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In this paper, a new analytical formula is proposed to determine the mean outcrossing rate of nonstationary non-Gaussian performance functions. The performance function is firstly transform...

Structural reliability analysis without the exclusion of random variables with unknown distributions has attracted increasing attention, and many endeavors have been devoted to this aspect....

In determining structural responses under random excitation using the Monte Carlo method, simulation of non-Gaussian processes is always a challenge. Various methods have been developed to ...

The effects of erosion mode, erosion age, and concentration of sulfate solution on mechanical properties of concrete were investigated. The dimensionless relationship model of the stress-strain of concrete on the basis of randomness was proposed. The variation of the elasticity modulus and Poisson’s ratio of the concrete surface attacked by sulfate was studied, and a novel method of using a superficial parameter to characterize the performance change of the concrete surface was recommended. The results showed that the dimensionless relationship model of stress-strain of concrete could be used to represent the variations of mechanical properties of concrete. The differences of load-displacement of concrete before and after sulfate attack were reflected as the change of curve’s slope and ultimate bearing capacity, and the slope of a straight section of the lateral and longitudinal strain curves of concrete surface also varied. The increment rates of ultimate bearing capacity of concrete attacked by 1% and saturated sulfate solution were about 30% and 10%, respectively. However, the decreasing ratio of the ultimate bearing capacity of concrete attacked by saturated sulfate solution was approximately 25%. The damage factor of the elasticity modulus of the concrete surface of C20 and C40 was 0.185 and −0.19, respectively. The obtained results could provide a support for investigating the variations of stress-strain relationship and mechanical performance of concrete under a sulfate environment.

In this article, an analytical moment-based procedure is developed for estimating the first passage probability of stationary non-Gaussian structural responses for practical applications. In the procedure, an improved explicit third-order polynomial transformation (fourth-moment Gaussian transformation) is proposed, and the coefficients of the third-order polynomial transformation are first determined by the first four moments (i.e. mean, standard deviation, skewness, and kurtosis) of the structural response. The inverse transformation (the equivalent Gaussian fractile) of the third-order polynomial transformation is then used to map the marginal distributions of a non-Gaussian response into the standard Gaussian distributions. Finally, the first passage probabilities can be calculated with the consideration of the effects of clumping crossings and initial conditions. The accuracy and efficiency of the proposed transformation are demonstrated through several numerical examples for both the “softening” responses (with wider tails than Gaussian distribution; for example, kurtosis > 3) and “hardening” responses (with narrower tails; for example, kurtosis

Probability distributions of basic random variables are essential for the accurate evaluation of structural reliability. In engineering practice, the probability distributions of some random variables are often unknown and the only available information about these may be their statistical moments. To conduct structural reliability analysis without the exclusion of random variables with unknown probability distributions, the fourth-moment normal transformation (FMNT) has been proposed. However, the applicability of expression of the FMNT has not been sufficiently investigated. Furthermore, the monotonic regions of the FMNT are not defined without which the application of the transformation is inconvenient, or even unreliable in reliability analysis. In the present paper, a complete expression of the FMNT including six cases with different combinations of skewness and kurtosis is derived, and the monotonicity of each case of the FMNT expression is confirmed. Literature suggests that the complete monotonic expression of the fourth-moment normal transformation is the first time to be successfully accomplished up to date. Through the numerical examples, the FMNT is found to be quite efficient for normal transformation and to be sufficiently accurate to include random variables with unknown probability distributions in structural reliability analysis.

Analyzing the uncertainty of flutter propagated from different sources especially flutter derivatives is important for the risk assessment of long-span bridges. The uncertainty of flutter derivatives is often characterized based on the experimental data. Dependence and sampling error would therefore exist which may influence the onset prediction of bridges. This paper devotes to developing a framework towards the uncertainty analysis of bridge flutter considering the dependence and sampling error in flutter derivative measurements. In this framework, the dependence among different flutter derivatives is identified with Spearman’s rho instead of conventional Pearson’s product-moment correlation coefficient. A modified procedure based on bootstrap is proposed to characterize the sampling error by representing distribution parameters of flutter derivatives as interval variables. Regarding the probability-interval hybrid uncertainty, a double-layer analysis is employed. In the inner layer, probability of onset velocity is estimated using the point estimate-based probability analysis. In the outer one, the distribution envelope is obtained by the genetic algorithm-assisted interval analysis. Application of the framework to reliability assessment of bridge against flutter is also presented. In the numerical study, the influence of dependence and sampling error is verified in detail. By this framework, the distribution envelope of onset velocity or the interval of reliability index against flutter are expected to contain the true estimates. Consequently, the potential risk with respect to bridge flutter can be assessed in a conservative way.

In practical engineering, the probability distributions of some random variables are often unknown, and the only available information about these may be their statistical moments. To conduct structural reliability assessment without the exclusion of random variables with unknown probability distributions, an explicit fourth-moment standardization function has been proposed, and a single expression of its inverse transformation, that is, normal transformation, with limitations of inputting sets of the third and fourth moments (skewness and kurtosis) of random variables was derived. However, the clear definition of the complete expressions of the inverse transformation of fourth-moment standardization function under different combinations of skewness and kurtosis of random variables has not been provided yet. It is in this regard that four criteria are proposed to derive the complete inverse transformation of fourth-moment standardization function, and then the complete expressions of the inverse transformation are formulated. Through the numerical examples presented, the proposed complete expressions are found to be quite efficient for normal transformations and to be sufficiently accurate to include random variables with unknown probability distributions in structural reliability assessment.

In this paper, an efficient and explicit technique is proposed for transforming correlated non-normal random variables into independent standard normal variables based on the three-parameter (3P) lognormal distribution. In contrast with the classic Nataf transformation, the derived equivalent correlation coefficient in non-orthogonal standard normal space of the proposed transformation is expressed as an explicit formula, thereby avoiding tedious iteration algorithm or multifarious empirical formulas. Meanwhile, the applicable range of the original correlation coefficient is determined based on fundamental properties of the proposed expression of correlation distortion and the definition of correlation coefficient. The proposed transformation requires only the first three moments (i.e., mean, standard deviation, and skewness) of basic random variables, as well as their correlation matrix. Therefore, the proposed transformation can also be applied even when the joint distribution or marginal distributions of the basic random variables are unknown. Several numerical examples are presented to demonstrate the user-friendliness, efficiency, and accuracy of the proposed transformation applied in structural reliability analysis involving correlated non-normal random variables.

In this paper, a new method for normal transformation is proposed to transform correlated non-normal random variables into independent standard normal ones based on their first four linear moments (L-moments), standard deviations and correlation matrix. The complete monotonic expressions of the equivalent correlation coefficient are proposed and the applicable range of the original correlation coefficient to ensure the transformation's executability is identified. Numerical studies demonstrate that the admissible range of the normal transformation for correlated random variables based on L -moments is larger in scope than that based on ordinary central moments (C-moments), and the proposed method is effective for normal transformations and sufficiently accurate for reliability engineering practices.

In this paper, an efficient method is proposed for constructing the seismic fragility curves of bridge piers, in which the dynamical reliability assessment is developed from methods of moment. The main procedure for constructing seismic fragility curves consists of four steps. First, the non-stationary seismic acceleration process samples are simulated by the random function-spectrum representation model (RFSRM) with one elementary random variable at a given PGA value. Second, the first four moments (i.e., mean value, standard deviation, skewness and kurtosis) of extreme structural response are then evaluated from the extremum of each nonlinear time-history seismic response based on RFSRM. Third, the failure probability is then estimated by the fourth-moment reliability index, which is derived from an explicit fourth-moment transformation. Finally, the seismic fragility curves can be constructed by repeating the three steps and considering different PGA values and damage levels (i.e., slight, moderate, extensive, and complete). The efficiency and accuracy of the proposed methodology are demonstrated by a numerical example for constructing the fragility curves of a high-speed railway bridge pier in which the direct incremental dynamic analysis (IDA) method based on Monte Carlo simulations are employed for comparison.

In this paper, a fourth-moment transformation technique is proposed to transform correlated nonnormal random variables into independent standard normal ones. The procedure mainly includes two steps: First, the correlated nonnormal random variables are transformed into correlated standard normal ones using the fourth-moment transformation, where the complete mathematical formula of the correlation coefficient in standard normal space, i.e., equivalent correlation coefficient, is proposed and the upper and lower bounds of original correlation coefficient are identified to ensure the transformation executable; Second, the correlated standard normal random variables are transformed into independent standard normal ones using Cholesky decomposition. For the cases of original correlation matrix with very small eigenvalues, the equivalent correlation matrix might become a nonpositive semidefinite matrix. A recently developed method for solving the problem is adopted to make Cholesky decomposition ready. A first-order reliability method (FORM) for structural reliability analysis involving correlated random variables is developed using the proposed transformation technique. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method for structural reliability assessment considering correlated random variables.

In structural reliability analysis, the input variables are often nonnormal and correlated. A procedure for efficient normal transformation, i.e., transforming dependent nonnormal random va...

Estimation of statistical moments of performance functions is one of the main topics in structural reliability analysis by moment methods. It is possible to apply the well-developed multipl...

The reliability-based load and resistance factors design (LRFD) has been widely used in the structural design codes. In almost all of the current reliability methods for the determination of the load and resistance factors, the basic random variables are assumed to have known probability distributions. However, in reality, the probability distributions of some of the basic random variables are often unknown due to the lack of statistical data. In this paper, the high-order moment methods for LRFD including random variables with unknown probability distributions are proposed. From the investigation of the present paper, it can be concluded that: 1) The load and resistance factors can be determined even when the probability distributions of the basic random variables are unknown; 2) The present method is convenient and more effective in estimating the load and resistance factors in practical engineering since it needs neither the iterative computation of derivatives nor any design points; 3) In the applicable range of the high-order moment method, although the load and resistance factors obtained by the proposed method may be different from those obtained by first order reliability method (FORM), the target mean resistances obtained by both methods are essentially the same.

Load and resistance factors are generally obtained using the first order reliability method (FORM) in which the design point should be determined and derivative-based iterations used. In this article, the third-moment reliability index, based on the three-parameter lognormal (3P-lognormal) distribution, is investigated. A simple method based on the third-moment method for estimating load and resistance factors is then proposed, and a simple formula for the target mean resistance is also presented to avoid iterative computations. Unlike the currently used method, the proposed method can be used to determine load and resistance factors, even when the probability density functions (PDFs) of the basic random variables are not available. Moreover, the proposed method does not require the iterative computation of derivatives or any design points. Thus, the method provides a more convenient and effective way to estimate load and resistance factors in practical engineering applications. Numerical examples are presented to demonstrate the advantages of the proposed third moment method for determining load and resistance factors.

The load and resistance factors are generally obtained using the First Order Reliability Method (FORM), in which the design point should be determined and derivative-based iterations have to be used. In this paper, a simple method for estimating the load and resistance factors using the first four moments of the basic random variables is proposed and a simple formula for the target mean resistance is also proposed to avoid iteration computation. Unlike the currently used method, the load and resistance factors can be determined using the proposed method even when the probability density functions (PDFs) of the basic random variables are not available. Moreover, the proposed method does not need either the iterative computation of derivatives or any design points. Thus, the present method provides a more convenient and effective way to estimate the load and resistance factors in practical engineering. Numerical examples are presented to demonstrate the advantages of the proposed fourth moment method for determining the load and resistance factors.

Information on the distribution of the basic random variable is essential for the accurate analysis of structural reliability. The usual method for determining the distributions is to fit a candidate distribution to the histogram of available statistical data of the variable and perform approximate goodness-of-fit tests. Generally, such candidate distribution would have parameters that may be evaluated from the statistical moments of the statistical data. In the present paper, a cubic normal distribution, whose parameters are determined using the first four moments of available sample data, is investigated. A parameter table based on the first four moments, which simplifies parameter estimation, is given. The simplicity, generality, flexibility and advantages of this distribution in statistical data analysis and its significance in structural reliability evaluation are discussed. Numerical examples are presented to demonstrate these advantages.

In order to avoid the difficulties involved in searching the design point by iteration using the derivatives of the performance function, a principle shortcoming of reliability methods based on the first-order reliability method (FORM), an explicit fourth-moment reliability index, which is derived from the fourth-moment standardization of performance function, is presented and investigated in the present paper. The proposed formula is simple, and does not require integration or solve nonlinear equations, which are necessary in current used formulas. Through the investigations of the present paper, it is found that the third-moment method can be significantly improved by the proposed simple fourth moment reliability index and the fourth-moment reliability index has sufficient accuracy as a reliability index based on the fourth-moment standardization of performance function.

The load and resistance factors are generally determined using the first order reliability method, in which the “design point” must be determined and derivative-based iteration has to be used. In the present paper, the general formula of the load and resistance factor based on the method of moments is deduced based on the review of basic principle of the load and resistance factor format for structural design. A simple formula for determining the load and resistance factors using the fourth moment method in which neither the design point nor derivative-based iteration is necessary, is proposed. From the investigation of the present paper, it is found that the present method gives good improvement upon the method based on the third moment method and although the load and resistance factors obtained using the method of moments are different from those obtained through the first order reliability method, the required structural resistances under a specific design condition are almost the same.

In structural reliability analysis, the uncertainties related to resistance and load are generally expressed as random variables that have known cumulative distribution functions. However, in practical applications, the cumulative distribution functions of some random variables may be unknown, and the probabilistic characteristics of these variables may be expressed using only statistical moments. In the present paper, in order to conduct structural reliability analysis without the exclusion of random variables having unknown distributions, the third-order polynomial normal transformation technique using the first four central moments is investigated, and an explicit fourth-moment standardization function is proposed. Using the proposed method, the normal transformation for indepen- dent random variables with unknown cumulative distribution functions can be realized without using the Rosenblatt transformation or Nataf transformation. Through the numerical examples presented, the proposed method is found to be sufficiently accurate in its inclusion of the independent random variables which have unknown cumulative distribution functions, in structural reliability analyses with minimal additional computational effort.

It is important for obtaining the relationship between seismic energies of single degree-of-freedom (SDOF) systems and multiple degree-of-freedom (MDOF) structures in engineering. In this paper, the formula of hysteretic energy between the MDOF structures and equivalent SDOF systems is developed. Here is also presented the procedure for estimating hysteretic energy of MDOF structures subjected to severe ground motions employing the energy relation equation based on equivalent SDOF systems. Eight examples for two regular and six irregular MDOF structures show that the procedure to obtain the hysteretic energy demands of MDOF structures may be used as a simple and effective energy estimation method.

In this paper, the applicable range of the fourth–moment method for structural reliability is investigated and a simple fourth–moment reliability index is suggested. In the applicable range of the ...

In this paper, a new method for estimating the load and resistance factors using the first three moments of random variables is proposed. Unlike the currently used method, the load and resistance factors can be determined without using distributions of random variables, and the present method needs neither the iterative computation of derivatives nor any design points. The present method can effectively reflect the characteristics of the skewness of random variables and the performance function, and generally provide much more accurate results than the second moment method. Thus, the present method should be convenient and more effective in estimating the load and resistance factors in practical engineering. Numerical examples are presented to demonstrate the advantages of the present method.

The objectives of the present paper are to investigate the applicable range of the third-moment method for structural reliability and to suggest a simple third-moment method for practical application in engineering. The applicable range of the second-moment method is also given. The applicable range of the third-moment method is obtained through investigation of the differences among several third-moment methods. Within the applicable range, it is found that the simple reliability index has a good agreement with the original one, and it is therefore suggested as a simple third-moment reliability index. Since only the first three central moments of the performance functions are used, and since it is unnecessary to know the probability distribution of the basic random variables, the present method should be practical in engineering. In order to investigate the efficiency of the proposed method, several examples are examined under different conditions.