Reliability analysis of underground excavation in elastic-strain-softening rock mass

This paper deals with the reliability analysis of a circular tunnel in elastic-strain-softening rock mass. Dilatancy angle which varies with softening parameter in different stress conditions is accounted for. Deterministic and probabilistic analyses of the circular tunnel in elastic-strain-softening rock mass are performed. Computational procedures for the first-order and second-order reliability methods (FORM/SORM) are used in the reliability analyses of the elastic-strain-softening model. The results are in good agreement with those from Monte Carlo simulations incorporating importance sampling. Reliability-based design of the required support pressure for the circular tunnel is efficiently conducted. The effect of positive correlation between compressive strength and elastic modulus of the rock mass on the reliability of the tunnel is discussed. The influence of in situ field stress and support pressure as random variables on the probability of failure of the tunnel is investigated.     

Reliability analysis of roof wedges and rockbolt forces in tunnels

The ambiguous nature of the factor of safety is first discussed in the context of a symmetric roof wedge of a circular tunnel, where two different definitions of the factor of safety are shown to be reconcilable when using the reliability index computed with the first-order reliability method (FORM). The probabilities of failure based on the second-order reliability method (SORM) are also obtained for comparison with those of FORM and Monte Carlo simulations. The FORM and SORM analyses are then applied to a circular tunnel supported with elastic rockbolts in a homogeneous and isotropic elasto-plastic ground with the Coulomb failure criterion. The similarities and differences between the ratios of mean values to design-point values, on the one hand, and the partial factors of limit state design, on the other hand, are discussed. Finally, all this is used to show how a reliability-based design can be performed to obtain the length and spacing of rockbolts for a target reliability index.     

Reliability analysis of tunnel using least square support vector machine

In the reliability analysis of tunnels, the limited state function is implicit and nonlinear, and is difficult to apply based on the traditional reliability method, especially for large-scale projects. Least squares support vector machines (LS-SVM) are capable of approximating the limited state function without the need for additional assumptions regarding the function form, in comparison to traditional polynomial response surfaces. In the present work, the LS-SVM method was adapted to obtain the limited state function. An LS-SVM-based response surface method (RSM), combined with the first-order reliability method (FORM), is proposed for use in tunnel reliability analysis and implementation of the method is described. The reliability index obtained from the proposed method applied to particular tunnel configurations under different conditions shows excellent agreement with Low and Tang’s (2007) method and traditional RSM results, and indicates that the LS-SVM-based RSM is an efficient and effective approach for reliability analysis in tunnel engineering.     

Reliability analysis of laterally loaded piles using response surface methods

Response surface methods have been applied to the reliability analysis of laterally loaded piles. Beam elements and a series of discrete springs are used to model the pile–soil system. The pile head displacement and the maximum bending moment in the piles are used as the performance criteria in this study. It is shown in the illustrative example that the CDF and PDF curves of the pile head displacement and the maximum bending moment in the pile obtained from the proposed methods are in good agreement with Monte Carlo simulation. The failure probabilities of the pile under specified performance criteria, the probabilistic responses of the pile-soil system, and the effect of pile-soil parameters to the failure probability of the pile are also studied.     

An adaptive response surface method for reliability analysis of structures with multiple loading sequences

The response surface method (RSM) has been widely used in conjunction with non-linear finite element (NLFE) analysis to predict the reliability levels of structures that do not have explicit failure functions. In the present study, the solution of the reliability analysis initially diverged when the loading was applied in sequence in the NLFE analysis. A case study was performed in order to investigate the cause of the divergence. It was found that the divergence was mainly due to the non-smoothness of the response surface, and the quality and validity of the experimental design for numerical or NLFE analysis. An adaptive design approach is proposed to overcome these problems in reliability analysis, and several suggestions are made to improve the robustness of the RSM. Three numerical examples have been chosen to demonstrate the proposed method, which was verified by an independent Monte Carlo Simulation.     

An improvement of the response surface method

The coupling of the Monte Carlo method and the finite element method for the reliability analysis of structures leads often to a prohibitive computational cost. The response surface method is a powerful reliability method that approximates the limit state function with a polynomial expression using the values of the function at specific points. This type of analytical function replaces the exact limit state function in the Monte Carlo simulation. Therefore, the computational effort required for the assessment of the reliability of structural systems can be reduced significantly. The position of the sample points and the type of polynomial response surface have been investigated by several authors and the performance of the response surface method is still under discussion. In this paper an improvement of the response surface method is proposed. An iterative strategy is used to determine a response surface that is able to fit the limit state function in the neighbourhood of the design point. The locations of the sample points used to evaluate the free parameters of the response surface are chosen according to the importance sensitivity of each random variable. Several analytical and structural examples are considered to demonstrate the advantages of the proposed improvement.     

基于区间变量的响应面可靠性分析方法

通过对响应面方法、区间变量和地下结构不确定性特点的研究发现:(1)响应面方法在具有不确定性结构的静力响应分析中具有相当高的准确性;(2)地下岩体结构物理、力学参数及力学响应的不确定性特征非常适合通过区间数进行表达,但结构功能函数解析表达式往往很难直接求出。因此,对基于数理统计特征的响应面方法进行了改进,提出了基于区间变量的响应面函数形式选取,试验设计方法及函数解析式的拟合程序。在此基础上,根据函数中自变量的特点,研究了区间变量函数值域区间的求解方法。其中结合响应面函数形式,重点研究了同一自变量多次出现的函数形式的值域求解方法和具体计算程序。从而形成了完整的基于区间变量的响应面非概率可靠性分析方法,利用该方法分析了某一工程实例,展示了其有效性和可行性。     

Structural reliability under monotony: Properties of FORM, simulation or response surface methods and a new class of Monotonous Reliability Methods (MRM)

In many mechanical or physical systems, the failure function, albeit complex and computationally intensive, happens to be monotonous with respect to its uncertain model inputs. In that case, some properties can be derived on the partial robustness of classical methods such as FORM, Monte–Carlo or response surfaces to estimate rare failure probabilities under the constraint of the number of expensive model runs, such as robust probability bounds, saved number of model runs or guaranteed variance reduction. A new formulation taking full advantage of monotony is introduced along with a family of associated Monotonous Reliability Methods (MRM). They consist in narrowing progressively some robust upper and lower bounds on the failure probability through an adaptive design of experiment. A number of variants can be considered: they comprise adaptive Monte–Carlo, dedicated response surfaces and deterministic design of experiments or hybrid variants with classical FORM or simulation methods. Their common advantages are that the prediction accuracy of the failure probability is guaranteed with certainty; additional model runs always ameliorate the accuracy; a change of the uncertainty model is possible without additional runs. Performance is compared on benchmarks including a nuclear finite-element mechanical study and a simple flooding model. Simple MRM variants appear quite promising in low input dimensions with highly efficient computation of a bounding established with certainty. Research perspectives are given to extend the efficiency of the methods in higher dimensions and address the relaxing of full monotony hypotheses.     

Prediction of cyclic freeze–thaw damage in concrete structures based on response surface method

The initiation and growth process of cyclic ice body in porous systems are affected by thermo-physical and mass transport properties as well as by gradients of temperature and chemical potentials. Furthermore, the diffusivity of deicing chemicals reaches a significantly higher value under cyclic freeze–thaw conditions. Moreover, disintegration of concrete structures is aggravated by marine environments, higher altitudes, and northern areas. A serious concern for concrete engineers is that the property of cyclic freeze–thaw with crack growth and the deterioration, caused by accumulated damages hard to be identified by testing. In order to predict the accumulated damages by cyclic freeze–thaw, an optimized regression analysis by response surface method (RSM) is performed. Such important parameters for cyclic freeze–thaw-deterioration of concrete structures as water to cement ratio, entrained air pores, and the number of cycles of freezing and thawing are used to construct the limit state function of RSM. The regression equation fitted to the important deterioration criteria such as accumulated plastic deformation, relative dynamic modulus and equivalent plastic deformations served as the probabilistic evaluation of the performance to resist the structural degradation. The prediction of relative dynamic modulus and residual strains after 300 cycles of freeze–thaw showed good agreements with the experimental results, showing that the RSM result can be used to predict the probability of failure for the accumulated damage by cyclic freeze–thaw using designer specified critical values. Hence, it is possible to evaluate the life cycle management of concrete structures by the proposed prediction method in consideration of the accumulated damage due to cyclic freeze–thaw.     

CQ2RS: a new statistical approach to the response surface method for reliability analysis

Assessing the reliability of a complex structure requires a deal between reliability algorithms and numerical methods used to model the mechanical behavior. The Response Surface Method (RSM) represents a convenient way to achieve this purpose. The interest of such a method is that the user is allowed to choose and check the computed mechanical experiments. Nevertheless, this choice in an optimal way turns out to be not always an easy task. This paper proposes a response surface method named CQ2RS (Complete Quadratic Response Surface with ReSampling) allowing to take into account the knowledge of the engineer on one hand and to reduce the cost of the reliability analysis using a statistical formulation of the RSM problem on the other hand. Some academic and industrial examples illustrate the efficiency of the method.     

基于HDMR响应面的结构可靠度计算方法

介绍了响应面法在可靠度分析中的应用进展,提出了一种基于高维模型表示(HDMR)扩展的迭代响应面法。在详述采用HDMR响应面法计算结构可靠度的基本原理后,采用Matlab语言对若干数值算例进行计算,同时给出了其它计算方法的结果以对比分析。结果表明:本文方法具有较高的效率和可靠的精度,在结构可靠度分析中具有一定的实用性。     

建筑结构可靠度分析方法

介绍了建筑结构可靠度研究的意义,详细阐述了结构可靠度的基本计算方法,对矩法、响应面法、蒙特卡罗法等可靠度计算分析方法进行了分析,指出了各种方法的优点及适用范围,并对结构体系可靠度研究进行了展望.     

High-order limit state functions in the response surface method for structural reliability analysis

The stochastic response surface method (SRSM) is a technique for the reliability analysis of complex systems with low failure probabilities, for which Monte Carlo simulation (MCS) is too computationally intensive and for which approximate methods are inaccurate. Typically, the SRSM approximates a limit state function with a multi-dimensional quadratic polynomial by fitting the polynomial to a number of sampling points from the limit state function. This method can give biased approximations of the failure probability for cases in which the quadratic response surface can not conform to the true limit state function’s nonlinearities. In contrast to recently proposed algorithms which focus on the positions of sample points to improve the accuracy of the quadratic SRSM, this paper describes the use of higher order polynomials in order to approximate the true limit state more accurately. The use of higher order polynomials has received relatively little attention to date because of problems associated with ill-conditioned systems of equations and an approximated limit state which is very inaccurate outside the domain of the sample points. To address these problems, an algorithm using orthogonal polynomials is proposed to determine the necessary polynomial orders. Four numerical examples compare the proposed algorithm with the conventional quadratic polynomial SRSM and a detailed MCS.     

Response surface augmented moment method for efficient reliability analysis

The reliability analysis procedure based on design of experiment (DOE) is combined with the response surface method (RSM) for numerical efficiency. Instead of using the inefficient full factorial DOE, a response surface is constructed initially based on the data on the axial experimental points and updated successively by adding one more experimental point selected using an influence index, until the probability is converged. It is calculated using the Pearson system and the four statistical moments obtained from the experimental data complemented by the response surface. During the update of a response surface, cross product terms can be added into the formulation. The number of updating steps is finite, since the points to be added are selected among the set of points of full factorial design. The performance of the proposed method is tested with several examples containing various types of distributions. It is shown that the probability converges early in the process and thus the amount of calculation is only a small fraction of that of a full factorial design. This is comparable or even better than any other methods including FORM for a moderate number of random variables tested.     

Probabilistic analysis of consolidation that considers spatial variability using the stochastic response surface method

To obtain more accurate and reasonable results in the analyses of soil consolidation, the spatial variability of the soil properties should be considered. In this study, we analyzed the consolidation by vertical drains for soil improvement considering the spatial variability of the coefficients of consolidation. The coefficients for the variation in the vertical and horizontal coefficients of consolidation in Yeonjongdo, South Korea were evaluated, and the probability density function (PDF) was assumed by the Anderson–Darling goodness-of-fit test. Standard Gaussian random fields were generated based on a Karhunen–Loeve expansion, and then transformed using Hermite polynomials in the random field with the log-Gaussian PDF of the coefficient of consolidation. The average degree of consolidation was subsequently calculated using the finite difference method coupled with log-Gaussian random fields. In addition, the stochastic response surface method (SRSM) was applied for the efficient probabilistic uncertainty propagation. A sensitivity analysis was performed for the input parameters of the random field, and the spatial variability was considered using random variables from the Karhunen–Loeve expansion as the input data for the SRSM. The results indicated that when considering the spatial variability of soil properties, the probability of failure for the target degree of consolidation was smaller when the correlation distance was taken into account than when it was not. Additionally, the probability of failure decreased when the correlation distance decreased. Compared with the Monte Carlo simulation (MCS) results, the SRSM analysis can achieve results of similar accuracy to those obtained using the MCS analysis with a sample size of 100,000 (numerical runs), and a third-order SRSM expansion with only 333 numerical runs is sufficient for obtaining the probability with errors less than 0.01.     

Mechanical analysis of circular tunnels supported by steel sets embedded in primary linings

A series of analytical studies have been performed to investigate the support characteristics of the steel sets embedded in the tunnel primary lining. The analytical models for various modes of loading on an infinite cylinder carrying uniform line load, carrying uniform surface load, reinforced by a single circular ring and reinforced by equidistant rings are proposed respectively. Comparing the deformation characteristics of the infinite cylinder carrying uniform line load and uniform surface load reveals that for a practical tunnel application, the difference between the deformations induced by line load and equivalent surface load is less than 1.5%. The steel set reinforcements in the primary lining can therefore be represented by plane rings in the cylindrical shell. A study of the infinite cylinder reinforced by a single ring reveals that the most prominent effect of a steel set is attributed to its role as a cantilever fixing. It helps carry a certain length of the surrounding ground both ahead and behind before shotcreting and during shotcrete hardening. A subsequent study of the infinite cylinder reinforced by equidistant rings suggests that if the steel set spacing is smaller than a threshold value, the differences of the total stiffness calculated by the proposed method at the locations of the steel sets and other parts of the liner are negligible. The results obtained by the proposed method and the classical method have negligible differences. If the steel set spacing is larger than a threshold value, the classical method should be used with caution. Besides, the proposed method is more accurate than the classical method for calculating the support characteristics of the composite liner.     

基于SVM响应面的人员疏散可靠度分析

为解决火灾中人员疏散可靠性分析面临的没有显式极限状态方程和计算量过大的问题,提出了一种基于支持向量机(SVM)的响应面可靠度分析技术。采用均匀试验设计初始计算输入参数组,分别使用CFAST和EVACNET计算不同参数组下人员可用疏散时间ASET和必需疏散时间RSET,在此基础上采用SVM回归出ASET和RSET关于输入参数的响应面,进一步在响应面基础上使用直接Monte Carlo模拟技术获得ASET和RSET的概率分布和人员疏散的失效概率。     

Reliability analysis of tunnels using a metamodeling technique based on augmented radial basis functions

Metamodeling techniques have been developed and used for years in engineering reliability analysis involving expensive response simulations. In practical tunnel engineering problems where finite element (FE) simulations are required, the limited state/performance functions are in general implicit and nonlinear, and it is difficult to apply traditional gradient-based or sampling-based reliability methods, especially for large-scale problems. There is a need to develop accurate and efficient metamodels for practical tunnel engineering applications. In this paper, a metamodeling technique for reliability analysis of tunnels was studied based on augmented radial basis functions (RBFs). With a relatively small size of samples, the RBFs were used to create accurate approximate functions for different types of responses including linear and higher-order nonlinear functions. With the RBF-based metamodel constructed to express a limit state/performance function, Monte Carlo simulations (MCS) were applied to evaluate failure probability. The failure probability and reliability index obtained using the RBF-based metamodeling method were found to have good accuracy with a reasonable number of sample points. The reliability analyses of two existing tunnel examples showed that the augmented RBF metamodeling approach was efficient and effective for tunnel engineering problems.     

Optimization of the coagulation-flocculation process for pulp mill wastewater treatment using a combination of uniform design and response surface methodology

Pulp mill wastewater was treated using the coagulation-flocculation process with aluminum chloride as the coagulant and a modified natural polymer, starch-g-PAM-g-PDMC [polyacrylamide and poly (2-methacryloyloxyethyl) trimethyl ammonium chloride], as the flocculant. A novel approach with a combination of response surface methodology (RSM) and uniform design (UD) was employed to evaluate the effects and interactions of three main influential factors, coagulant dosage, flocculant dosage and pH, on the treatment efficiency in terms of the supernatant turbidity and lignin removals as well as the water recovery. The optimal conditions obtained from the compromise of the three desirable responses, supernatant turbidity removal, lignin removal and water recovery efficiency, were as follows: coagulant dosage of 871 mg/L, flocculant dosage of 22.3 mg/L and pH 8.35. Confirmation experiments demonstrated that such a combination of the UD and RSM is a powerful and useful approach for optimizing the coagulation-flocculation process for the pulp mill wastewater treatment.     

Variable complexity design of composite fuselage frames by response surface techniques

Curved frame structures are often used as part of the internal skeletal structure in aircraft. Laminated composite materials offer potential weight savings for such structures, but composite frames have different and more complex failure mechanisms than metallic frames. In particular, failure mechanisms involving interlaminar stresses are important in composite structures. Interlaminar stresses can be directly computed from three-dimensional finite element models, but the computational expense of these models is prohibitive. In this work, two- and three-dimensional (2D and 3D) finite element models are combined to reduce the computational expense associated with designing composite frames. A response surface design approach is used to approximate the failure response of curved composite C-section frames subjected to an axial tensile loading using a minimum number of finite element analyses. Results are presented for two examples with two and five design variables, respectively.