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• A unified DPIM-COM approach for time-invariant and time-variant RBDO is proposed. • DPIM together with Heaviside function is devised to compute failure probability directly. • DPIM-COM can attack time-invariant and time-variant RBDO with MPPs effectively. • DPIM combines with extreme value distribution to assess time-variant probabilistic constraint. Reliability-based design optimization (RBDO) can fulfill performance-based design of static and dynamic structures considering various uncertainties. For time-invariant and time-variant RBDO problems with multiple most probable points (MPPs, also termed as design points), the existing approaches encounter some difficulties in searching for all MPPs and performing time-variant RBDO which requires complicated time-variant reliability analysis. In this paper, the direct probability integral method (DPIM) combining with the change of probability measure (COM), i.e., DPIM-COM approach, is proposed to attack time-invariant and time-variant RBDO problems with multiple MPPs in a unified framework. Firstly, the DPIM in conjunction with Heaviside function is developed to estimate time-invariant and time-variant probabilistic constraints directly and explicitly. Then, an efficient formula for calculating the sensitivity of probabilistic constraints with respect to random design variables is derived based on the DPIM and COM strategy, where the representative points for reliability computation keep unchanged and only the corresponding assigned probabilities need to be updated. Accordingly, the corresponding structural responses of representative points at the current design variables can be reused to evaluate the sensitivity of probabilistic constraints, which saves much computational cost, especially for time-variant RBDO problems. Finally, three numerical examples, including the ten-story building with tuned mass damper under deterministic earthquake ground motion and near-fault stochastic impulsive ground motions, demonstrate the effectiveness and versatility of the proposed DPIM-COM approach for addressing time-invariant and time-variant RBDO problems with multiple MPPs. [ABSTRACT FROM AUTHOR]

Random vibration analysis of structures subjected to combined Gaussian and Poisson white noise excitation is a challenging issue. In this paper, a novel direct probability integral method (DPIM) is suggested to address stochastic responses and dynamic reliability of nonlinear multi-degree-of-freedom (MDOF) systems under combined excitation. Firstly, probability density integral equation (PDIE) of MDOF system under combined excitation is derived based on the principle of probability conservation. Then, DPIM is proposed to achieve the probability density function of stochastic response by solving deterministic motion equation of MDOF system and PDIE in sequence. To solve PDIE, two techniques, i.e., partition of input probability space and smoothing of Dirac delta function, are introduced. From the perspective of probability conservation, furthermore, the equivalent relationship between the PDIE and the corresponding probability density differential equation of a Markov system under combined Gaussian and Poisson noise is established. Since Dirac delta function is analytically integrated as Heaviside function, the first-passage dynamic reliability of MDOF system under combined excitation is readily evaluated by introducing the extreme value mapping of stochastic response. Finally, two nonlinear MDOF systems, including multiple-span bridge under combined vehicle load and non-stationary seismic excitation, are solved. Results demonstrate that the proposed DPIM is effective for random vibration and dynamic reliability analyses of MDOF structures excited by combined Gaussian and Poisson noise, and considering the randomness of vehicle load and seismic excitation simultaneously in the bridge design benefits the safe operation of bridge. [ABSTRACT FROM AUTHOR]

Structural system reliability analysis is of important significance for evaluating the safety of structures with many components. Since a structural system can be acted by static or dynamic load, a unified and efficient method is required to assess the system reliability of static or dynamic load induced structures. In this study, the direct probability integral method (DPIM) is proposed to uniformly attack system reliability problems of static and dynamic structures. Firstly, the static and first-passage dynamic reliability formulas of the series, parallel and mixed systems are established by the joint probability density function (PDF) of multiple performance functions. Based on the probability density integral equation (PDIE) of performance functions, the DPIM is proposed along the two approaches, i.e., DPIM-S and DPIM-H. The former computes the system reliability using the PDF of extreme value mapping of performance functions, which is obtained by smoothing Dirac delta function. In the latter, the system reliability formulas with Heaviside function are analytically derived by the PDIE of multiple performance functions. Specially, the role of smoothing of Dirac delta function in DPIM for stochastic response and reliability analyses is revealed. Finally, five typical examples, including two mathematical examples and three static and dynamic structural systems, demonstrate high efficiency and accuracy of the DPIM for system reliability computation. Because of omitting the smoothing of Dirac delta function, the DPIM-H takes less CPU time for solving system reliabilities of static and dynamic structures than DPIM-S, while the DPIM-S has a significant advantage of obtaining the PDF of performance function. • System reliability analyses of static and dynamic structures are attacked uniformly. • Probability density integral equation for joint PDF of performance functions is established. • Reliability formulas of series, parallel and mixed systems are analytically derived. • Extreme value mapping is utilized to calculate static and dynamic system reliabilities. • Examples indicate high efficiency and accuracy of direct probability integral method. [ABSTRACT FROM AUTHOR]

• Direct probability integral method (DPIM) for multimodal distributions is proposed. • DPIM presents high efficiency and accuracy for static/dynamic reliability analysis. • GF-discrepancy method is superior in solving probability density integral equation. • Uncertainty propagation in structures with multimodal distributions is revealed. For practical structures, some input random variables follow multimodal distributions, and conventional reliability analysis methods may result in large computational errors. In this paper, a novel direct probability integral method (DPIM), which decouples governing equation of structure and the probability density integral equation (PDIE), is proposed to address the static and dynamic reliability assessment of structures involving random variables with multimodal distributions. Firstly, the multimodal probability density functions (PDFs) of input random variables are established by the Gaussian mixture model. Then, using numerical integration and smoothing technique of Dirac delta function, the PDFs of structural responses with multimodal random variables are achieved by solving the PDIE, in which three numerical integration algorithms, namely, the Quasi-Monte Carlo approach, the sparse grid approach, and Generalized F-discrepancy-based point selection approach are employed. Further, the reliability of static structure can be readily obtained by integrating the PDF of response function, while the dynamic reliability can be evaluated by DPIM combining with extreme value distribution of stochastic process. Finally, several examples demonstrate the superiority of DPIM, and the Generalized F-discrepancy-based point selection approach has the highest accuracy and efficiency for solving PDIE. The characteristics of uncertainty propagation in structures involving multimodal distributions of input random variables are revealed. [ABSTRACT FROM AUTHOR]

• A unified framework of static/dynamic reliability analysis is established based on direct probability integral method (DPIM). • New formula to determine smoothing parameter of Dirac function is suggested. • Two DPIM-based approaches for dynamic reliability analysis are proposed. • Example of nonlinear dynamic structure indicates superiority of unified framework. Generally, the static and dynamic reliabilities of structures are addressed separately in the existing methods except the computationally expensive stochastic sampling-based approaches. This study establishes a unified framework of reliability analysis for static and dynamic structures based on the direct probability integral method (DPIM). Firstly, the probability density integral equations (PDIEs) of performance functions for static and dynamic structures are presented based on the principle of probability conservation. The DPIM decouples the physical mapping (i.e., performance function) of structure and PDIE, and involves the partition of probability space and the smoothing of Dirac delta function. This study proposes a new adaptive formula of smoothing parameter based on kernel density estimation. Then, the improved DPIM is utilized to obtain the probability density function (PDF) of performance functions by solving the corresponding representative values and the PDIE successively. Furthermore, the reliability of static structure is calculated by integrating the PDF of performance function within safety domain. To overcome the difficulty of evaluating first passage dynamic reliability, the two approaches, namely the DPIM-based absorbing condition (DPIM-AC) and the DPIM-based extreme value distribution (DPIM-EVD), are also proposed. Finally, three engineering examples with stochastic parameters and random excitation indicate the desired efficiency and accuracy of the established framework for unified reliability analysis. Specifically, the challenging issue of dynamic reliability assessment for nonlinear structural system is attacked based on DPIM rather than Monte Carlo simulation or other sampling-based method. The proposed method is beneficial for propagation analysis of aleatory or/and epistemic uncertainties, as well as for stochastic model updating. [ABSTRACT FROM AUTHOR]

This paper proposes a unified and efficient direct probability integral method (DPIM) to calculate the probability density function (PDF) of responses for linear and nonlinear stochastic structures under static and dynamic loads. Firstly, based on the principle of probability conservation, the probability density integral equation (PDIE) equivalent to the probability density differential equation is derived for stochastic system. We highlight that, for time dependent stochastic system the PDIE is satisfied at each time instant. Secondly, the novel DPIM is proposed to solve PDIE directly by means of the point selection technique based on generalized F discrepancy and the smoothing of Dirac delta function. Moreover, the difference and connection among the DPIM, the existing probability density evolution method and probability transformation method are examined. Finally, four typical examples for stochastic response analysis, including the linear and nonlinear systems subjected to static and dynamic loads, demonstrate the high computational efficiency and accuracy of proposed DPIM. • Direct probability integral method is proposed based on probability conservation. • Probability density integral equation is satisfied at each instant of dynamic system. • Typical examples indicate high efficiency and accuracy of the proposed method. • The present method facilitates stochastic response analysis of general systems. [ABSTRACT FROM AUTHOR]

• New framework for layout and size optimization of dampers in frames is established. • Sequential approximate mixed integer programming for damper design is proposed. • Direct probability integral method efficiently calculates reliability and its sensitivity. • STM-based probability constraint target relaxation scheme alleviates iterative oscillation. • The larger PGA of stochastic ground motions is, the more cost of dampers is required. Viscous dampers, as effective energy-dissipation devices, have been widely used for seismic mitigation of building structures. Due to the intrinsic uncertainties of structural parameters and earthquake ground motions, to obtain an optimal performance of structural control, the stochastic design optimization of viscous dampers for buildings is essential. This paper establishes a new dynamic reliability-based optimization framework to address the simultaneous layout and size design of nonlinear viscous dampers in frame buildings, considering the dual randomness of both nonstationary seismic excitations and uncertain structural model parameters. Firstly, a mixed integer optimization problem for nonlinear viscous dampers, including discrete and continuous design variables, is formulated, and the design target is to minimize the cost of the dampers subjected to performance constraints on dynamic reliability of building structures. Then, sequential approximate mixed integer programming with trust region is proposed to solve the optimization problem. Moreover, direct probability integral method is suggested to assess the dynamic reliability and its sensitivity with respect to design variables. Finally, accounting for both stochastic nonstationary seismic excitations and random structural parameters, the optimized results of two numerical examples indicate that the proposed method is a competitive choice for realizing the layout and size optimization of limiting types of nonlinear viscous dampers in frame buildings. To ensure the same target performance of buildings in terms of the probability of exceeding the target value of inter-story drift, the larger peak ground acceleration of stochastic ground motions is, the more cost of dampers is required for seismic protection of buildings. [ABSTRACT FROM AUTHOR]

• New hybrid model is proposed by combining probabilistic/non-probabilistic exponential models. • A new hybrid exponential probability integral method is developed for reliability analysis. • Lower and upper bounds of failure probability of static and dynamic systems are computed. • Examples verify the generality, accuracy and efficiency of proposed model and methods. Uncertainty propagation and reliability evaluation, being the crucial parts of engineering system analysis, play vital roles in safety assessment. How to reasonably consider the complex multisource uncertainty behavior in both static and dynamic systems is paramount to ensuring their safe operation. However, there is a significant lack of research on aleatory and epistemic uncertainties for both static and dynamic systems. To this end, a new hybrid exponential model is proposed by combining probabilistic and non-probabilistic exponential models, which aims to accurately measure the uncertainty propagation and reliability evaluation problem with aleatory and epistemic uncertainties for static and dynamic systems. The proposed hybrid exponential model consists of nested double optimization loops. The outer loop performs a probabilistic analysis based on the direct probability integral method, and the inner loop performs a non-probabilistic computation. Then, a new hybrid exponential probability integral method is developed to effectively perform uncertainty propagation and reliability analysis. Finally, four examples, including two static and two dynamic examples with complex performance functions, are tested. The results indicate that the proposed hybrid exponential model offers a universal tool for uncertainty quantification in static and dynamic systems. Moreover, the hybrid exponential probability integral method can accurately and efficiently obtain the upper and lower bounds of the probability density function and cumulative distribution function. [ABSTRACT FROM AUTHOR]

• Reliability-based stochastic optimal control method via DPIM of building is proposed. • The method is effective for vibration reduction of frame under near-fault ground motions. • The control scheme achieves a proper balance between control efficacy and control force. • Velocity pulses of near-fault motions affect greatly control scheme of frame building. Most of stochastic optimal control methods were developed on the basis of Itô stochastic differential equation, which assumes that the external excitation is white Gaussian noise or filtered white Gaussian noise. However, this assumption is far from the real excitations, which hinders the applications of stochastic optimal control theory. In this paper, the reliability-based stochastic optimal control via direct probability integral method (DPIM) for building structure is proposed, which is applicable to performance-based design of general control systems of linear structures under non-stationary and non-white random excitations. Firstly, the DPIM is utilized to accurately and efficiently compute dynamic reliability of the controlled system. Then, the reliability-based objective function is suggested to optimize the parameters of the placed control devices of building structure, and reliability-based maximum story controllability index is established to determine the optimal placement of control devices, thus fulfilling the stochastic optimal control of structure. Finally, a 10-story shear frame building controlled by active tendons under random excitations of near-fault earthquake ground motions verifies the effectiveness of the proposed optimal control method. Moreover, the optimal control schemes of frame structure under near-fault non-pulse and impulsive ground motions are compared, indicating that the velocity pulses of near-fault ground motions impose significant effect on structural responses, and the more control energy is needed for guaranteeing structural safety. [ABSTRACT FROM AUTHOR]

Stochastic finite element method (SFEM) for uncertainty quantification is widely applied for analysis of structures with intrinsic randomness. For determining the geometrically nonlinear deformations of laminated composite plates with random fields, the existing intrusive SFEMs have the limitations of low applicability, insufficient accuracy, or low efficiency. To this end, this paper proposes a novel and efficient non-intrusive SFEM incorporating direct probability integral method to achieve probability density functions (PDFs) of stochastic responses and reliabilities of laminated composite plates with geometric nonlinearity. Firstly, the von Kármán strain-displacement relation based on the third-order shear deformation theory is employed to model the geometric nonlinearity of laminated plates. The random field is discretized via Karhunen–Loève expansion. Secondly, the probability density integral equation (PDIE) is derived from the new perspective of probability conservation. The proposed non-intrusive SFEM decouples the equilibrium equation and PDIE to compute the response PDFs and reliabilities of uncertain laminated composite plates in a unified way. Moreover, the criterion which can judge the applicability of geometrically nonlinear theory is suggested for performing uncertainty quantification. Finally, comparisons of the results in terms of Monte Carlo simulation and the literature demonstrate the high accuracy and efficiency of the proposed method. For stochastic laminated plates, the response statistical moments vary nonlinearly with linear increase of load amplitude due to geometric nonlinearity, the deflection variability increases and structural reliability decreases with the increase in variability and correlation length of random field, and the stacking angle significantly affects the stochastic nonlinear deflections and reliabilities. [ABSTRACT FROM AUTHOR]

In practical applications, some random variables follow multimodal distributions. However, conventional reliability analysis methods in reliability-based topology optimization (RBTO), such as the first order reliability method, often result in considerable computational errors when dealing with problems involving multimodal distributions. Consequently, the corresponding RBTO design is incredible. Moreover, the RBTO problem also faces the challenge of high computational cost. To this end, this paper proposes an efficient two-phase approach for RBTO of continuum structures with multimodal distributions combining sequential approximate integer programming with trust region (SAIP-TR) and direct probability integral method (DPIM). Firstly, DPIM is advanced to address the difficult problems of failure probability estimation and efficient sensitivity analysis under multimodal distributions. Secondly, a reliability-based discrete variable topology optimization framework based on SAIP-TR and DPIM is established, which yields clear topology configurations and facilitates engineering manufacturing. Owing to the merit of SAIP-TR, the original RBTO process is divided into two phases: the first phase performs deterministic topology optimization, and the second phase focuses on RBTO. Moreover, an adaptive selection strategy of representative points, considering structural compliance as performance function, is devised to further enhance computational efficiency. Finally, several examples illustrate high efficiency and accuracy of the proposed approach. The multimodal random variable and the random field are employed separately to describe global and local uncertainties in materials. In contrast, the optimized result considering global material uncertainty is more suitable for additive manufacturing. The proposed approach also presents potential in handling complex RBTO problems with random fields. • Novel two-phase approach to reliability-based topology optimization is proposed. • Efficient adaptive representative points selection strategy is established. • Direct probability integral method assesses reliability with multimodal distribution. • Material uncertainties from local and global perspectives are examined. [ABSTRACT FROM AUTHOR]

• A new non-intrusive SFEM is proposed for efficient uncertainty propagation of plates. • Probability density functions of stochastic responses and structural reliabilities are achieved. • For Gaussian random thickness, linearly elastic plate yields non-Gaussian response. • Increasing correlation length and variability of random field reduces plate's reliability. Currently, several intrusive stochastic finite element methods (SFEMs) such as perturbation method and spectral SFEM are widely applied for stochastic response analysis of continuous structures. However, the intrusive SFEMs need to modify conventional finite element formulations to establish the stochastic stiffness matrix, and cannot calculate the probability density function of structural response and the reliability straightforwardly. This paper proposes a new non-intrusive SFEM for efficiently computing stochastic responses and reliabilities of plates in a unified way. Firstly, the direct probability integral method (DPIM) is developed to obtain the probability density function of stochastic response by solving probability density integral equation (PDIE). Secondly, the non-intrusive SFEM based on DPIM decouples the deterministic finite element analysis and PDIE to calculate the stochastic responses and reliabilities of uncertain plate structures, and the discretization and quantification of random fields of elastic modulus and thickness are implemented through Karhunen-Loève expansion. Finally, several examples of uncertain Kirchhoff and Mindlin plates demonstrate the efficiency and versatility of the proposed non-intrusive method by comparing with the results from Monte Carlo simulation and literature. The effects of correlation length, mean and variability of random field on the probability distribution of responses and the reliabilities of plates are revealed. For Gaussian random thickness, the linearly elastic plate yields non-Gaussian distributed responses. Increasing the correlation length and variability of random field reduces the reliabilities of plates. [ABSTRACT FROM AUTHOR]