Optimum values of structural safety factors for a predefined reliability level with extension to multiple limit states

In the field of deterministic structural optimization, the designer reduces the structural cost without taking into account uncertainties concerning materials, geometry and loading. This way, the resulting optimum solution may represent a lower level of reliability and thus a higher risk of failure. It is the objective of reliability-based design optimization (RBDO) to design structures that should be both economic and reliable. The coupling between mechanical modeling, reliability analyses and optimization methods leads to very high computational costs and weak convergence stability. Since the traditional RBDO solution is achieved by alternating between reliability and optimization iterations, the structural designers performing deterministic optimization do not consider the RBDO model as a practical tool for the design of real structures. Fortunately, a hybrid method based on simultaneous solution of the reliability and the optimization problem, has successfully reduced the computational time problem. The hybrid method allows us to satisfy a required reliability level, but the vector of variables here contains both deterministic and random variables. The hybrid RBDO problem is thus more complex than that of deterministic design. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a special optimization procedure. In this paper a new methodology is presented with the aim of finding a global solution to RBDO problems without additional computing cost for the reliability evaluation. The safety factor formulation for a single limit state case has been used to efficiently reduce the computational time . This technique is fundamentally based on a study of the sensitivity of the limit state function with respect to the design variables. In order to demonstrate analytically the efficiency of this methodology, the optimality condition is then used. The efficiency of this technique is also extended to multiple limit state cases. Two numerical examples are presented at the end of the paper to demonstrate the applicability of the new methodology.     

Time-variant reliability-based design optimization using sequential kriging modeling

This paper presents a sequential Kriging modeling approach (SKM) for time-variant reliability-based design optimization (tRBDO) involving stochastic processes. To handle the temporal uncertainty, time-variant limit state functions are transformed into time-independent domain by converting the stochastic processes and time parameter to random variables. Kriging surrogate models are then built and enhanced by a design-driven adaptive sampling scheme to accurately identify potential instantaneous failure events. By generating random realizations of stochastic processes, the time-variant probability of failure is evaluated by the surrogate models in Monte Carlo simulation (MCS). In tRBDO, the first-order score function is employed to estimate the sensitivity of time-variant reliability with respect to design variables. Three case studies are introduced to demonstrate the efficiency and accuracy of the proposed approach.     

Temporal and spatial multi-parameter dynamic reliability and global reliability sensitivity analysis based on the extreme value moments

Aiming at efficiently estimating the dynamic failure probability with multiple temporal and spatial parameters and analyzing the global reliability sensitivity of the dynamic problem, a method is presented on the moment estimation of the extreme value of the dynamic limit state function. Firstly, two strategies are proposed to estimate the dynamic failure probability. One strategy is combining sparse grid technique for the extreme value moments with the fourth-moment method for the dynamic failure probability. Another is combining dimensional reduction method for fractional extreme value moments and the maximum entropy for dynamic failure probability. In the proposed two strategies, the key step is how to determine the temporal and spatial parameters where the dynamic limit state function takes their minimum value. This issue is efficiently addressed by solving the differential equations satisfying the extreme value condition. Secondly, three-point estimation is used to evaluate the global dynamic reliability sensitivity by combining with the dynamic failure probability method. The significance and the effectiveness of the proposed methods for estimating the temporal and spatial multi-parameter dynamic reliability and global sensitivity indices are demonstrated with several examples.     

On the Stochastic Sensitivity and Noise-Induced Transitions of a Kaldor-Type Business Cycle Model

In the paper, we consider a Kaldor-type model of the business cycle with external additive and internal parametric disturbances. We study analytically and numerically the probability properties of stochastically forced equilibria and limit cycles via stochastic sensitivity function technique. In particular, we discuss the effects of additive and parametric noises on the economic variables and we detect some stochastic bifurcations such as a P-bifurcation, i.e a phenomenon of noise-induced transition from monostability to bistability. This stochastic bistability causes a new trigger regime in economic dynamics.     

Stochastic optimum design criterion for linear damper devices for seismic protection of buildings

This paper concerns a new approach for the reliability-based optimum design of linear multistory frames seismically protected by viscous dampers. A deterministic objective function, defined as the total added damping, is minimized, while stochastic restraints impose a limit to system failure probability. This latter is here related to a maximum interstory drift crossing over a given value, which is associated with damage control requirements. System failure probability is evaluated by means of the covariance analysis of a frame subject to a seismic load represented by a nonstationary modulated Kanai–Tajimi stochastic process. Numerical examples are developed, concerning a plane-shear-type frame with three floors equipped with viscous damper devices. Two different possible design solutions are performed, considering different earthquake intensities: more in detail, the first adds constant damping at each story, while the second considers a variable added damping distribution.     

Maximum probable life time analysis under the required time-dependent failure probability constraint and its meta-model estimation

Time-dependent reliability (failure probability) aims at measuring the probability of the normal (abnormal) operation for structure/mechanism within the given time interval. To analyze the maximum probable life time under a required time-dependent failure probability (TDFP) constraint, an inverse process corresponding to the time-dependent reliability is proposed by taking the randomness of the input variables into consideration. The proposed inverse process employs the monotonicity between the TDFP and the upper boundary of the given time interval which reflects the life time, and an adaptive single-loop sampling meta-model for the time-dependent limit state function is presented to estimate the TDFP at the given time interval flexibly. Since the TDFP is generally monotonic to the upper boundary of the given time interval, thus by adjusting the probable upper and lower boundaries of the time interval in which the corresponding TDFPs include the required TDFP constraint, the proposed approach can always search the maximum probable life time at the required TDFP by the dichotomy. By introducing the time variable as an input which is the same level as the input random variables and constructing the adaptive single-loop sampling meta-model for the time-dependent limit state function in a longer time interval with the TDFP bigger than the required TDFP, the TDFP in any subintervals of the time interval involved in the constructed meta-model can be estimated as a byproduct of the constructed meta-model without any additional actual limit state evaluations. Then the efficiency for analyzing the maximum probable life time is improved by the dichotomy and the unified meta-model of the time-dependent limit state function. Two examples are employed to illustrate the accuracy and the efficiency of the proposed approach.     

Reliability-based structural optimization under hazard loads

The present paper studies the reliability-based structural optimization of the civil engineering in the seismic zone. The objective is to minimize the sum of construction material cost and the expected failure loss under severe earthquake, which is obtained by the sum of the products of the failure probability and its failure losses for the important failure modes. The set of constraints includes the deterministic constraints, and the constraints based on structural reliability—the reliability index constraints of structural element failure for the serviceability state under minor earthquake and the failure probability of the structural system for the ultimate limit state under severe earthquake. By introducing the load roughness index, the structural system reliability computation under hazard load can be greatly simplified, which is approximately determined by its weakest failure mode. Finally, the numerical example of high rising shear RC frame is calculated.     

Saddlepoint approximation based structural reliability analysis with non-normal random variables

The saddlepoint approximation (SA) can directly estimate the probability distribution of linear performance function in non-normal variables space. Based on the property of SA, three SA based methods are developed for the structural system reliability analysis. The first method is SA based reliability bounds theory (RBT), in which SA is employed to estimate failure probability and equivalent normal reliability index for each failure mode firstly, and then RBT is employed to obtain the upper and the lower bo...     

A fast approximate method for parameter sensitivity estimation in Monte Carlo structural reliability

In estimating the effect of a change in a random variable parameter on the (time-invariant) probability of structural failure estimated through Monte Carlo methods the usual approach is to carry out a duplicate simulation run for each parameter being varied. The associated computational cost may become prohibitive when many random variables are involved. Herein a procedure is proposed in which the numerical results from a Monte Carlo reliability estimation procedure are converted to a form that will allow the basic ideas of the first order reliability method to be employed. Using these allows sensitivity estimates of low computational cost to be made. Illustrative examples with sensitivities computed both by conventional Monte Carlo and the proposed procedure show good agreement over a range of probability distributions for the input random variables and for various complexities of the limit state function.     

Intervening variables and constraint approximations in safety index and failure probability calculations

The emphasis of this paper is on developing suitable intervening variables and constraint approximations for structural reliability analysis. Traditionally, these procedures are used in structural optimization, whereas this research work adopts these concepts to safety index and failure probability computations. The use of these concepts enables the development of an efficient and stable iteration algorithm for identifying the most probable failure points (MPPs) of the limit state functions. An approximate second-order failure probability is calculated at this MPP with no extra computations of the limit state function and gradients. The efficiency and accuracy of the proposed algorithm are demonstrated by several examples with highly nonlinear, complex, explicit/implicit performance functions.     

Extend the quickest path problem to the system reliability evaluation for a stochastic-flow network

From the point of view of quality management, it is an important issue to reduce the transmission time in the network. The quickest path problem is to find the path in the network to send a given amount of data from the source to the sink such that the transmission time is minimized. Traditionally, this problem assumed that the capacity of each arc in the network is deterministic. However, the capacity of each arc is stochastic due to failure, maintenance, etc. in many real-life networks. This paper proposes a simple algorithm to evaluate the probability that d units of data can be sent from the source to the sink through the stochastic-flow network within T units of time. Such a probability is called the system reliability. The proposed algorithm firstly generates all lower boundary points for (d,T) and the system reliability can then be computed in terms of such points.Scope and purposeThe shortest path problem is a well-known problem in operations research, computer science, etc. Chen and Chin have proposed a variant of the shortest path problem, termed the quickest path problem. It is to find a path in the network to send a given amount of data from the source to the sink with minimum transmission time. More specifically, the capacity of each arc in the network is assumed to be deterministic. However, in many real-life networks such as computer systems, telecommunication systems, etc., the capacity of each arc is stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. This paper proposes a simple algorithm to evaluate the probability that the specified amount of data can be sent from the source to the sink through the network within a given time. Such a probability is called the system reliability.     

Computational strategies for reliability-based structural optimization of aeroelastic limit cycle oscillations

Gradient-based optimization, via the adjoint method, is needed to realistically enable the reliability-based design of a nonlinear unsteady aeroelastic system with many random and/or deterministic design variables. The adjoint derivatives of a time-marched system entail a cumbersome reverse-time integration, and so a time-periodic spectral element scheme is used here to efficiently capture the gradients of the limit cycle oscillations. Further reductions in the computational cost of the monolithic-time adjoint vector are obtained with proper orthogonal decomposition, which projects the large system onto a reduced basis. Design reliability is computed with the first order reliability method, which provides an estimate of the failure probability without resorting to sampling-based approaches (infeasible for large systems). Analytical gradients are needed to obtain the most probable point (in the random variable space), as well as the reliability design derivatives. These computational strategies are utilized to locate the optimal thickness distribution of a cantilevered wing operating beyond its flutter point in supersonic flow (via piston theory). Specifically, the wing mass is minimized under both deterministic and non-deterministic limit cycle oscillation amplitude constraints, with both structural and flow uncertainties considered in the latter.     

Design sensitivity and reliability-based structural optimization by univariate decomposition

This paper presents a new univariate decomposition method for design sensitivity analysis and reliability-based design optimization of mechanical systems subject to uncertain performance functions in constraints. The method involves a novel univariate approximation of a general multivariate function in the rotated Gaussian space for reliability analysis, analytical sensitivity of failure probability with respect to design variables, and standard gradient-based optimization algorithms. In both reliability and sensitivity analyses, the proposed effort has been reduced to performing multiple one-dimensional integrations. The evaluation of these one-dimensional integrations requires calculating only conditional responses at selected deterministic input determined by sample points and Gauss–Hermite integration points. Numerical results indicate that the proposed method provides accurate and computationally efficient estimates of the sensitivity of failure probability, which leads to accurate design optimization of uncertain mechanical systems.     

Optimal signaling and detector design for M-ary communication systems in the presence of multiple additive noise channels

An M-ary communication system is considered in which the transmitter and the receiver are connected via multiple additive (possibly non-Gaussian) noise channels, any one of which can be utilized for the transmission of a given symbol. Contrary to deterministic signaling (i.e., employing a fixed constellation), a stochastic signaling approach is adopted by treating the signal values transmitted for each information symbol over each channel as random variables. In particular, the joint optimization of the channel switching (i.e., time sharing among different channels) strategy, stochastic signals, and decision rules at the receiver is performed in order to minimize the average probability of error under an average transmit power constraint. It is proved that the solution to this problem involves either one of the following: (i) deterministic signaling over a single channel, (ii) randomizing (time sharing) between two different signal constellations over a single channel, or (iii) switching (time sharing) between two channels with deterministic signaling over each channel. For all cases, the optimal strategies are shown to employ corresponding maximum a posteriori probability (MAP) decision rules at the receiver. In addition, sufficient conditions are derived in order to specify whether the proposed strategy can or cannot improve the error performance over the conventional approach, in which a single channel is employed with deterministic signaling at the average power limit. Finally, numerical examples are presented to illustrate the theoretical results.     

Reliability analysis of structures with complex limit state functions using probability density evolution method

In reliability-based structural analysis and design optimization, there exist some limit state functions exhibiting disjoint failure domains, multiple design points and discontinuous responses. This study addresses this type of challenging problem of reliability assessment of structures with complex limit state functions based on the probability density evolution method (PDEM). Probability density function (PDF) of stochastic structures under static and dynamic loads can be acquired, which is independent of the specific form of limit state functions. Numerical results of several typical examples illustrate that, the time-invariant and instantaneous PDF curves and failure probabilities of stochastic structures with disjoint failure domains, multiple design points and discontinuous responses are calculated effectively and accurately. Moreover, the PDEM is validated to be more efficient than the Monte Carlo simulation and the subset simulation, and is a feasible and general approach to tackle the reliability analysis of complicated problems. In addition, the influence of random design parameters of structures on uncertainty propagation is also scrutinized.     

Reliability calculation of time-consuming problems using a small-sample artificial neural network-based response surface method

An important step when designing and assessing the reliability of existing structures and/or structural elements is to calculate the reliability level described by failure probability or reliability index. Since calculating the structural response of complex systems such as bridges is usually a time-consuming task, the utilization of approximation methods with a view to reducing the computational effort to an acceptable level is an appropriate solution. The paper introduces a small-sample artificial neural network-based response surface method. An artificial neural network is used as an approximation (a so-called response surface) of the original limit state function. In order to be as effective as possible with respect to computational effort, a stratified Latin hypercube sampling simulation method is utilized to properly select training set elements. Subsequently, the artificial neural network-based response surface is utilized to calculate failure probability. To increase the accuracy of the determined failure probability, the response surface can be updated close to the failure region. This is performed by finding a new anchor point, which lies close to the design point of the limit state function. The new anchor point is then used to prepare the updated training set. The efficiency of the proposed method is tested for different training set sizes using a nonlinear limit state function taken from the literature, and the reliability assessment of three concrete bridges, one with explicit and two with implicit limit state functions in the form of finite element method models.

     

A simulation method for time-dependent structural reliability

A one-dimensional simulation procedure is developed for use in estimating structural reliability in multi-dimensional load and resistance space with the loads represented as stochastic process. The technique employed is based on the idea of using ‘strips’ of points parallel to each other and sampled on the limit state hyperplanes. The ‘local’ outcrossing rate and the zero time failure probability P_f(0) associated with the narrow strips are derived using the conditional reliability index. When the domain boundary consists of a set of limit states, second order bounds are used to obtain a lower bound approximation of the outcrossing rate and P_f(0) associated with the union of a set of λ strips. It is shown by examples that for high reliability problems, λ may be much less than the number of limit states without significant loss of accuracy and with considerable saving in computation time. It was also found that the rate of convergence of the simulations is quite fast even without using importance sampling.     

Conditional probability Markov chain simulation based reliability analysis method for nonnormal variables

Based on fast Markov chain simulation for generating the samples distributed in failure region and saddlepoint approximation(SA) technique,an efficient reliability analysis method is presented to evaluate the small failure probability of non-linear limit state function(LSF) with non-normal variables.In the presented method,the failure probability of the non-linear LSF is transformed into a product of the failure probability of the introduced linear LSF and a feature ratio factor.The introduced linear LSF wh...     

Reliability analysis of structures using stochastic response surface method and saddlepoint approximation

Stochastic response surface method (SRSM) is a technique used for reliability analysis of complex structural systems having implicit or time consuming limit state functions. The main aspects of the SRSM are the collection of sample points, the approximation of response surface and the estimation of the probability of failure. In this paper, sample points are selected close to the most probable point of failure and the actual limit state surface (LSS). The response surface is fitted using the weighted regression technique, which allows the fitting points to be weighted based on their distance from the LSS. The cumulant generating function (CGF) of the response surface is derived analytically. The saddlepoint approximation (SPA) method is utilized to compute the probability of failure of the structural system. Finally, four numerical examples compare the proposed algorithm with the traditional quadratic polynomial SRSM, Kriging based SRSM and direct MCS.     

Inverse analysis method using MPP-based dimension reduction for reliability-based design optimization of nonlinear and multi-dimensional systems

There are two commonly used analytical reliability analysis methods: linear approximation - first-order reliability method (FORM), and quadratic approximation - second-order reliability method (SORM), of the performance function. The reliability analysis using FORM could be acceptable in accuracy for mildly nonlinear performance functions, whereas the reliability analysis using SORM may be necessary for accuracy of nonlinear and multi-dimensional performance functions. Even though the reliability analysis using SORM may be accurate, it is not as much used for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is rather difficult to develop.This paper proposes an inverse reliability analysis method that can be used to obtain accurate probability of failure calculation without requiring the second-order sensitivities for reliability-based design optimization (RBDO) of nonlinear and multi-dimensional systems. For the inverse reliability analysis, the most probable point (MPP)-based dimension reduction method (DRM) is developed. Since the FORM-based reliability index (β) is inaccurate for the MPP search of the nonlinear performance function, a three-step computational procedure is proposed to improve accuracy of the inverse reliability analysis: probability of failure calculation using constraint shift, reliability index update, and MPP update. Using the three steps, a new DRM-based MPP is obtained, which estimates the probability of failure of the performance function more accurately than FORM and more efficiently than SORM. The DRM-based MPP is then used for the next design iteration of RBDO to obtain an accurate optimum design even for nonlinear and/or multi-dimensional system. Since the DRM-based RBDO requires more function evaluations, the enriched performance measure approach (PMA+) with new tolerances for constraint activeness and reduced rotation matrix is used to reduce the number of function evaluations.