Improved response surface method and its application to stiffened plate reliability analysis

The response surface method (RSM) is a powerful approach for carrying out reliability analysis for complicate engineering with implicit limit state functions. The quality of a response surface mainly depends on the choice of response surface function and the selection of sample points. To investigate the influence of the types of response surface functions and to reduce the computation efforts, two new sampling methods and a hybrid RSM are proposed. In the hybrid RSM, four types of response surface functions and three sampling methods are involved, and each Response surface function can be connected to each sampling method. The four response surface functions are quadratic polynomial without cross terms (PN1), quadratic polynomial with cross terms, radial basic function network (RBFN) and support vector machine (SVM). The three RSMs using the traditional sampling method, the new iterative sampling method and the new experiment design method are RSM1, RSM2 and RSM3, respectively. The accuracy and efficiency of different RSMs are illustrated through three examples. When an iterative method is used for locating sample points, the PN1-based RSM2 is proposed for its accuracy and efficiency. And when an experiment design method is used for locating sample points, the RBFN- or SVM-based RSM3 is suggested because the RBFN or SVM is suitable for global fitting.

A new method for welded joint fatigue reliability analysis on bogie frame of high-speed electric multiple unit (EMU) is proposed according to the International Union of Railways (UIC) standard, the Britain Standard Institute (BSI) standard and the Miner rule. Firstly, the fatigue calculation condition is loaded according to the UIC standard. The welded joint damage ratio is calculated by the BSI standard. Fatigue reliability analysis is conducted for a typical welded joint based on probabilistic design system (PDS) module in software ANSYS 14.0. In the system, the automatic program is prepared by ANSYS parametric design language (APDL) to calculate the welded joint fatigue damage ratio. Meantime, the relationship between the input variables and the output response is fitted by the response surface (RS) method. Then, the RS method is improved by the orthogonal design. Specifically, the selecting process optimization for the RS method based on the orthogonal design is conducted to improve the fitting efficiency of the RS method. Finally, a program of the automatic analysis documents is generated by Microsoft foundation class (MFC) technique. Through the self-defined interface, the researchers can easily import the automatic generation of the orthogonal table into the PDS module and analyze the fatigue reliability. Results show that the improved RS method can better fit the RS which covers the sample area, and the distribution law agrees well with the actual situation. At the same time, the improved RS method reduces the calculation time.

In order to analyze the reliability and failure probability of the repaired shaft engineering, this article makes fail-safe analysis and instability risk prediction of the shaft engineering. A reliability analysis method was applied combining the improved response surface method which has improved sample points’ selection based on the traditional response surface method to decreasing the number of the structural analysis under the premise of the accuracy of the calculated results with numerical simulation. The limit state whose failure criteria was safe factor was constructed to analyze the reliability of the repaired shaft engineering, the reliability index of the repaired shaft engineering was computed by JC method. The results show the reliability index value of the repaired shaft engineering is 3.1127 slight fewer than design objective value and it’s failure probability value is 0.0929%, which show that the repaired shaft is stable and it’s deformation should be monitored to ensure it is safety application because of it is reliability index value slight fewer than design objective value.

Reliability analysis of pipe whip impacts

The reliability of a simple turbomachinery model was calculated to demonstrate the application of a newly developing system integration tool, Probabilistic Design and Analysis Framework(PRODAF), along with efficient probabilistic methods using a response surface method. The model represents a system consisting of hypothetical turbine components. The parts include a blade, disk, and shaft with an applied angular velocity. All the components were modeled with the properties of the nickel alloy, Inconel 718. A response surface was calculated for the system of components to improve probabilistic computational efficiency. In addition, a fast probability integration method, Advanced First Order Reliability Method (AFORM), was used for the probabilistic analysis in order to provide an efficient analysis as possible. Geometric dimensions, the applied load, and material yield strength were varied for this study. The probability of failure was determined using the maximum first principal stress response and the material yield strength. A simple G function using the difference between strength and loading stress was used to determine failure limits. The probabilistic sensitivity of the failure response relative to the individual variables was determined also with material yield strength having the greatest influence. The model was recreated with every iteration of the probabilistic analysis in order to vary the geometry. As a result, the response surface method has a significant impact on improving computational efficiency and enabling reliability analysis with rapid turnaround.

The seismic safety of reinforced concrete containment building can be evaluated by probabilistic analysis considering randomness of earthquake, which is more rational than deterministic analysis. In the safety assessment of earthquake-resistant structures by the deterministic theory, it is not easy to consider the effects of random variables but the reliability theory and random vibration theory are useful to assess the seismic safety with considering random effects. The reliability assessment of reinforced concrete containment building subjected to earthquake load includes the structural analysis considering random variables such as load, resistance and analysis method, the definition of limit states and the reliability analysis. The reliability analysis procedure requires much time and labor and also needs to get the high confidence in results. In this study, random vibration analysis of containment building is performed with random variables as earthquake load, concrete compressive strength, modal damping ratio. The seismic responses of critical elements of structure are approximated at the most probable failure point by the response surface method. The response surface method helps to figure out the quantitative characteristics of structural response variability. And the limit state is defined as the failure surface of concrete under multi-axial stress, finally the limit state probability of failure can be obtained simply by first-order second moment method. The reliability analysis for the multiaxial strength limit state and the uniaxial strength limit state is performed and the results are compared with each other. This study concludes that the multiaxial failure criterion is a likely limit state to predict concrete failure strength under combined state of stresses and the reliability analysis results are compatible with the fact that the maximum compressive strength of concrete under biaxial compression state increases.

To ameliorate reliability analysis efficiency for aeroengine components, such as compressor blade, support vector machine response surface method (SRSM) is proposed. SRSM integrates the advantages of support vector machine (SVM) and traditional response surface method (RSM), and utilizes experimental samples to construct a suitable response surface function (RSF) to replace the complicated and abstract finite element model. Moreover, the randomness of material parameters, structural dimension and operating condition are considered during extracting data so that the response surface function is more agreeable to the practical model. The results indicate that based on the same experimental data, SRSM has come closer than RSM reliability to approximating Monte Carlo method (MCM); while SRSM (17.296 s) needs far less running time than MCM (10958 s) and RSM (9840 s). Therefore, under the same simulation conditions, SRSM has the largest analysis efficiency, and can be considered a feasible and valid method to analyze structural reliability.

Objectives: Quantification of uncertainties and its implications on stability assessment of geotechnical problems can very well be captured through probabilistic analysis in which parameters are considered to be random variables. Thus, randomness of the response of the system is also captured and safety assessment is reported in terms of a safety index, commonly called as “reliability index”. In this study, probabilistic approach in conjunction with conventional deterministic approach is used for the analysis of a deep excavation problem and advantages of probabilistic approach are highlighted. Methods/Statistical Analysis: Probabilistic analysis requires establishment of connection between input parameters and output responses, for which response surface method (RSM) concepts can effectively be utilized. RSM helps developing regression model between input and output which can be used for the probabilistic analysis through First Order Reliability Method (FORM). Advanced finite element numerical tool, i.e., PLAXIS 3D is utilized for the purpose of the analysis. Findings: Considering this typical case of deep excavation problem, it is highlighted that probabilistic analysis when used in conjunction with the deterministic approach, it brings rationality in decision making while dealing with uncertainty and provides mathematical basis for treating uncertainty. Application/Improvements: Conventionally, stability analyses of a deep excavation system are carried out through limit equilibrium based approaches and the parameter used to define extent of stability is factor of safety. With advent of computers and finite element or finite difference based computing facilities, strength reduction technique is adopted to estimate the factor of safety. These numerical tools also help in predicting the deformation pattern and judging the serviceability requirements. In either case, the soil parameters used as input in the numerical tools are taken a constant value uniformly. Choice of appropriate value of design input parameter is based on past experience, good engineering judgment and confidence in the adopted analysis methodology. Due to various uncertain sources involved in the estimation of the input properties of a natural material like soil, such as, natural variability, testing errors, model transformation, it is impractical to assign a single value to design parameter. Deterministic approach, which involves single value of design parameters for stability assessment is considered to be simple and straight forward. It is demonstrated that quantification of uncertainties and its influence on stability assessment of deep excavation problem can be captured in probabilistic framework.

This study focused on modeling and optimization of the concentration of poly (vinyl alcohol) (PVA), cellulose nanocrystals (CNC), snail shell nanoparticles (SSN), and glycerol for the development of bioplastic films. The response surface methodology using Box–Behnken experimental design was used to investigate the effect of the independent parameters (additives concentrations) on the ultimate tensile strength and Young's modulus of fabricated bioplastic films. A varied ultimate tensile strength and Young's modulus with different component loadings was observed, proving the effect of nanoparticles loading effect on the mechanical properties of bioplastic films. The quadratic polynomial model experiment data provided a coefficient of determination (R2) of 0.795 for ultimate tensile strength and 0.732 for Young's modulus, evidencing the fitness of the models to pilot the optimization space. The optimum parameters were PVA (7.820%), CNC (1079%), SSN (1241%), and glycerol (2.657%). The ultimate tensile strength and Young's modulus of 27.2 MPa and 31.2 MPa were obtained for the developed bioplastic film with optimized concentrations of each component. The bioplastic films showed improved thermal stability and degradation. The scanning electron microscopy(SEM) imaging revealed a homogeneous dispersion of SSN and CNC in the matrix, which explained the improved properties observed.

The present study outlines a sequentially integrated finite-element method (FEM)–response surface method (RSM)–genetic algorithm (GA) framework and implements this to predict the crack parameters, namely, crack location and crack depth ratio. A central composite face centered response surface design of the RSM technique is used to establish the direct relationships between the input parameters (crack location and crack depth ratio) and responses (natural frequencies) to build the response surface function (RSF). Multiple edge cracks are considered, which exist on the top flange of a thin-walled channel section cantilever beam and modeled as line spring elements. In order to obtain RSFs of the first five natural frequencies in terms of process factors such as crack position and crack depth ratio, a number of numerical experiments based on FEM are conducted by using the design-of-experiment approach. An objective function obtained as the square of the difference between RSF and experimentally measured natural frequency has been minimized using GA to find out the optimum crack parameters. Twenty-four steel channel beam specimens have been tested in the laboratory to extract the crack parameters from measured natural frequencies using the proposed approach. The results of the study indicate that the proposed approach performs remarkably, yielding crack parameters with great precision. Copyright © 2015 John Wiley & Sons, Ltd.

In order to evaluate the failure probability of a complicated structure, the structural responses usually need to be estimated by some numerical analysis methods such as finite element method (FEM). The response surface method (RSM) can be used to reduce the computational effort required for reliability analysis when the performance functions are implicit. However, the conventional RSM is time-consuming or cumbersome if the number of random variables is large. This paper proposes a Legendre orthogonal neural network (LONN)-based RSM to estimate the structural reliability. In this method, the relationship between the random variables and structural responses is established by a LONN model. Then the LONN model is connected to a reliability analysis method, i.e. first-order reliability methods (FORM) to calculate the failure probability of the structure. Numerical examples show that the proposed approach is applicable to structural reliability analysis, as well as the structure with implicit performance functions.

This paper presents a reliability-based analysis of a shallow foundation subjected to a central vertical load, using the limit equilibrium theory. The reliability index is calculated by applying the first order reliability method (FORM). A deterministic model based on the Meyerhof equations is employed to compute the ultimate bearing capacity of the foundation. The concept of response surface method (RSM) is used for the assessment of the Hasofer–Lind reliability indexes by generating an approximate polynomial function for the performance function. In this study, unit weight and the soil shear strength parameters are considered as random variables. The optimization of the Hasofer-Lind reliability index is performed using a combination of response surface model and the multi-objective genetic algorithm.KeywordsReliability analysisShallow foundationRSMOptimizationGenetic algorithm

Reliability has been incorporated in many pavement design procedures to account for the effects of inputs variabilities and uncertainties on predicted performance. The American Association of State Highway and Transportation Officials (AASHTO) mechanistic empirical pavement design guide (MEPDG) computes the reliability of pavement sections with the assumption that the variability of predicted distresses follows normal distribution. This approach does not account for the systematic contribution of each design input variability on the overall output variance. This paper evaluates a two-component reliability analysis methodology for pavement application. The two-component reliability analysis methodology uses a response surface method (RSM) for a surrogate model generation and the first order reliability method (FORM) for reliability computation. Three different response surface methods (central composite, Box–Behnken and Doehlert designs) were implemented and statistically verified for their suitability for surrogate model generation. The two-component reliability analysis methodology was further utilized for the generation of partial safety factors for the development of a load and resistance factor design (LRFD) procedure for pavement applications. Field pavement sections with a wide range in design inputs and target reliabilities were used to evaluate the proposed reliability analysis methodology. The results have shown that the three RSM can be used effectively for pavement reliability problems.

In the past, residual factor R in strain-softening soil slopes has been included, either directly or indirectly, as a deterministic variable in both deterministic and probabilistic studies. This paper discusses the uncertainties associated with R and outlines a systematic approach for the reliability analysis of a natural slope in which shear strength parameters and pore pressure ratio are random variables, each assumed with a lognormal probability distribution. For the residual factor R, seven probability distribution options under the generalized beta-distribution system are considered. Slope reliability is computed based on the first order reliability method (FORM) and validated against Monte-Carlo simulation (MCS). Results obtained from two illustrative examples indicate that the probability of failure, with R as one of six random variables, can be orders of magnitude higher than that based on five random variables with R considered as a deterministic parameter. The magnitude of influence of R as a random variable is, however, highly dependent on its probability distribution, the left-skewed triangular distribution having the most significant influence in both the examples. Results of sensitivity analyses reveal that, for almost all of its assumed probability distributions, R is the most dominant among the six random variables. Effects of variation of some of the statistical and correlation properties of the other random variables, viz. the shear strength parameters and the pore pressure ratio, on the results of reliability analyses are also studied.

Aiming at the problem of big calculation error in solving reliability index by the traditional response surface method, an improved response surface method based on the linear gradient iteration criterion is proposed. First, the linear gradient iteration criterion is proposed to reduce the iteration step size with the increase of iteration times. It will improve the fitting accuracy of response surface and lead to a better convergence while approaching the limit state surface. Then, the reduction coefficient of the linear gradient iterative criterion is studied. The optimal value of coefficient is 0.2. The improved response surface method will get a more accurate reliability index quickly. Examples show that the proposed method has obvious advantages of high accuracy and efficiency. The application of this method can also be expanded in other similar engineering structure.