integrating the unscented Kalman filter (UKF) and the probability density evolution method (PDEM). The UKF estimates the displacement, velocity, stiffness, and damping parameters of a structure at each time step subjected to dynamic loading for structural damage quantification. The estimated parameters at each time step are then input into the PDEM to calculate the time‐varying probability density function (PDF) of the estimated states. The estimated PDF is used to update the uncertainty matrix of the estimated states in each iteration and to determine the time‐varying reliability curves of the structure. To demonstrate the effectiveness of the proposed method, we applied it to a numerical model of a three‐degree‐of‐freedom system and a full‐scale seven‐story building with different damage scenarios. The method is used to estimate the level of damage and calculate the corresponding reliability curve of the system over time for each damage scenario, utilizing the estimated structural responses and stiffness values. The extracted reliability values for each damage scenario follow the level of damage over time. This study shows that the newly developed method is computationally efficient for building a digital twin and enables real‐time damage identification and reliability analysis in various structural systems. The method's applicability to different types of structures highlights its versatility and potential for widespread use in assessing the integrity of buildings and infrastructure.

Solving coupled differential equation groups using PINO-CDE

As a fundamental mathematical tool in many engineering disciplines, coupled differential equation groups are being widely used to model complex structures containing multiple physical quantities. Engineers constantly adjust structural parameters at the design stage, which requires a highly efficient solver. The rise of deep learning technologies has offered new perspectives on this task. Unfortunately, existing black-box models suffer from poor accuracy and robustness, while the advanced methodologies of single-output operator regression cannot deal with multiple quantities simultaneously. To address these challenges, we propose PINO-CDE, a deep learning framework for solving coupled differential equation groups (CDEs) along with an equation normalization algorithm for performance enhancing. Based on the theory of physics-informed neural operator (PINO), PINO-CDE uses a single network for all quantities in a CDEs, instead of training dozens, or even hundreds of networks as in the existing literature. We demonstrate the flexibility and feasibility of PINO-CDE for one toy example and two engineering applications: vehicle-track coupled dynamics (VTCD) and reliability assessment for a four-storey building (uncertainty propagation). The performance of VTCD indicates that PINO-CDE outperforms existing software and deep learning-based methods in terms of efficiency and precision, respectively. For the uncertainty propagation task, PINO-CDE provides higher-resolution results in less than a quarter of the time incurred when using the probability density evolution method (PDEM). This framework integrates engineering dynamics and deep learning technologies and may reveal a new concept for CDEs solving and uncertainty propagation.

An adaptive hybrid deep learning-based reliability assessment framework for damping track system considering multi-random variables

Reliability assessment is essential in the design and management of rail transit system (RTS), with a heightened focus on damping tracks that feature low-stiffness structures and various uncertainties. However, existing research on RTS has notable gaps: one is the unexplored nonlinear track irregularities in surrogate model, while the other is the significant computational burden of current reliability analysis. Therefore, Firstly, based on a database acquired from simulations of rigid-flexible coupling dynamic models, a hybrid deep learning (DL) surrogate model is developed and trained. Notably, it accommodates both time-series and feature parameters data as inputs, and integrates a novel algorithm featuring slide window with variational decomposition mode-sample entropy, optimization algorithm and adaptive learning strategy (AS). Subsequently, an advanced reliability assessment framework is proposed, utilizing AS hybrid DL-based Probability Density Evolution Method (PDEM), which thoughtfully designed to analyze the two aforementioned categories of random variables and then applied to a train-steel spring floating slab track. Experimental studies show the proposed surrogate model effectively and robustly predicts four safety metrics, with each component's excellence confirmed. Furthermore, this framework shows higher accuracy than the Monte Carlo Method and enhances computational efficiency by 10 ∼ 30 times compared to traditional mechanism-based PDEM. Consequently, reliability assessments indicate values of 0.9312 and 0.9214 for wheel-load reduction rate and rail displacement, respectively, with other metrics showing zero safety hazards. Findings of this study have practical applicability in the field of RTS design and maintenance.

Reliability-based design optimization for seismic structures considering randomness associated with ground motions

When addressing the dynamic reliability analysis of structures, it becomes necessary to account for multiple limit state functions or their combinations. In scenarios where structures are subjected to random excitation, this can lead to intricate inter-dependencies among different limit states, and the computational workload can pose a substantial challenge in ensuring sufficient precision. Code-based design primarily ensures safety at the member level, while deterministic optimization fails to accommodate the inherent uncertainties associated with external excitation or the system as a whole. Therefore, in such cases, to address both the uncertainties in excitations and the presence of multiple limit states while mitigating computational challenges, equivalent extreme-value criteria are employed within the framework of the probability density evolution method to calculate the global reliability of the structure subjected to stochastic ground motions generated from the physically motivated stochastic ground motion model. Numerical optimization is subsequently conducted using genetic algorithms, aiming to minimize the cost of the superstructure while adhering to the design performance criteria related to the inter-story drift ratio and considering global reliability. Additionally, multi-objective optimization is carried out using NSGA-II, permitting the generation of multiple solutions, from which one can select the most suitable solution as needed. The numerical results illustrate the effectiveness of this technique in achieving an optimal balance between the cost of the structure and the consideration of global reliability, providing a comprehensive solution for dynamic reliability analysis and design optimization of structures under random excitations.

Reliability assessment of stochastic dynamical systems using physics informed neural network based PDEM

In the recent decade, the reliability analysis of a stochastic system coupled with the uncertainty related to the system’s parameter has attracted much attention. Probability density evolution method (PDEM) is one of the viable options that estimates the probability density function of the structural response by solving generalized density evolution equations (GDEEs). The advantage of PDEM is that it is derived based on the principle of probability conservation, where GDEEs are decoupled from the physical system. In general, GDEEs in PDEM are solved using a finite difference scheme in which the accuracy of the numerical solution depends on the number of temporal and spatial discretizations, leading to computationally inefficient for high-fidelity models. With this in view, this study proposes a physics-informed neural network (PINN), a novel deep learning method, based PDEM, for solving the GDEEs. PINN utilizes physical information in the form of differential equations to enhance the performance of the neural networks. This method does not need any interpolation or coordinate transformation, which is often seen in any numerical scheme, thus the computational budget is reduced. Three numerical examples are presented in this study to illustrate the proposed PINN-based PDEM, including a Van-der-Pol oscillator subjected to Gaussian white noise, a one-storey moment resisting frame coupled with a nonlinear energy sink with negative stiffness and sliding friction, and a high-rise timber building coupled with shape memory alloy-based outriggers. The first example is utilized to show the accuracy of the proposed method by comparing results with the Fokker–Planck–Kolmogorov equation and Monte Carlo simulation. The rest two examples are investigated for estimating time-dependent probability of failure. Numerical results show that the proposed PINN-based PDEM can estimate the probability of failure efficiently.

Stochastic analysis and reliability assessment of critical RC structural components considering material properties uncertainty

The unavoidable heterogeneity in the mechanical characteristics of concrete is widely acknowledged. Although it is widely considered as either perfectly correlated or entirely independent random variables in engineering practice; however, such treatment is illogical, and the outcomes may be deceptive. In high-rise buildings comprised of multiple structural components, it is crucial to consider the material properties’ spatial variability (MPSV) to obtain a reliable structural response and avoid damage to these structures. To this end, three main components, including column, rectangular shear wall, and U-shaped shear wall, are considered herein to investigate their stochastic response. The MPSV is represented by a covariance matrix decomposition-based random field generator combined with a GF-discrepancy-based point selection strategy to generate samples efficiently. A simplified strategy is developed to represent the random field for the U-shaped wall. Moreover, the probability density evolution method combined with the extreme value event is employed to obtain the failure probability of the studied components, where failure probabilities of 18%, 23%, and 32% are recorded for the studied RC column, rectangular shear wall, and U-shaped shear wall, respectively. Furthermore, different failure modes were identified and could not be determined through the deterministic analysis, highlighting the importance of accounting for material uncertainty. The proposed framework proved that the stochastic response and non-linear behavior of the considered components could be well captured and provide full perspective about the uncertainty quantification and reliability assessment and can be further implemented to capture the stochastic response and safety assessment of high-rise buildings.

Reliability-based design optimization of tuned mass-damper-inerter for mitigating structural vibration

The tuned mass-damper-inerter (TMDI), comprising mass blocks, springs, dampers, and two-terminal inertial elements, has garnered significant attention as an advanced iteration of the tuned mass damper (TMD) in recent years. However, previous studies of TMDI's design are primarily conducted according to the theoretical formulas based on deterministic analysis assuming harmonic excitations or stochastic analysis assuming Gaussian white noise type excitations. This treatment falls short of capturing the non-stationary and non-Gaussian characteristics exhibited by engineering excitations and thus cannot ensure the safety of controlled structure. A few studies considering non-stationary excitations and based on reliability metrics, in conjunction with stochastic simulation methods, often confront excessive computational overhead. Aiming at such a challenge, the reliability-based design optimization (RBDO) of TMDI system, combining the probability density evolution method (PDEM) and genetic algorithm (GA), for mitigating structural vibration under non-stationary excitations is performed in this study. For the purposes of illustration, a single-degree-of-freedom system attached the TMDI and subjected to stochastic ground motions is studied. Considering the cases of different levels of inertance-to-mass ratios, the RBDO of the TMDI is carried out. The solutions by the theoretical formulas and by the variance-based design optimization are compared to verify the superiority of the proposed method. For assessing the seismic mitigation performance of the optimized TMDI system, probability density evolution analysis of structure-TMDI system with three levels of inertance-to-mass ratios is performed. Comparison between the RBDO of TMDI and that of TMD under consistent mass ratio is further addressed. Numerical results reveal the technical advantages of TMDI in a probabilistic sense, i.e., high vibration mitigation performance, considerable mass reduction, and less stroke demand.