Study of Computation for Structural Reliability Index Based on Penalty Function Method

In this paper, a novel approach is proposed to determine target reliability index based on modeling performance degradation of concrete structures as a stochastic process. First, the Wiener process with drift is used to simulate the degradation process of relative dynamic elastic modulus of concrete, and the drift coefficient and diffusion coefficient of Wiener process are estimated according to the experimental data of degradation of relative dynamic elastic modulus. Then, based on the relations between concrete compressive strength, dynamic elastic modulus, and relative dynamic elastic modulus, the relationship between compressive strength and the amount of degradation of relative dynamic elastic modulus are deduced, and then, the statistical parameters of concrete compressive strength under different service time can be calculated. Finally, based on the established limit state equation of flexural capacity of the weak section and the statistical parameters of compressive strength at different times, the reliability index of a structure under different service life is calculated. On the other hand, according to the drift Wiener process model of relative dynamic elastic modulus degradation of concrete, the time‐varying failure probability of structural durability degradation is calculated based on the first‐passage failure probability formula of drift Wiener process. Taking a concrete plant structure as an example, the time‐varying reliability index calculated based on the limit state equation and the time‐varying failure probability obtained based on the first‐passage failure probability analysis of drift Wiener process are compared, and then the recommended values of target reliability index of structural bearing capacity are comprehensively determined.

This chapter discusses the method and application in practical engineering of structural reliability assessment. Reliability theory is the basis for measuring structural safety level under a probability-based framework. This chapter starts from the definition of structural reliability and reliability index, followed by the computation methods of reliability index, including first-order second-moment method, first-order reliability method, simulation-based method and moment-based method. Subsequently, some simple models for system reliability are presented, namely series system, parallel system and k-out-of-n system. The probability-based modelling of structural resistance and external loads are also addressed, followed by the probability-based limit state design approaches. The reliability theory can be further used to optimize structural design and conduct life-cycle cost analysis. Finally, the evaluation of structural reliability in the presence of imprecisely-informed random variables is considered.

This paper considers calculation of structural reliability indices (SRI) for electrical power system (EPS), when «cutoff» failures are taken into account. Calculations are based on «exclusion-renewal» technique. In this paper, the design scheme slightly differs from the electrical scheme. Stochastic network reduction (SNR) technique is considered as the main tool for network reduction. SNR also accounts for directional elements, which have different SRI in forward and backward direction of power flow. The paper shows that difference in resulting SRI of diverse software solutions (SS) arise from different approaches to SNR of series-connected elements. The article compares «p» and «gamma» network reduction techniques (NRT). Taking into account specific nature of EPS, the paper proposes the «gamma» NRT for practical calculations of the EPS structural reliability. The paper also considers implementation of the rhomb NRT at the stage of renewal of the rank 2 nodes. This method is based on parallel calculus of the SRI, which permits substantial speed-up of the design procedure.

Introduction:Reliability analysis is a good tool to deal with the uncertainty and has been widely used in the engineering system. The first order reliability method (FORM) is generally used to calculate the reliability index but FORM is time-consuming and requires derivative computing.Methods:Artificial Bee Colony (ABC) algorithm is a very simple, robust and population-based stochastic optimization algorithm. In this study, an ABC-based reliability analysis was proposed to calculate the reliability index of engineering system through combining Artificial Bee Colony (ABC) algorithm with FORM. FORM was adopted to calculate the reliability index and design point. ABC is used to solve the constrained optimization about FORM. The procedure of ABC-based reliability analysis was presented in detail.Results and Conclusion:The proposed method was verified by two classic examples and then applied to geotechnical engineering. The results show that the ABC algorithm can effectively solve the global optimization problem in FORM. Results demonstrate that ABC-based reliability analysis is a good approach to obtain the reliability index and design point with a good accuracy so that it can be applied to analyze the reliability of a complex engineering system.

The joint eigenvalue decomposition algorithm based on first-order Taylor expansion via the exterior penalty function method

Reliability analysis of rammed earth structures

A two-phase constraint-handling technique is integrated into the evolutionary algorithms to solve constrained optimization problems (called TPDE) in this article. In phase one, denoted as the exploration phase, an exterior penalty function method with the dynamic penalty coefficients is developed to compare any two candidate solutions, which aims to push the population into the feasible region. To reduce the computational burden, in phase two, denoted as the exploitation phase, an interior penalty function method with the dynamic penalty coefficients is developed, which enhances the search ability by using the information of constraints in feasible solutions. During the optimization process, differential evolution is adopted as the search algorithm to produce the offspring population. Experiment results on four benchmark test suites, namely, IEEE CEC 2006, IEEE CEC 2010, IEEE CEC 2017, and IEEE CEC 2020, indicate that TPDE is competitive with other popular algorithms.

In this work, a novel framework is proposed for the risk based design optimization of engineering systems by minimizing the demand on the system components’ accuracy (which directly relates to their cost). The fundamental development of this work is an analytical upper bound for calculating the probability of failure. This is in contrast with First Order Reliability Method (FORM), where a lower bound is used in calculating the probability of failure. FORM is one of the most popular methods for reliability analysis of engineering systems. In this paper, we show that FORM results in an optimistic measure of risk, hence potentially catastrophic in engineering design. A more accurate measure of failure is proposed by utilizing an analytical upper bound for the distribution of reliability index (the length of the most probable point vector to origin). This distribution is a function of the eigenvalues of the linearized limit state function in the normal space which results in a better understanding of failure phenomenon. The proposed formulation is computationally efficient and straightforward to solve, since it only involves finding eigenvalues in each iteration. This algorithm is applicable to any linearizable continuous limit state function with any type of distribution for the design variables. The method is applied to two examples and its accuracy is compared with the Monte Carlo simulation and FORM, demonstrating its effectiveness and value.

For floating production storage and offloading (FPSO) vessels, a dynamic positioning controller is necessary because using only a mooring system is not possible to keep the ship within a predefined region. Position control of the FPSO vessel is extremely challenging due to model uncertainties and unknown control coefficients. This paper develops a new robust adaptive positioning controller consisting of several components: adaptive law, dynamic surface control (DSC) technology, sigmoid tracking differentiator (STD), Nussbaum gain function, and structural reliability index. Model uncertainties can be estimated by the adaptive law derived from the Lyapunov theory. The DSC technology is used to eliminate repeated differentiation by introducing first-order filtering of the virtual control. The chattering-free STD with the characteristics of global fast convergence can estimate the derivatives of model uncertainties that are difficult to calculate directly. Therefore, the DSC and STD techniques make the proposed controller simpler to compute and easier to implement in engineering practice. Most of the traditional controllers require the information about the control coefficients to guarantee the stability of the closed-loop system while the Nussbaum gain function can remove the requirement for a priori knowledge of the sign of control coefficients. The capacity of the mooring system can be fully utilized to position the FPSO vessel by adjusting the structural reliability index on the premise of ensuring the safety of mooring lines, and hence less control effort is needed for the positioning controller. Simulations using two sets of system parameters demonstrate the proposed controller’s effectiveness. In addition, a qualitative comparison with the adaptive backstepping controller shows that our proposed controller is computationally more efficient and does not require a priori knowledge of the sign of control coefficients. A quantitative comparison with robust adaptive controller without the structural reliability shows that less control effort is needed using our proposed controller.

In this paper, we develop a generalized Fibonacci search method for one-dimensional unconstrained non-linear optimization of unimodal functions. This method uses the idea of the “ratio length of 1” from the golden section search. Our method takes successive lower Fibonacci numbers as the initial ratio and does not specify beforehand, the number of iterations to be used. We evaluated the method using Microsoft Excel with nine one-dimensional benchmark functions. We found that our generalized Fibonacci search method out-performed the golden section and other Fibonacci-type search methods such as the Fibonacci, Lucas and Pell approaches.

A neural dynamics model for structural optimization—Theory

System reliability-based seismic collapse assessment of steel moment frames using incremental dynamic analysis and Bayesian probability network

Structural health monitoring data has been widely acknowledged as a significant source for evaluating the performance and health conditions of structures. However, a holistic framework that efficiently incorporates monitored data into structural identification and, in turn, provides a realistic life-cycle performance assessment of structures is yet to be established. There are different sources of uncertainty, such as structural parameters, computer model bias and measurement errors. Neglecting to account for these factors results in unreliable structural identifications, consequent financial losses, and a threat to the safety of structures and human lives. This paper proposes a new framework for structural performance assessment that integrates a comprehensive probabilistic finite element model updating approach, which deals with various structural identification uncertainties and structural reliability analysis. In this framework, Gaussian process surrogate models are replaced with a finite element model and its associate discrepancy function to provide a computationally efficient and all-round uncertainty quantification. Herein, the structural parameters that are most sensitive to measured structural dynamic characteristics are investigated and used to update the numerical model. Sequentially, the updated model is applied to compute the structural capacity with respect to loading demand to evaluate its as-is performance. The proposed framework’s feasibility is investigated and validated on a large lab-scale box girder bridge in two different health states, undamaged and damaged, with the latter state representing changes in structural parameters resulted from overloading actions. The results from the box girder bridge indicate a reduced structural performance evidenced by a significant drop in the structural reliability index and an increased probability of failure in the damaged state. The results also demonstrate that the proposed methodology contributes to more reliable judgment about structural safety, which in turn enables more informed maintenance decisions to be made.

FPSO is widely used during the deep‐sea oil and gas exploration operations, for which it is an effective way to keep their position by means of positioning mooring (PM) technology to ensure the long‐term reliability of operations, even in extreme seas. Here, a kind of dynamic positioning (DP) controller in terms of structural reliability is presented for the single‐point turret‐moored FPSOs. Firstly, the mathematical model of the moored FPSO in terms of kinematics and dynamics is established. Secondly, the catenary method is applied to analyze the mooring line dynamics, and mathematical model of one single mooring line is set up based on the catenary equation. Thereafter, mathematical model for the whole turret mooring system is established. Thirdly, a structural reliability index is defined to evaluate the breaking strength of each mooring line. At the same time, control constraints are also considered to design a state feedback controller using the backstepping technique. Finally, a series of simulation tests are carried out for a certain turret‐moored FPSO with eight mooring lines. It is shown in the simulation results that the moored FPSO can keep its position well in extreme seas. Besides, the FPSO mooring line tension is reduced effectively to ensure mooring lines safety to a large extent in harsh sea environment.

Application of the homotopy analysis method to determine the analytical limit state functions and reliability index for large deflection of a cantilever beam subjected to static co-planar loading