Probabilistic Design Methods Research Articles

Monte Carlo Analysis and Optimization of Inertial Switch Tolerance of a Proximity Fuze

Abstract Aiming at the low operation rate of the firing mechanism of ammunition fuze, robustness analysis and probabilistic design methods were introduced into the fuze design to carry out system simulation research on performance prediction and optimization of the firing mechanism of ammunition fuze. In this paper, the dynamic model of the inertial impact switch of a proximity fuze is established, and the mechanism can be reliably triggered under nominal values by simulation method. Based on the Monte Carlo simulation of dimensional tolerance distribution, the trigger performance is evaluated. The contribution rate of each dimension variable to the target value is obtained by using the optimal Latin hypercube experiment design, and the variable tolerance with greater contribution rate is optimized by Pointer algorithm. After optimization, the range of variable Jdxh_f was optimized from 0.1274 to 0.1666 to 0.1274 to 0.1556. Finally, the optimized results were validated, with a uniform distribution yielding 0.003193517N, a bimodal distribution yielding 0.2230769N, and a normal distribution yielding 0.2778532N. The validation analysis of these results shows that the target value distribution data after optimization meets the design requirements. This study seeks a balance between performance reliability and manufacturing cost, which is of great significance to improve product performance and reduce manufacturing cost.

Probabilistic Design Methods for Gust-Based Loads on Wind Turbines

The IEC 61400-1 standard specifies design load cases (DLCs) to be considered in the design of wind turbine structures. Specifically, DLC 2.3 considers the occurrence of a gust while the turbine shuts down due to an electrical fault. Originally, this load case used a deterministic wind event called the extreme operating gust (EOG), but the standard now also includes an approach for calculating the extreme response based on stochastic simulations with turbulent wind. This study presents and compares existing approaches with novel probabilistic design approaches for DLC 2.3 based on simulations with turbulent wind. First, a semiprobabilistic approach is proposed, where the inverse first-order reliability method (iFORM) is used for the extrapolation of the response for electrical faults occurring at a given rate. Next, three probabilistic approaches are formulated for the calculation of the reliability index, which differs in how the aggregation is performed over wind conditions and whether faults are modeled using a Poisson distribution or just by the rate. An example illustrates the methods considering the tower fore-aft bending moment at the tower base and shows that the approach based on iFORM can lead to reductions in material usage compared to the existing methods. For reliability assessment, the probabilistic approach using the Poisson process is needed for high failure rates, and the reliabilities obtained for designs using all semiprobabilistic methods are above the target level, indicating that further reductions may be obtained via the use of probabilistic design methods.

A probabilistic framework of creep life assessment of structural components at elevated temperatures

Current creep life assessment methods of components at elevated temperatures are mainly based on deterministic analysis strategies, which could not achieve the goal of probabilistic evaluation on creep failure. Based on this, a probabilistic framework of creep life assessment for components at elevated temperatures was provided. A method of determining distribution characteristics of material parameters was provided by randomly selecting a group of results at each stress level. Monte Carlo simulation was combined with finite element analysis technology to capture the distribution characteristic of creep rupture life of one typical structural component. Effect of standard derivation of material parameters on creep reliability assessment was discussed. Comparisons between probabilistic and deterministic creep design methods were made. Results indicated that the probabilistic analysis strategy can calculate the specific value of failure probability at various loading conditions, not two values of failure probability (i.e. 100% and 0%) by deterministic analysis strategy. The effect of standard derivation on mean values of effective stress and creep rupture life of the component is dependent on distribution characteristics of material parameters and related variables. A small standard derivation reduces the data scatter of effective stress and creep rupture life of the component.

Focus on software, data acquisition, and instrumentation

Focus on software, data acquisition, and instrumentation

Reliability based design of shaft for gearbox

In this article, the author presents research on the reliability-based design methods of mechanical systems, includes 3 stages: the first, the fault tree analysis (FTA) method is utilized for reliability analysis of single and double reducer into the logic diagram; the second that apply the probabilistic design methods, including moment matching method (MMM), first-order reliability method (FORM), and Monte Carlo simulation (MCS) for reliability-based design and to analyze the components of the mechanical transmission systems, and the last sensitivity analysis is performed in the post-design stage after a design solution is identified. The study also aims to set up RADME (Reliability-based analysis and design of the mechanical systems) computer programs applying the aforementioned methods. The reliability design problem was surveyed as reliability-based design optimization with one design variable.

A systematic and probabilistic approach for optimal design and on-site adaptive balancing of building central cooling systems concerning uncertainties

In current design practice, chillers and pumps are often oversized due to conservative consideration of uncertainties using safety factors to avoid the risk of undersizing, which often results in significant energy waste in operation. In recent years, probabilistic optimal design methods have been proposed for the components of cooling systems, enabling risk-based decision-making rather than sizing systems with safety margins to consider uncertainties. However, approaches for probabilistic optimal design and balancing of entire cooling systems are still absent. This article therefore presents a systematic approach of probabilistic optimal design and adaptive balancing for central cooling systems of buildings to minimize the impacts (energy waste and increased life-cycle cost) of oversizing in operation. The probabilistic optimal design considers both the uncertainties of design inputs and the flexibility of on-site adaptive balancing, while adaptive balancing enables flexible balancing to maximize energy saving according to characteristics of constructed systems. A case study is conducted to test and validate the proposed approach. Results show that significant cost reduction and energy saving were achieved for chillers and pumps, respectively, through the systematic approach of probabilistic design and adaptive balancing. Energy consumption of pumps was reduced by 41% through coordinating pump design with probabilistic chiller design.

Seismic Design and Evaluation Methods for Small-to-Medium Span Highway Girder Bridges Based on Machine Learning and Earthquake Damage Experience

This study summarized the application field of machine learning to explore the basic seismic thinking of machine learning methods used for bridges. The development and actuality of bridge seismic analysis theories and technology were briefly reviewed, particularly in relation to the field of civil engineering. This study introduced the concept of machine learning, summarized its key factors and current software platforms, and illustrated the common methods and representative algorithms of machine learning using simple examples. First, the normal types of simply supported and continuous girder bridges, which are used for highway bridges with small and medium spans in China, were summarized. The earthquake damage phenomena had been observed for bridge types during the Wenchuan earthquake. Data from these phenomena were assessed, including the damage grade divisions of piers, bearings, shear keys, and abutments. Second, a series of seismic performance tests had been conducted by international and domestic academics for bearings, shear keys, piers, and abutments. These tests were summarized in this study to obtain the constitutive relationships for seismic analysis and determine the seismic design parameters of bridge components (including foundations). Finally, an overall analysis methodology based on machine learning was introduced into the bridge seismic analysis. This methodology explained that machine learning for bridge seismic tasks had two aspects. The first was the collection of considerable bridge design data and set up data sets, and the second was data reduction, including raw data processing and debugging or developing a reasonable machine learning algorithm model. This study also discussed the shortcomings of existing performance-based probabilistic seismic design and evaluation methods that are currently used in analyzing bridges in China. Results indicated the potential future major concerns for bridge seismic analysis technology. The establishment of computational simulations based on artificial intelligence method was also recommended. In addition, the mutual integration between disciplines and mutual communication among different professionals were advocated in this study.

An ANOVA approach for accounting for life-cycle uncertainty reduction measures in RBDO: the FSAE brake pedal case study

Accounting for uncertainty reduction measures (URMs) is critical to maximize the potential benefits of probabilistic design methods such as reliability-based design optimization (RBDO) and tackle the challenges in the design and construction of lightweight, high quality and reliable products. This work formulates and solves the RBDO of a Formula SAE (FSAE) brake pedal model with two failure modes (stress-Smax and buckling-fbuck) accounting for uncertainty reduction measures (URMs) throughout the product lifecycle while establishing the URMs global relative contributions to weight savings (expected value and variability) and computational expense. Given a set of URMs such as number of coupon tests, mesh refinement and manufacturing control, the solution approach includes: i) modeling structural analysis errors, ii) construction of surrogate models for the functions of interest, e.g., mass-M, Smax, fbuck and the corresponding error functions, iii) modeling pre-design and post-design URMs, such as material property density functions from coupon tests, and manufacturing tolerances (quality control), iv) solving the RBDO problems associated with each of the entries in a DOE with replication, and v) using ANOVA to compute main effects of most significant URMs on selected performance measures, i.e., mean and standard deviation of brake pedal mass, and computational expense. Results show that in the context of the brake pedal case study: the adoption of URMs led to reductions of up to 15 and 85% of mass mean and standard deviation, respectively, design and post-design URMs were responsible for 77 and 19% of the maximum mass reduction, respectively, and it was possible to set preliminary guidelines for URMs allocation and meet a particular performance objective under alternative URMs.

Robust knockdown factors for the design of cylindrical shells under axial compression: Analysis and modeling of stiffened and unstiffened cylinders

For the design of thin-walled cylindrical shells under axial compression empirical knockdown factors are applied. These knockdown factors are based on experimental results from the beginning of the 20th century and have been shown to be very conservative for modern shell structures.In order to determine less conservative and physically based knockdown factors for the design of axially loaded shells, different analytical and numerical design approaches have been developed. In this paper common as well as new shell design approaches are presented in detail and evaluated regarding the lower-bound buckling load. Among these design approaches are the EN 1993 1–6, the reduced energy method, linear buckling eigenmode imperfections, perturbation approaches and the new threshold knockdown factors.Important analysis and modeling details of each design approach are described and test examples are given and validated. Advantages and disadvantages of each approach are listed and design recommendations are given.A comparison of deterministic design approaches with modern probabilistic design methods is shown and the range of application of both design philosophies is discussed.Orthogrid stiffened cylinders with weld lands from NASAs Shell Buckling Knockdown Factor Project (SBKF) are modeled, analyzed and lower-bound buckling load calculations for improved knockdown factors are shown.

A Probabilistic Design Method for Fatigue Life of Metallic Component

In the present study, a general probabilistic design framework is developed for cyclic fatigue life prediction of metallic hardware using methods that address uncertainty in experimental data and computational model. The methodology involves: (i) fatigue test data conducted on coupons of Ti6Al4V material, (ii) continuum damage mechanics (CDM) based material constitutive models to simulate cyclic fatigue behavior of material, (iii) variance-based global sensitivity analysis, (iv) Bayesian framework for model calibration and uncertainty quantification, and (v) computational life prediction and probabilistic design decision making under uncertainty. The outcomes of computational analyses using the experimental data prove the feasibility of the probabilistic design methods for model calibration in the presence of incomplete and noisy data. Moreover, using probabilistic design methods results in assessment of reliability of fatigue life predicted by computational models.

Robust and Reliability-Based Aeroelastic Design of Composite Plate Wings

Probabilistic design methods may be used to account for inherent variability in the manufacturing quality of aerospace structures. Two different probabilistic approaches are compared alongside a deterministic method for the aeroelastic design of composite plate wings with uncertain ply orientations. The instability speed is found to be a discontinuous function of the lamination parameters, which are themselves functions of the ply orientations. A surrogate modeling approach is presented in which Gaussian processes are combined with support vector machines to emulate the discontinuous instability speed to efficiently propagate uncertainty through the model. The surrogate model is used to calculate the objectives of two optimization strategies: a reliability-based design, in which the probability of aeroelastic instability occurring within the design envelope is minimized, and a robust design, in which the mean and standard deviation of the instability speed are traded off. A genetic algorithm is used to optimize the layup, and reductions in the probability of failure of 94% are achieved in the reliability-based design using a design speed, relative to benchmark deterministic designs. A 74% reduction in the standard deviation is achieved in the robust design, at the expense of a 1% reduction in the mean.

Reliability based calibration of safety factors for unstiffened cylindrical composite shells

The feasibility of probabilistic design methods to overcome conservatism associated with deterministic design methods for unstiffened cylindrical composite shells has been shown, but studies are frequently based on assumptions regarding relevant uncertainties due to the lack of available data. The reliability based calibration method suggested here relies on extensive measurements regarding the statistical characteristics of thickness, fibre volume fraction, fibre orientation, material stiffness and geometric imperfections of 11 previously tested cylinders. Furthermore, the stiffness of the potting of tested cylinders is estimated using Finite Element Analysis and the introduction of a pre-stress state through the potting process investigated. Following a sensitivity study, Bayesian inference is used to update the information of relevant uncertain variables. Monte Carlo simulation is employed to compute the distribution functions of the buckling load, and these are used to calibrate structural safety factors at a chosen reliability level. A multiplicative error model is established to compute safety factors covering the model uncertainty inherent to the approach. In combining these two safety factors in a Bayesian sense, a transparent, updatable safety-factor concept is developed, leading to less conservative design loads than currently considered.

Effect of Geometric Uncertainties on the Aerodynamic Characteristic of Offshore Wind Turbine Blades

Offshore wind turbines operate in a complex unsteady flow environment which causes unsteady aerodynamic loads. The unsteady flow environment is characterized by a high degree of uncertainty. In addition, geometry variations and material imperfections also cause uncertainties in the design process. Probabilistic design methods consider these uncertainties in order to reach acceptable reliability and safety levels for offshore wind turbines. Variations of the rotor blade geometry influence the aerodynamic loads which also affect the reliability of other wind turbine components. Therefore, the present paper is dealing with geometric uncertainties of the rotor blades. These can arise from manufacturing tolerances and operational wear of the blades. First, the effect of geometry variations of wind turbine airfoils on the lift and drag coefficients are investigated using a Latin hypercube sampling. Then, the resulting effects on the performance and the blade loads of an offshore wind turbine are analyzed. The variations of the airfoil geometry lead to a significant scatter of the lift and drag coefficients which also affects the damage-equivalent flapwise bending moments. In contrast to that, the effects on the power and the annual energy production are almost negligible with regard to the assumptions made.

Sample size dependent probabilistic design of axially compressed cylindrical shells

The fact that the buckling load of cylindrical shells depends on imperfections raised the idea of applying probabilistic design methods for these structures. Whether a probabilistically motivated design load may be regarded as a representative for a type of shell depends, among others, on the samples that build the data basis for the probabilistic methodology. In the current paper a methodology is presented that takes into account the samples size within a fast probabilistic design. The method presented leads to a lower design load, as the sample size decreases. The methodology is applied to a set of 33 beer cans, which have been measured and tested at the Delft University of Technology. The 33 cans are subdivided into groups which are then analyzed probabilistically in order to observe how the probabilistically motivated design load varies for different samples sizes. The results indicate that the method presented provides a useful tool for designers to ensure that a small data basis does not yield a unsafe probabilistic design.

직립방파제의 케이슨 활동에 대한 최초통과확률법의 허용기준 산정

Probabilistic design methods can consider uncertainties of design variables and are widely used in the design of vertical breakwaters. The probabilistic design methods include a partial safety factor method, reliability-based design method, and performance-based design method. Especially the performance-based design method calculates the accumulated sliding distance during the lifetime of the breakwater or during a design storm. Recently a time-dependent performance-based design method has been developed based on the first-passage probability of individual sliding distance during a design storm. However, because the allowable criteria in the first-passage probability method are not established, the stability of structures cannot be quantitatively evaluated. In this study, the allowable first-passage probabilities for two limit states are proposed by calculating the first-passage probabilities for the cross-sections designed with various water depths and characteristics of extreme wave height distributions. The allowable first-passage probabilities are proposed as 5% and 1%, respectively, for the repairable limit state (allowable individual sliding distance of 0.03 m) and ultimate limit state (allowable individual sliding distance of 0.1 m). The proposed criteria are applied to the evaluation of the effect of wave-height increase due to climate change on the stability of the breakwater. 핵심용어: vertical breakwater, first-passage probability method, allowable first-passage probability, probabilistic design method, ì§ë¦½ë°©íŒŒì œ, ìµœì´ˆí†µê³¼í™•ë¥ ë²•, í—ˆìš©ìµœì´ˆí†µê³¼í™•ë¥ , í™•ë¥ ë¡ ì  설계법

PEER Performance-Based Earthquake Engineering Methodology, Revisited

A performance-based earthquake engineering (PBEE) methodology was developed at the Pacific Earthquake Engineering Research (PEER) Center. The method is based on explicit determination of performance, e.g., monetary losses, in a probabilistic manner where uncertainties in earthquake ground motion, structural response, damage, and losses are explicitly considered. There is an increasing trend towards use of probabilistic performance-based design (PPBD) methods in practice. Therefore, the International Federation for Structural Concrete (fib) initiated a task group to disseminate PPBD methods. This article is a contribution to this task group summarizing and demonstrating the PEER PBEE methodology in a useful manner to practicing engineers.

The effects of the cement thickness on the failure probability of the cemented hip prosthesis

Finite element analysis has been widely used to determine the behavior of implants placed in a body. However, uncertainties in loading acting on the implant, material properties and implant geometry are not taken into account during these analyses. Thus, it can result in either an unreliable design or unnecessarily overdesigned implants. Therefore, probabilistic design methods, in which uncertainties related to design variables are taken into account during the solution, have been recently considered as an important approach to design more reliable and long-lasting implant. Since the failure of the cemented hip prostheses mostly occurs at the cement mantle, determining the failure probability becomes more important. Therefore, in this study, the failure probability of the cement mantle which bonds the hip prosthesis into bone was calculated using the reliability methods by considering the uncertainties in the related design variables. In order to calculate the failure probability, three-dimensional solid models of the cement mantle, femur and hip prosthesis was obtained. The same model was used to obtain the maximum principal stress from the deterministic stress analysis in order to compare the failures state of the cemented hip prosthesis using the both approaches. The deterministic stress analysis carried out considering the mean values of the design parameters does not indicate any failure for the cement mantle thickness considered in this study. On the other hand, the probabilistic analysis yields a failure probability up to 0.3. Therefore these results indicate that the probabilistic analysis should be considered to design a more reliable and long-lasting implant.

PROBABILISTIC DESIGN OF AXIALLY COMPRESSED COMPOSITE CYLINDERS WITH GEOMETRIC AND LOADING IMPERFECTIONS

The discrepancy between the analytically determined buckling load of perfect cylindrical shells and experimental test results is traced back to imperfections. The most frequently used guideline for design of cylindrical shells, NASA SP-8007, proposes a deterministic calculation of a knockdown factor with respect to the ratio of radius and wall thickness, which turned out to be very conservative in numerous cases and which is not intended for composite shells. In order to determine a lower bound for the buckling load of an arbitrary type of shell, probabilistic design methods have been developed. Measured imperfection patterns are described using double Fourier series, whereas the Fourier coefficients characterize the scattering of geometry. In this paper, probabilistic analyses of buckling loads are performed regarding Fourier coefficients as random variables. A nonlinear finite element model is used to determine buckling loads, and Monte Carlo simulations are executed. The probabilistic approach is used for a set of six similarly manufactured composite shells. The results indicate that not only geometric but also nontraditional imperfections like loading imperfections have to be considered for obtaining a reliable lower limit of the buckling load. Finally, further Monte Carlo simulations are executed including traditional as well as loading imperfections, and lower bounds of buckling loads are obtained, which are less conservative than NASA SP-8007.

Investigation of variability in fatigue crack nucleation and propagation in alpha+beta Ti-6Al-4V

The understanding of fatigue variability in turbine engine materials is vital to permit a reduction of US Air Force sustainment costs through fatigue life extension and/or inspection interval extension. Additionally, the US Air Force is currently moving further towards the use of probabilistic damage tolerance design methods. These probabilistic models require not only a good understanding of the variability in specimen data, but also an understanding of the microstructural sources of variability to allow scaling to component analysis.The objective of this work was to study the fatigue variability of a common turbine engine alloy Ti-6Al-4V. Typical testing consisted of smooth bar fatigue tests at multiple stress ratios and stress levels in order to generate a fully populated stress-life curve. These tests, however, typically do not consist of many repeats. The approach of this work was to conduct a statistically significant number of repeated fatigue tests at several loading conditions. A similar approach has been performed on several other turbine engine material systems often revealing bimodal life distributions consisting of a number of low life specimens that may fail due to a separate mechanism. This paper discusses the Ti-6Al-4V life distributions and sources of variability. Crack propagation using small crack growth data was used to predict the lower tail of the life distributions.

Probabilistic analysis of an uncemented total hip replacement

This paper describes the application of probabilistic design methods to the analysis of the behaviour of an uncemented total hip replacement femoral component implanted in a proximal femur. Probabilistic methods allow variations in factors which control the behaviour of the implanted femur (the input parameters) to be taken into account in determining the performance of the construct. Monte Carlo sampling techniques were applied and the performance indicator was the maximum strain in the bone. The random input parameters were the joint load, the angle of the applied load and the material properties of the bone and the implant. Two Monte Carlo based simulations were applied, direct sampling and latin hypercube sampling. The results showed that the convergence of the mean value of the maximum strain improved gradually as a function of the number of simulations and it stabilised around a value of 0.008 after 6200 simulations. A similar trend was observed for the cumulative distribution function of the output. The strain output was most sensitive to the bone stiffness, followed very closely by the magnitude of the applied load. The application of latin hypercube sampling with 1000 simulations gave similar results to direct sampling with 10,000 simulations in a much reduced time. The results suggested that the number of simulations and the selection of parameters and models are important for the reliability of both the probability values and the sensitivity analyses.